*> \brief \b DCOPY
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DCOPY(N,DX,INCX,DY,INCY)
*
* .. Scalar Arguments ..
* INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION DX(*),DY(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DCOPY copies a vector, x, to a vector, y.
*> uses unrolled loops for increments equal to 1.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> number of elements in input vector(s)
*> \endverbatim
*>
*> \param[in] DX
*> \verbatim
*> DX is DOUBLE PRECISION array, dimension ( 1 + ( N - 1 )*abs( INCX ) )
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> storage spacing between elements of DX
*> \endverbatim
*>
*> \param[out] DY
*> \verbatim
*> DY is DOUBLE PRECISION array, dimension ( 1 + ( N - 1 )*abs( INCY ) )
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> storage spacing between elements of DY
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup copy
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> jack dongarra, linpack, 3/11/78.
*> modified 12/3/93, array(1) declarations changed to array(*)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DCOPY(N,DX,INCX,DY,INCY)
*
* -- Reference BLAS level1 routine --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION DX(*),DY(*)
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I,IX,IY,M,MP1
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
IF (N.LE.0) RETURN
IF (INCX.EQ.1 .AND. INCY.EQ.1) THEN
*
* code for both increments equal to 1
*
*
* clean-up loop
*
M = MOD(N,7)
IF (M.NE.0) THEN
DO I = 1,M
DY(I) = DX(I)
END DO
IF (N.LT.7) RETURN
END IF
MP1 = M + 1
DO I = MP1,N,7
DY(I) = DX(I)
DY(I+1) = DX(I+1)
DY(I+2) = DX(I+2)
DY(I+3) = DX(I+3)
DY(I+4) = DX(I+4)
DY(I+5) = DX(I+5)
DY(I+6) = DX(I+6)
END DO
ELSE
*
* code for unequal increments or equal increments
* not equal to 1
*
IX = 1
IY = 1
IF (INCX.LT.0) IX = (-N+1)*INCX + 1
IF (INCY.LT.0) IY = (-N+1)*INCY + 1
DO I = 1,N
DY(IY) = DX(IX)
IX = IX + INCX
IY = IY + INCY
END DO
END IF
RETURN
*
* End of DCOPY
*
END
*> \brief \b DISNAN tests input for NaN.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DISNAN + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* LOGICAL FUNCTION DISNAN( DIN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION, INTENT(IN) :: DIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DISNAN returns .TRUE. if its argument is NaN, and .FALSE.
*> otherwise. To be replaced by the Fortran 2003 intrinsic in the
*> future.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIN
*> \verbatim
*> DIN is DOUBLE PRECISION
*> Input to test for NaN.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup isnan
*
* =====================================================================
LOGICAL FUNCTION DISNAN( DIN )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION, INTENT(IN) :: DIN
* ..
*
* =====================================================================
*
* .. External Functions ..
LOGICAL DLAISNAN
EXTERNAL DLAISNAN
* ..
* .. Executable Statements ..
DISNAN = DLAISNAN(DIN,DIN)
RETURN
END
*> \brief \b DLAISNAN tests input for NaN by comparing two arguments for inequality.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAISNAN + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION, INTENT(IN) :: DIN1, DIN2
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is not for general use. It exists solely to avoid
*> over-optimization in DISNAN.
*>
*> DLAISNAN checks for NaNs by comparing its two arguments for
*> inequality. NaN is the only floating-point value where NaN != NaN
*> returns .TRUE. To check for NaNs, pass the same variable as both
*> arguments.
*>
*> A compiler must assume that the two arguments are
*> not the same variable, and the test will not be optimized away.
*> Interprocedural or whole-program optimization may delete this
*> test. The ISNAN functions will be replaced by the correct
*> Fortran 03 intrinsic once the intrinsic is widely available.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIN1
*> \verbatim
*> DIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] DIN2
*> \verbatim
*> DIN2 is DOUBLE PRECISION
*> Two numbers to compare for inequality.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laisnan
*
* =====================================================================
LOGICAL FUNCTION DLAISNAN( DIN1, DIN2 )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION, INTENT(IN) :: DIN1, DIN2
* ..
*
* =====================================================================
*
* .. Executable Statements ..
DLAISNAN = (DIN1.NE.DIN2)
RETURN
END
!> \brief \b DLARTG generates a plane rotation with real cosine and real sine.
!
! =========== DOCUMENTATION ===========
!
! Online html documentation available at
! http://www.netlib.org/lapack/explore-html/
!
! Definition:
! ===========
!
! SUBROUTINE DLARTG( F, G, C, S, R )
!
! .. Scalar Arguments ..
! REAL(wp) C, F, G, R, S
! ..
!
!> \par Purpose:
! =============
!>
!> \verbatim
!>
!> DLARTG generates a plane rotation so that
!>
!> [ C S ] . [ F ] = [ R ]
!> [ -S C ] [ G ] [ 0 ]
!>
!> where C**2 + S**2 = 1.
!>
!> The mathematical formulas used for C and S are
!> R = sign(F) * sqrt(F**2 + G**2)
!> C = F / R
!> S = G / R
!> Hence C >= 0. The algorithm used to compute these quantities
!> incorporates scaling to avoid overflow or underflow in computing the
!> square root of the sum of squares.
!>
!> This version is discontinuous in R at F = 0 but it returns the same
!> C and S as ZLARTG for complex inputs (F,0) and (G,0).
!>
!> This is a more accurate version of the BLAS1 routine DROTG,
!> with the following other differences:
!> F and G are unchanged on return.
!> If G=0, then C=1 and S=0.
!> If F=0 and (G .ne. 0), then C=0 and S=sign(1,G) without doing any
!> floating point operations (saves work in DBDSQR when
!> there are zeros on the diagonal).
!>
!> Below, wp=>dp stands for double precision from LA_CONSTANTS module.
!> \endverbatim
!
! Arguments:
! ==========
!
!> \param[in] F
!> \verbatim
!> F is REAL(wp)
!> The first component of vector to be rotated.
!> \endverbatim
!>
!> \param[in] G
!> \verbatim
!> G is REAL(wp)
!> The second component of vector to be rotated.
!> \endverbatim
!>
!> \param[out] C
!> \verbatim
!> C is REAL(wp)
!> The cosine of the rotation.
!> \endverbatim
!>
!> \param[out] S
!> \verbatim
!> S is REAL(wp)
!> The sine of the rotation.
!> \endverbatim
!>
!> \param[out] R
!> \verbatim
!> R is REAL(wp)
!> The nonzero component of the rotated vector.
!> \endverbatim
!
! Authors:
! ========
!
!> \author Edward Anderson, Lockheed Martin
!
!> \date July 2016
!
!> \ingroup lartg
!
!> \par Contributors:
! ==================
!>
!> Weslley Pereira, University of Colorado Denver, USA
!
!> \par Further Details:
! =====================
!>
!> \verbatim
!>
!> Anderson E. (2017)
!> Algorithm 978: Safe Scaling in the Level 1 BLAS
!> ACM Trans Math Softw 44:1--28
!> https://doi.org/10.1145/3061665
!>
!> \endverbatim
!
subroutine DLARTG( f, g, c, s, r )
use LA_CONSTANTS, &
only: wp=>dp, zero=>dzero, half=>dhalf, one=>done, &
safmin=>dsafmin, safmax=>dsafmax
!
! -- LAPACK auxiliary routine --
! -- LAPACK is a software package provided by Univ. of Tennessee, --
! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! February 2021
!
! .. Scalar Arguments ..
real(wp) :: c, f, g, r, s
! ..
! .. Local Scalars ..
real(wp) :: d, f1, fs, g1, gs, u, rtmin, rtmax
! ..
! .. Intrinsic Functions ..
intrinsic :: abs, sign, sqrt
! ..
! .. Constants ..
rtmin = sqrt( safmin )
rtmax = sqrt( safmax/2 )
! ..
! .. Executable Statements ..
!
f1 = abs( f )
g1 = abs( g )
if( g == zero ) then
c = one
s = zero
r = f
else if( f == zero ) then
c = zero
s = sign( one, g )
r = g1
else if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then
d = sqrt( f*f + g*g )
c = f1 / d
r = sign( d, f )
s = g / r
else
u = min( safmax, max( safmin, f1, g1 ) )
fs = f / u
gs = g / u
d = sqrt( fs*fs + gs*gs )
c = abs( fs ) / d
r = sign( d, f )
s = gs / r
r = r*u
end if
return
end subroutine
*> \brief \b DLAS2 computes singular values of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAS2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION F, G, H, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAS2 computes the singular values of the 2-by-2 matrix
*> [ F G ]
*> [ 0 H ].
*> On return, SSMIN is the smaller singular value and SSMAX is the
*> larger singular value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> The smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is DOUBLE PRECISION
*> The larger singular value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup las2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Barring over/underflow, all output quantities are correct to within
*> a few units in the last place (ulps), even in the absence of a guard
*> digit in addition/subtraction.
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows, or is within a few ulps of overflow.
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION F, G, H, SSMAX, SSMIN
* ..
*
* ====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
FA = ABS( F )
GA = ABS( G )
HA = ABS( H )
FHMN = MIN( FA, HA )
FHMX = MAX( FA, HA )
IF( FHMN.EQ.ZERO ) THEN
SSMIN = ZERO
IF( FHMX.EQ.ZERO ) THEN
SSMAX = GA
ELSE
SSMAX = MAX( FHMX, GA )*SQRT( ONE+
$ ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
END IF
ELSE
IF( GA.LT.FHMX ) THEN
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
AU = ( GA / FHMX )**2
C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
SSMIN = FHMN*C
SSMAX = FHMX / C
ELSE
AU = FHMX / GA
IF( AU.EQ.ZERO ) THEN
*
* Avoid possible harmful underflow if exponent range
* asymmetric (true SSMIN may not underflow even if
* AU underflows)
*
SSMIN = ( FHMN*FHMX ) / GA
SSMAX = GA
ELSE
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
$ SQRT( ONE+( AT*AU )**2 ) )
SSMIN = ( FHMN*C )*AU
SSMIN = SSMIN + SSMIN
SSMAX = GA / ( C+C )
END IF
END IF
END IF
RETURN
*
* End of DLAS2
*
END
*> \brief \b DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASCL + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TYPE
* INTEGER INFO, KL, KU, LDA, M, N
* DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASCL multiplies the M by N real matrix A by the real scalar
*> CTO/CFROM. This is done without over/underflow as long as the final
*> result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
*> A may be full, upper triangular, lower triangular, upper Hessenberg,
*> or banded.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TYPE
*> \verbatim
*> TYPE is CHARACTER*1
*> TYPE indices the storage type of the input matrix.
*> = 'G': A is a full matrix.
*> = 'L': A is a lower triangular matrix.
*> = 'U': A is an upper triangular matrix.
*> = 'H': A is an upper Hessenberg matrix.
*> = 'B': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the lower
*> half stored.
*> = 'Q': A is a symmetric band matrix with lower bandwidth KL
*> and upper bandwidth KU and with the only the upper
*> half stored.
*> = 'Z': A is a band matrix with lower bandwidth KL and upper
*> bandwidth KU. See DGBTRF for storage details.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The lower bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The upper bandwidth of A. Referenced only if TYPE = 'B',
*> 'Q' or 'Z'.
*> \endverbatim
*>
*> \param[in] CFROM
*> \verbatim
*> CFROM is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CTO
*> \verbatim
*> CTO is DOUBLE PRECISION
*>
*> The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
*> without over/underflow if the final result CTO*A(I,J)/CFROM
*> can be represented without over/underflow. CFROM must be
*> nonzero.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The matrix to be multiplied by CTO/CFROM. See TYPE for the
*> storage type.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
*> TYPE = 'B', LDA >= KL+1;
*> TYPE = 'Q', LDA >= KU+1;
*> TYPE = 'Z', LDA >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> 0 - successful exit
*> <0 - if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lascl
*
* =====================================================================
SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, KL, KU, LDA, M, N
DOUBLE PRECISION CFROM, CTO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
INTEGER I, ITYPE, J, K1, K2, K3, K4
DOUBLE PRECISION BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH, DISNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
*
IF( LSAME( TYPE, 'G' ) ) THEN
ITYPE = 0
ELSE IF( LSAME( TYPE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( TYPE, 'U' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( TYPE, 'H' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( TYPE, 'B' ) ) THEN
ITYPE = 4
ELSE IF( LSAME( TYPE, 'Q' ) ) THEN
ITYPE = 5
ELSE IF( LSAME( TYPE, 'Z' ) ) THEN
ITYPE = 6
ELSE
ITYPE = -1
END IF
*
IF( ITYPE.EQ.-1 ) THEN
INFO = -1
ELSE IF( CFROM.EQ.ZERO .OR. DISNAN(CFROM) ) THEN
INFO = -4
ELSE IF( DISNAN(CTO) ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR.
$ ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN
INFO = -7
ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN
INFO = -9
ELSE IF( ITYPE.GE.4 ) THEN
IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN
INFO = -2
ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR.
$ ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) )
$ THEN
INFO = -3
ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR.
$ ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR.
$ ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN
INFO = -9
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASCL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
CFROMC = CFROM
CTOC = CTO
*
10 CONTINUE
CFROM1 = CFROMC*SMLNUM
IF( CFROM1.EQ.CFROMC ) THEN
! CFROMC is an inf. Multiply by a correctly signed zero for
! finite CTOC, or a NaN if CTOC is infinite.
MUL = CTOC / CFROMC
DONE = .TRUE.
CTO1 = CTOC
ELSE
CTO1 = CTOC / BIGNUM
IF( CTO1.EQ.CTOC ) THEN
! CTOC is either 0 or an inf. In both cases, CTOC itself
! serves as the correct multiplication factor.
MUL = CTOC
DONE = .TRUE.
CFROMC = ONE
ELSE IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN
MUL = SMLNUM
DONE = .FALSE.
CFROMC = CFROM1
ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN
MUL = BIGNUM
DONE = .FALSE.
CTOC = CTO1
ELSE
MUL = CTOC / CFROMC
DONE = .TRUE.
IF (MUL .EQ. ONE)
$ RETURN
END IF
END IF
*
IF( ITYPE.EQ.0 ) THEN
*
* Full matrix
*
DO 30 J = 1, N
DO 20 I = 1, M
A( I, J ) = A( I, J )*MUL
20 CONTINUE
30 CONTINUE
*
ELSE IF( ITYPE.EQ.1 ) THEN
*
* Lower triangular matrix
*
DO 50 J = 1, N
DO 40 I = J, M
A( I, J ) = A( I, J )*MUL
40 CONTINUE
50 CONTINUE
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* Upper triangular matrix
*
DO 70 J = 1, N
DO 60 I = 1, MIN( J, M )
A( I, J ) = A( I, J )*MUL
60 CONTINUE
70 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* Upper Hessenberg matrix
*
DO 90 J = 1, N
DO 80 I = 1, MIN( J+1, M )
A( I, J ) = A( I, J )*MUL
80 CONTINUE
90 CONTINUE
*
ELSE IF( ITYPE.EQ.4 ) THEN
*
* Lower half of a symmetric band matrix
*
K3 = KL + 1
K4 = N + 1
DO 110 J = 1, N
DO 100 I = 1, MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
100 CONTINUE
110 CONTINUE
*
ELSE IF( ITYPE.EQ.5 ) THEN
*
* Upper half of a symmetric band matrix
*
K1 = KU + 2
K3 = KU + 1
DO 130 J = 1, N
DO 120 I = MAX( K1-J, 1 ), K3
A( I, J ) = A( I, J )*MUL
120 CONTINUE
130 CONTINUE
*
ELSE IF( ITYPE.EQ.6 ) THEN
*
* Band matrix
*
K1 = KL + KU + 2
K2 = KL + 1
K3 = 2*KL + KU + 1
K4 = KL + KU + 1 + M
DO 150 J = 1, N
DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
140 CONTINUE
150 CONTINUE
*
END IF
*
IF( .NOT.DONE )
$ GO TO 10
*
RETURN
*
* End of DLASCL
*
END
*> \brief \b DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ1 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ1 computes the singular values of a real N-by-N bidiagonal
*> matrix with diagonal D and off-diagonal E. The singular values
*> are computed to high relative accuracy, in the absence of
*> denormalization, underflow and overflow. The algorithm was first
*> presented in
*>
*> "Accurate singular values and differential qd algorithms" by K. V.
*> Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
*> 1994,
*>
*> and the present implementation is described in "An implementation of
*> the dqds Algorithm (Positive Case)", LAPACK Working Note.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, D contains the diagonal elements of the
*> bidiagonal matrix whose SVD is desired. On normal exit,
*> D contains the singular values in decreasing order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, elements E(1:N-1) contain the off-diagonal elements
*> of the bidiagonal matrix whose SVD is desired.
*> On exit, E is overwritten.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm failed
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 100*N
*> iterations (in inner while loop) On exit D and E
*> represent a matrix with the same singular values
*> which the calling subroutine could use to finish the
*> computation, or even feed back into DLASQ1
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq1
*
* =====================================================================
SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO
DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLASQ1', -INFO )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
D( 1 ) = ABS( D( 1 ) )
RETURN
ELSE IF( N.EQ.2 ) THEN
CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX )
D( 1 ) = SIGMX
D( 2 ) = SIGMN
RETURN
END IF
*
* Estimate the largest singular value.
*
SIGMX = ZERO
DO 10 I = 1, N - 1
D( I ) = ABS( D( I ) )
SIGMX = MAX( SIGMX, ABS( E( I ) ) )
10 CONTINUE
D( N ) = ABS( D( N ) )
*
* Early return if SIGMX is zero (matrix is already diagonal).
*
IF( SIGMX.EQ.ZERO ) THEN
CALL DLASRT( 'D', N, D, IINFO )
RETURN
END IF
*
DO 20 I = 1, N
SIGMX = MAX( SIGMX, D( I ) )
20 CONTINUE
*
* Copy D and E into WORK (in the Z format) and scale (squaring the
* input data makes scaling by a power of the radix pointless).
*
EPS = DLAMCH( 'Precision' )
SAFMIN = DLAMCH( 'Safe minimum' )
SCALE = SQRT( EPS / SAFMIN )
CALL DCOPY( N, D, 1, WORK( 1 ), 2 )
CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
$ IINFO )
*
* Compute the q's and e's.
*
DO 30 I = 1, 2*N - 1
WORK( I ) = WORK( I )**2
30 CONTINUE
WORK( 2*N ) = ZERO
*
CALL DLASQ2( N, WORK, INFO )
*
IF( INFO.EQ.0 ) THEN
DO 40 I = 1, N
D( I ) = SQRT( WORK( I ) )
40 CONTINUE
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
ELSE IF( INFO.EQ.2 ) THEN
*
* Maximum number of iterations exceeded. Move data from WORK
* into D and E so the calling subroutine can try to finish
*
DO I = 1, N
D( I ) = SQRT( WORK( 2*I-1 ) )
E( I ) = SQRT( WORK( 2*I ) )
END DO
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, E, N, IINFO )
END IF
*
RETURN
*
* End of DLASQ1
*
END
*> \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ2( N, Z, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ2 computes all the eigenvalues of the symmetric positive
*> definite tridiagonal matrix associated with the qd array Z to high
*> relative accuracy are computed to high relative accuracy, in the
*> absence of denormalization, underflow and overflow.
*>
*> To see the relation of Z to the tridiagonal matrix, let L be a
*> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
*> let U be an upper bidiagonal matrix with 1's above and diagonal
*> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
*> symmetric tridiagonal to which it is similar.
*>
*> Note : DLASQ2 defines a logical variable, IEEE, which is true
*> on machines which follow ieee-754 floating-point standard in their
*> handling of infinities and NaNs, and false otherwise. This variable
*> is passed to DLASQ3.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> On entry Z holds the qd array. On exit, entries 1 to N hold
*> the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
*> trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
*> N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
*> holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
*> shifts that failed.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if the i-th argument is a scalar and had an illegal
*> value, then INFO = -i, if the i-th argument is an
*> array and the j-entry had an illegal value, then
*> INFO = -(i*100+j)
*> > 0: the algorithm failed
*> = 1, a split was marked by a positive value in E
*> = 2, current block of Z not diagonalized after 100*N
*> iterations (in inner while loop). On exit Z holds
*> a qd array with the same eigenvalues as the given Z.
*> = 3, termination criterion of outer while loop not met
*> (program created more than N unreduced blocks)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Local Variables: I0:N0 defines a current unreduced segment of Z.
*> The shifts are accumulated in SIGMA. Iteration count is in ITER.
*> Ping-pong is controlled by PP (alternates between 0 and 1).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASQ2( N, Z, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CBIAS
PARAMETER ( CBIAS = 1.50D0 )
DOUBLE PRECISION ZERO, HALF, ONE, TWO, FOUR, HUNDRD
PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
$ TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
LOGICAL IEEE
INTEGER I0, I1, I4, IINFO, IPN4, ITER, IWHILA, IWHILB,
$ K, KMIN, N0, N1, NBIG, NDIV, NFAIL, PP, SPLT,
$ TTYPE
DOUBLE PRECISION D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX,
$ QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL,
$ TOL2, TRACE, ZMAX, TEMPE, TEMPQ
* ..
* .. External Subroutines ..
EXTERNAL DLASQ3, DLASRT, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments.
* (in case DLASQ2 is not called by DLASQ1)
*
INFO = 0
EPS = DLAMCH( 'Precision' )
SAFMIN = DLAMCH( 'Safe minimum' )
TOL = EPS*HUNDRD
TOL2 = TOL**2
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLASQ2', 1 )
RETURN
ELSE IF( N.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
*
* 1-by-1 case.
*
IF( Z( 1 ).LT.ZERO ) THEN
INFO = -201
CALL XERBLA( 'DLASQ2', 2 )
END IF
RETURN
ELSE IF( N.EQ.2 ) THEN
*
* 2-by-2 case.
*
IF( Z( 1 ).LT.ZERO ) THEN
INFO = -201
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 2 ).LT.ZERO ) THEN
INFO = -202
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 3 ).LT.ZERO ) THEN
INFO = -203
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN
D = Z( 3 )
Z( 3 ) = Z( 1 )
Z( 1 ) = D
END IF
Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 )
IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN
T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) )
S = Z( 3 )*( Z( 2 ) / T )
IF( S.LE.T ) THEN
S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
ELSE
S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
END IF
T = Z( 1 ) + ( S+Z( 2 ) )
Z( 3 ) = Z( 3 )*( Z( 1 ) / T )
Z( 1 ) = T
END IF
Z( 2 ) = Z( 3 )
Z( 6 ) = Z( 2 ) + Z( 1 )
RETURN
END IF
*
* Check for negative data and compute sums of q's and e's.
*
Z( 2*N ) = ZERO
EMIN = Z( 2 )
QMAX = ZERO
ZMAX = ZERO
D = ZERO
E = ZERO
*
DO 10 K = 1, 2*( N-1 ), 2
IF( Z( K ).LT.ZERO ) THEN
INFO = -( 200+K )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
ELSE IF( Z( K+1 ).LT.ZERO ) THEN
INFO = -( 200+K+1 )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
END IF
D = D + Z( K )
E = E + Z( K+1 )
QMAX = MAX( QMAX, Z( K ) )
EMIN = MIN( EMIN, Z( K+1 ) )
ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) )
10 CONTINUE
IF( Z( 2*N-1 ).LT.ZERO ) THEN
INFO = -( 200+2*N-1 )
CALL XERBLA( 'DLASQ2', 2 )
RETURN
END IF
D = D + Z( 2*N-1 )
QMAX = MAX( QMAX, Z( 2*N-1 ) )
ZMAX = MAX( QMAX, ZMAX )
*
* Check for diagonality.
*
IF( E.EQ.ZERO ) THEN
DO 20 K = 2, N
Z( K ) = Z( 2*K-1 )
20 CONTINUE
CALL DLASRT( 'D', N, Z, IINFO )
Z( 2*N-1 ) = D
RETURN
END IF
*
TRACE = D + E
*
* Check for zero data.
*
IF( TRACE.EQ.ZERO ) THEN
Z( 2*N-1 ) = ZERO
RETURN
END IF
*
* Check whether the machine is IEEE conformable.
*
IEEE = ( ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 )
*
* Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
*
DO 30 K = 2*N, 2, -2
Z( 2*K ) = ZERO
Z( 2*K-1 ) = Z( K )
Z( 2*K-2 ) = ZERO
Z( 2*K-3 ) = Z( K-1 )
30 CONTINUE
*
I0 = 1
N0 = N
*
* Reverse the qd-array, if warranted.
*
IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN
IPN4 = 4*( I0+N0 )
DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( I4-3 )
Z( I4-3 ) = Z( IPN4-I4-3 )
Z( IPN4-I4-3 ) = TEMP
TEMP = Z( I4-1 )
Z( I4-1 ) = Z( IPN4-I4-5 )
Z( IPN4-I4-5 ) = TEMP
40 CONTINUE
END IF
*
* Initial split checking via dqd and Li's test.
*
PP = 0
*
DO 80 K = 1, 2
*
D = Z( 4*N0+PP-3 )
DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4
IF( Z( I4-1 ).LE.TOL2*D ) THEN
Z( I4-1 ) = -ZERO
D = Z( I4-3 )
ELSE
D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) )
END IF
50 CONTINUE
*
* dqd maps Z to ZZ plus Li's test.
*
EMIN = Z( 4*I0+PP+1 )
D = Z( 4*I0+PP-3 )
DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4
Z( I4-2*PP-2 ) = D + Z( I4-1 )
IF( Z( I4-1 ).LE.TOL2*D ) THEN
Z( I4-1 ) = -ZERO
Z( I4-2*PP-2 ) = D
Z( I4-2*PP ) = ZERO
D = Z( I4+1 )
ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND.
$ SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN
TEMP = Z( I4+1 ) / Z( I4-2*PP-2 )
Z( I4-2*PP ) = Z( I4-1 )*TEMP
D = D*TEMP
ELSE
Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) )
D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) )
END IF
EMIN = MIN( EMIN, Z( I4-2*PP ) )
60 CONTINUE
Z( 4*N0-PP-2 ) = D
*
* Now find qmax.
*
QMAX = Z( 4*I0-PP-2 )
DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4
QMAX = MAX( QMAX, Z( I4 ) )
70 CONTINUE
*
* Prepare for the next iteration on K.
*
PP = 1 - PP
80 CONTINUE
*
* Initialise variables to pass to DLASQ3.
*
TTYPE = 0
DMIN1 = ZERO
DMIN2 = ZERO
DN = ZERO
DN1 = ZERO
DN2 = ZERO
G = ZERO
TAU = ZERO
*
ITER = 2
NFAIL = 0
NDIV = 2*( N0-I0 )
*
DO 160 IWHILA = 1, N + 1
IF( N0.LT.1 )
$ GO TO 170
*
* While array unfinished do
*
* E(N0) holds the value of SIGMA when submatrix in I0:N0
* splits from the rest of the array, but is negated.
*
DESIG = ZERO
IF( N0.EQ.N ) THEN
SIGMA = ZERO
ELSE
SIGMA = -Z( 4*N0-1 )
END IF
IF( SIGMA.LT.ZERO ) THEN
INFO = 1
RETURN
END IF
*
* Find last unreduced submatrix's top index I0, find QMAX and
* EMIN. Find Gershgorin-type bound if Q's much greater than E's.
*
EMAX = ZERO
IF( N0.GT.I0 ) THEN
EMIN = ABS( Z( 4*N0-5 ) )
ELSE
EMIN = ZERO
END IF
QMIN = Z( 4*N0-3 )
QMAX = QMIN
DO 90 I4 = 4*N0, 8, -4
IF( Z( I4-5 ).LE.ZERO )
$ GO TO 100
IF( QMIN.GE.FOUR*EMAX ) THEN
QMIN = MIN( QMIN, Z( I4-3 ) )
EMAX = MAX( EMAX, Z( I4-5 ) )
END IF
QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) )
EMIN = MIN( EMIN, Z( I4-5 ) )
90 CONTINUE
I4 = 4
*
100 CONTINUE
I0 = I4 / 4
PP = 0
*
IF( N0-I0.GT.1 ) THEN
DEE = Z( 4*I0-3 )
DEEMIN = DEE
KMIN = I0
DO 110 I4 = 4*I0+1, 4*N0-3, 4
DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) )
IF( DEE.LE.DEEMIN ) THEN
DEEMIN = DEE
KMIN = ( I4+3 )/4
END IF
110 CONTINUE
IF( (KMIN-I0)*2.LT.N0-KMIN .AND.
$ DEEMIN.LE.HALF*Z(4*N0-3) ) THEN
IPN4 = 4*( I0+N0 )
PP = 2
DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( I4-3 )
Z( I4-3 ) = Z( IPN4-I4-3 )
Z( IPN4-I4-3 ) = TEMP
TEMP = Z( I4-2 )
Z( I4-2 ) = Z( IPN4-I4-2 )
Z( IPN4-I4-2 ) = TEMP
TEMP = Z( I4-1 )
Z( I4-1 ) = Z( IPN4-I4-5 )
Z( IPN4-I4-5 ) = TEMP
TEMP = Z( I4 )
Z( I4 ) = Z( IPN4-I4-4 )
Z( IPN4-I4-4 ) = TEMP
120 CONTINUE
END IF
END IF
*
* Put -(initial shift) into DMIN.
*
DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) )
*
* Now I0:N0 is unreduced.
* PP = 0 for ping, PP = 1 for pong.
* PP = 2 indicates that flipping was applied to the Z array and
* and that the tests for deflation upon entry in DLASQ3
* should not be performed.
*
NBIG = 100*( N0-I0+1 )
DO 140 IWHILB = 1, NBIG
IF( I0.GT.N0 )
$ GO TO 150
*
* While submatrix unfinished take a good dqds step.
*
CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX,
$ NFAIL,
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
$ DN2, G, TAU )
*
PP = 1 - PP
*
* When EMIN is very small check for splits.
*
IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN
IF( Z( 4*N0 ).LE.TOL2*QMAX .OR.
$ Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN
SPLT = I0 - 1
QMAX = Z( 4*I0-3 )
EMIN = Z( 4*I0-1 )
OLDEMN = Z( 4*I0 )
DO 130 I4 = 4*I0, 4*( N0-3 ), 4
IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR.
$ Z( I4-1 ).LE.TOL2*SIGMA ) THEN
Z( I4-1 ) = -SIGMA
SPLT = I4 / 4
QMAX = ZERO
EMIN = Z( I4+3 )
OLDEMN = Z( I4+4 )
ELSE
QMAX = MAX( QMAX, Z( I4+1 ) )
EMIN = MIN( EMIN, Z( I4-1 ) )
OLDEMN = MIN( OLDEMN, Z( I4 ) )
END IF
130 CONTINUE
Z( 4*N0-1 ) = EMIN
Z( 4*N0 ) = OLDEMN
I0 = SPLT + 1
END IF
END IF
*
140 CONTINUE
*
INFO = 2
*
* Maximum number of iterations exceeded, restore the shift
* SIGMA and place the new d's and e's in a qd array.
* This might need to be done for several blocks
*
I1 = I0
N1 = N0
145 CONTINUE
TEMPQ = Z( 4*I0-3 )
Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA
DO K = I0+1, N0
TEMPE = Z( 4*K-5 )
Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 ))
TEMPQ = Z( 4*K-3 )
Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 )
END DO
*
* Prepare to do this on the previous block if there is one
*
IF( I1.GT.1 ) THEN
N1 = I1-1
DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) )
I1 = I1 - 1
END DO
SIGMA = -Z(4*N1-1)
GO TO 145
END IF
DO K = 1, N
Z( 2*K-1 ) = Z( 4*K-3 )
*
* Only the block 1..N0 is unfinished. The rest of the e's
* must be essentially zero, although sometimes other data
* has been stored in them.
*
IF( K.LT.N0 ) THEN
Z( 2*K ) = Z( 4*K-1 )
ELSE
Z( 2*K ) = 0
END IF
END DO
RETURN
*
* end IWHILB
*
150 CONTINUE
*
160 CONTINUE
*
INFO = 3
RETURN
*
* end IWHILA
*
170 CONTINUE
*
* Move q's to the front.
*
DO 180 K = 2, N
Z( K ) = Z( 4*K-3 )
180 CONTINUE
*
* Sort and compute sum of eigenvalues.
*
CALL DLASRT( 'D', N, Z, IINFO )
*
E = ZERO
DO 190 K = N, 1, -1
E = E + Z( K )
190 CONTINUE
*
* Store trace, sum(eigenvalues) and information on performance.
*
Z( 2*N+1 ) = TRACE
Z( 2*N+2 ) = E
Z( 2*N+3 ) = DBLE( ITER )
Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 )
Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER )
RETURN
*
* End of DLASQ2
*
END
*> \brief \b DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ3 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL,
* ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
* DN2, G, TAU )
*
* .. Scalar Arguments ..
* LOGICAL IEEE
* INTEGER I0, ITER, N0, NDIV, NFAIL, PP
* DOUBLE PRECISION DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
* $ QMAX, SIGMA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
*> In case of failure it changes shifts, and tries again until output
*> is positive.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in,out] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*> Z holds the qd array.
*> \endverbatim
*>
*> \param[in,out] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> PP=2 indicates that flipping was applied to the Z array
*> and that the initial tests for deflation should not be
*> performed.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> Sum of shifts used in current segment.
*> \endverbatim
*>
*> \param[in,out] DESIG
*> \verbatim
*> DESIG is DOUBLE PRECISION
*> Lower order part of SIGMA
*> \endverbatim
*>
*> \param[in] QMAX
*> \verbatim
*> QMAX is DOUBLE PRECISION
*> Maximum value of q.
*> \endverbatim
*>
*> \param[in,out] NFAIL
*> \verbatim
*> NFAIL is INTEGER
*> Increment NFAIL by 1 each time the shift was too big.
*> \endverbatim
*>
*> \param[in,out] ITER
*> \verbatim
*> ITER is INTEGER
*> Increment ITER by 1 for each iteration.
*> \endverbatim
*>
*> \param[in,out] NDIV
*> \verbatim
*> NDIV is INTEGER
*> Increment NDIV by 1 for each division.
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*> IEEE is LOGICAL
*> Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).
*> \endverbatim
*>
*> \param[in,out] TTYPE
*> \verbatim
*> TTYPE is INTEGER
*> Shift type.
*> \endverbatim
*>
*> \param[in,out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN1
*> \verbatim
*> DN1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] DN2
*> \verbatim
*> DN2 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*> G is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*>
*> These are passed as arguments in order to save their values
*> between calls to DLASQ3.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq3
*
* =====================================================================
SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX,
$ NFAIL,
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1,
$ DN2, G, TAU )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER I0, ITER, N0, NDIV, NFAIL, PP
DOUBLE PRECISION DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
$ QMAX, SIGMA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CBIAS
PARAMETER ( CBIAS = 1.50D0 )
DOUBLE PRECISION ZERO, QURTR, HALF, ONE, TWO, HUNDRD
PARAMETER ( ZERO = 0.0D0, QURTR = 0.250D0, HALF = 0.5D0,
$ ONE = 1.0D0, TWO = 2.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
INTEGER IPN4, J4, N0IN, NN, TTYPE
DOUBLE PRECISION EPS, S, T, TEMP, TOL, TOL2
* ..
* .. External Subroutines ..
EXTERNAL DLASQ4, DLASQ5, DLASQ6
* ..
* .. External Function ..
DOUBLE PRECISION DLAMCH
LOGICAL DISNAN
EXTERNAL DISNAN, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
N0IN = N0
EPS = DLAMCH( 'Precision' )
TOL = EPS*HUNDRD
TOL2 = TOL**2
*
* Check for deflation.
*
10 CONTINUE
*
IF( N0.LT.I0 )
$ RETURN
IF( N0.EQ.I0 )
$ GO TO 20
NN = 4*N0 + PP
IF( N0.EQ.( I0+1 ) )
$ GO TO 40
*
* Check whether E(N0-1) is negligible, 1 eigenvalue.
*
IF( Z( NN-5 ).GT.TOL2*( SIGMA+Z( NN-3 ) ) .AND.
$ Z( NN-2*PP-4 ).GT.TOL2*Z( NN-7 ) )
$ GO TO 30
*
20 CONTINUE
*
Z( 4*N0-3 ) = Z( 4*N0+PP-3 ) + SIGMA
N0 = N0 - 1
GO TO 10
*
* Check whether E(N0-2) is negligible, 2 eigenvalues.
*
30 CONTINUE
*
IF( Z( NN-9 ).GT.TOL2*SIGMA .AND.
$ Z( NN-2*PP-8 ).GT.TOL2*Z( NN-11 ) )
$ GO TO 50
*
40 CONTINUE
*
IF( Z( NN-3 ).GT.Z( NN-7 ) ) THEN
S = Z( NN-3 )
Z( NN-3 ) = Z( NN-7 )
Z( NN-7 ) = S
END IF
T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) )
IF( Z( NN-5 ).GT.Z( NN-3 )*TOL2.AND.T.NE.ZERO ) THEN
S = Z( NN-3 )*( Z( NN-5 ) / T )
IF( S.LE.T ) THEN
S = Z( NN-3 )*( Z( NN-5 ) /
$ ( T*( ONE+SQRT( ONE+S / T ) ) ) )
ELSE
S = Z( NN-3 )*( Z( NN-5 ) / ( T+SQRT( T )*SQRT( T+S ) ) )
END IF
T = Z( NN-7 ) + ( S+Z( NN-5 ) )
Z( NN-3 ) = Z( NN-3 )*( Z( NN-7 ) / T )
Z( NN-7 ) = T
END IF
Z( 4*N0-7 ) = Z( NN-7 ) + SIGMA
Z( 4*N0-3 ) = Z( NN-3 ) + SIGMA
N0 = N0 - 2
GO TO 10
*
50 CONTINUE
IF( PP.EQ.2 )
$ PP = 0
*
* Reverse the qd-array, if warranted.
*
IF( DMIN.LE.ZERO .OR. N0.LT.N0IN ) THEN
IF( CBIAS*Z( 4*I0+PP-3 ).LT.Z( 4*N0+PP-3 ) ) THEN
IPN4 = 4*( I0+N0 )
DO 60 J4 = 4*I0, 2*( I0+N0-1 ), 4
TEMP = Z( J4-3 )
Z( J4-3 ) = Z( IPN4-J4-3 )
Z( IPN4-J4-3 ) = TEMP
TEMP = Z( J4-2 )
Z( J4-2 ) = Z( IPN4-J4-2 )
Z( IPN4-J4-2 ) = TEMP
TEMP = Z( J4-1 )
Z( J4-1 ) = Z( IPN4-J4-5 )
Z( IPN4-J4-5 ) = TEMP
TEMP = Z( J4 )
Z( J4 ) = Z( IPN4-J4-4 )
Z( IPN4-J4-4 ) = TEMP
60 CONTINUE
IF( N0-I0.LE.4 ) THEN
Z( 4*N0+PP-1 ) = Z( 4*I0+PP-1 )
Z( 4*N0-PP ) = Z( 4*I0-PP )
END IF
DMIN2 = MIN( DMIN2, Z( 4*N0+PP-1 ) )
Z( 4*N0+PP-1 ) = MIN( Z( 4*N0+PP-1 ), Z( 4*I0+PP-1 ),
$ Z( 4*I0+PP+3 ) )
Z( 4*N0-PP ) = MIN( Z( 4*N0-PP ), Z( 4*I0-PP ),
$ Z( 4*I0-PP+4 ) )
QMAX = MAX( QMAX, Z( 4*I0+PP-3 ), Z( 4*I0+PP+1 ) )
DMIN = -ZERO
END IF
END IF
*
* Choose a shift.
*
CALL DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1,
$ DN2, TAU, TTYPE, G )
*
* Call dqds until DMIN > 0.
*
70 CONTINUE
*
CALL DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, IEEE, EPS )
*
NDIV = NDIV + ( N0-I0+2 )
ITER = ITER + 1
*
* Check status.
*
IF( DMIN.GE.ZERO .AND. DMIN1.GE.ZERO ) THEN
*
* Success.
*
GO TO 90
*
ELSE IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND.
$ Z( 4*( N0-1 )-PP ).LT.TOL*( SIGMA+DN1 ) .AND.
$ ABS( DN ).LT.TOL*SIGMA ) THEN
*
* Convergence hidden by negative DN.
*
Z( 4*( N0-1 )-PP+2 ) = ZERO
DMIN = ZERO
GO TO 90
ELSE IF( DMIN.LT.ZERO ) THEN
*
* TAU too big. Select new TAU and try again.
*
NFAIL = NFAIL + 1
IF( TTYPE.LT.-22 ) THEN
*
* Failed twice. Play it safe.
*
TAU = ZERO
ELSE IF( DMIN1.GT.ZERO ) THEN
*
* Late failure. Gives excellent shift.
*
TAU = ( TAU+DMIN )*( ONE-TWO*EPS )
TTYPE = TTYPE - 11
ELSE
*
* Early failure. Divide by 4.
*
TAU = QURTR*TAU
TTYPE = TTYPE - 12
END IF
GO TO 70
ELSE IF( DISNAN( DMIN ) ) THEN
*
* NaN.
*
IF( TAU.EQ.ZERO ) THEN
GO TO 80
ELSE
TAU = ZERO
GO TO 70
END IF
ELSE
*
* Possible underflow. Play it safe.
*
GO TO 80
END IF
*
* Risk of underflow.
*
80 CONTINUE
CALL DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 )
NDIV = NDIV + ( N0-I0+2 )
ITER = ITER + 1
TAU = ZERO
*
90 CONTINUE
IF( TAU.LT.SIGMA ) THEN
DESIG = DESIG + TAU
T = SIGMA + DESIG
DESIG = DESIG - ( T-SIGMA )
ELSE
T = SIGMA + TAU
DESIG = SIGMA - ( T-TAU ) + DESIG
END IF
SIGMA = T
*
RETURN
*
* End of DLASQ3
*
END
*> \brief \b DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ4 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
* DN1, DN2, TAU, TTYPE, G )
*
* .. Scalar Arguments ..
* INTEGER I0, N0, N0IN, PP, TTYPE
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ4 computes an approximation TAU to the smallest eigenvalue
*> using values of d from the previous transform.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*> Z holds the qd array.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] N0IN
*> \verbatim
*> N0IN is INTEGER
*> The value of N0 at start of EIGTEST.
*> \endverbatim
*>
*> \param[in] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[in] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[in] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[in] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N)
*> \endverbatim
*>
*> \param[in] DN1
*> \verbatim
*> DN1 is DOUBLE PRECISION
*> d(N-1)
*> \endverbatim
*>
*> \param[in] DN2
*> \verbatim
*> DN2 is DOUBLE PRECISION
*> d(N-2)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> This is the shift.
*> \endverbatim
*>
*> \param[out] TTYPE
*> \verbatim
*> TTYPE is INTEGER
*> Shift type.
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*> G is DOUBLE PRECISION
*> G is passed as an argument in order to save its value between
*> calls to DLASQ4.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq4
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> CNST1 = 9/16
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
$ DN1, DN2, TAU, TTYPE, G )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I0, N0, N0IN, PP, TTYPE
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION CNST1, CNST2, CNST3
PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
$ CNST3 = 1.050D0 )
DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0,
$ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, HUNDRD = 100.0D0 )
* ..
* .. Local Scalars ..
INTEGER I4, NN, NP
DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* A negative DMIN forces the shift to take that absolute value
* TTYPE records the type of shift.
*
IF( DMIN.LE.ZERO ) THEN
TAU = -DMIN
TTYPE = -1
RETURN
END IF
*
NN = 4*N0 + PP
IF( N0IN.EQ.N0 ) THEN
*
* No eigenvalues deflated.
*
IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
*
B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
A2 = Z( NN-7 ) + Z( NN-5 )
*
* Cases 2 and 3.
*
IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
GAP2 = DMIN2 - A2 - DMIN2*QURTR
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
GAP1 = A2 - DN - ( B2 / GAP2 )*B2
ELSE
GAP1 = A2 - DN - ( B1+B2 )
END IF
IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
TTYPE = -2
ELSE
S = ZERO
IF( DN.GT.B1 )
$ S = DN - B1
IF( A2.GT.( B1+B2 ) )
$ S = MIN( S, A2-( B1+B2 ) )
S = MAX( S, THIRD*DMIN )
TTYPE = -3
END IF
ELSE
*
* Case 4.
*
TTYPE = -4
S = QURTR*DMIN
IF( DMIN.EQ.DN ) THEN
GAM = DN
A2 = ZERO
IF( Z( NN-5 ) .GT. Z( NN-7 ) )
$ RETURN
B2 = Z( NN-5 ) / Z( NN-7 )
NP = NN - 9
ELSE
NP = NN - 2*PP
GAM = DN1
IF( Z( NP-4 ) .GT. Z( NP-2 ) )
$ RETURN
A2 = Z( NP-4 ) / Z( NP-2 )
IF( Z( NN-9 ) .GT. Z( NN-11 ) )
$ RETURN
B2 = Z( NN-9 ) / Z( NN-11 )
NP = NN - 13
END IF
*
* Approximate contribution to norm squared from I < NN-1.
*
A2 = A2 + B2
DO 10 I4 = NP, 4*I0 - 1 + PP, -4
IF( B2.EQ.ZERO )
$ GO TO 20
B1 = B2
IF( Z( I4 ) .GT. Z( I4-2 ) )
$ RETURN
B2 = B2*( Z( I4 ) / Z( I4-2 ) )
A2 = A2 + B2
IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
$ GO TO 20
10 CONTINUE
20 CONTINUE
A2 = CNST3*A2
*
* Rayleigh quotient residual bound.
*
IF( A2.LT.CNST1 )
$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
END IF
ELSE IF( DMIN.EQ.DN2 ) THEN
*
* Case 5.
*
TTYPE = -5
S = QURTR*DMIN
*
* Compute contribution to norm squared from I > NN-2.
*
NP = NN - 2*PP
B1 = Z( NP-2 )
B2 = Z( NP-6 )
GAM = DN2
IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
$ RETURN
A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
*
* Approximate contribution to norm squared from I < NN-2.
*
IF( N0-I0.GT.2 ) THEN
B2 = Z( NN-13 ) / Z( NN-15 )
A2 = A2 + B2
DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
IF( B2.EQ.ZERO )
$ GO TO 40
B1 = B2
IF( Z( I4 ) .GT. Z( I4-2 ) )
$ RETURN
B2 = B2*( Z( I4 ) / Z( I4-2 ) )
A2 = A2 + B2
IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
$ GO TO 40
30 CONTINUE
40 CONTINUE
A2 = CNST3*A2
END IF
*
IF( A2.LT.CNST1 )
$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
ELSE
*
* Case 6, no information to guide us.
*
IF( TTYPE.EQ.-6 ) THEN
G = G + THIRD*( ONE-G )
ELSE IF( TTYPE.EQ.-18 ) THEN
G = QURTR*THIRD
ELSE
G = QURTR
END IF
S = G*DMIN
TTYPE = -6
END IF
*
ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
*
* One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
*
IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
*
* Cases 7 and 8.
*
TTYPE = -7
S = THIRD*DMIN1
IF( Z( NN-5 ).GT.Z( NN-7 ) )
$ RETURN
B1 = Z( NN-5 ) / Z( NN-7 )
B2 = B1
IF( B2.EQ.ZERO )
$ GO TO 60
DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
A2 = B1
IF( Z( I4 ).GT.Z( I4-2 ) )
$ RETURN
B1 = B1*( Z( I4 ) / Z( I4-2 ) )
B2 = B2 + B1
IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
$ GO TO 60
50 CONTINUE
60 CONTINUE
B2 = SQRT( CNST3*B2 )
A2 = DMIN1 / ( ONE+B2**2 )
GAP2 = HALF*DMIN2 - A2
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
ELSE
S = MAX( S, A2*( ONE-CNST2*B2 ) )
TTYPE = -8
END IF
ELSE
*
* Case 9.
*
S = QURTR*DMIN1
IF( DMIN1.EQ.DN1 )
$ S = HALF*DMIN1
TTYPE = -9
END IF
*
ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
*
* Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
*
* Cases 10 and 11.
*
IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
TTYPE = -10
S = THIRD*DMIN2
IF( Z( NN-5 ).GT.Z( NN-7 ) )
$ RETURN
B1 = Z( NN-5 ) / Z( NN-7 )
B2 = B1
IF( B2.EQ.ZERO )
$ GO TO 80
DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
IF( Z( I4 ).GT.Z( I4-2 ) )
$ RETURN
B1 = B1*( Z( I4 ) / Z( I4-2 ) )
B2 = B2 + B1
IF( HUNDRD*B1.LT.B2 )
$ GO TO 80
70 CONTINUE
80 CONTINUE
B2 = SQRT( CNST3*B2 )
A2 = DMIN2 / ( ONE+B2**2 )
GAP2 = Z( NN-7 ) + Z( NN-9 ) -
$ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
ELSE
S = MAX( S, A2*( ONE-CNST2*B2 ) )
END IF
ELSE
S = QURTR*DMIN2
TTYPE = -11
END IF
ELSE IF( N0IN.GT.( N0+2 ) ) THEN
*
* Case 12, more than two eigenvalues deflated. No information.
*
S = ZERO
TTYPE = -12
END IF
*
TAU = S
RETURN
*
* End of DLASQ4
*
END
*> \brief \b DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ5 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN,
* DNM1, DNM2, IEEE, EPS )
*
* .. Scalar Arguments ..
* LOGICAL IEEE
* INTEGER I0, N0, PP
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU, SIGMA, EPS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ5 computes one dqds transform in ping-pong form, one
*> version for IEEE machines another for non IEEE machines.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*> an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> This is the shift.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> This is the accumulated shift up to this step.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*> DNM1 is DOUBLE PRECISION
*> d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*> DNM2 is DOUBLE PRECISION
*> d(N0-2).
*> \endverbatim
*>
*> \param[in] IEEE
*> \verbatim
*> IEEE is LOGICAL
*> Flag for IEEE or non IEEE arithmetic.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*> EPS is DOUBLE PRECISION
*> This is the value of epsilon used.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq5
*
* =====================================================================
SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1,
$ DMIN2,
$ DN, DNM1, DNM2, IEEE, EPS )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER I0, N0, PP
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU,
$ SIGMA, EPS
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameter ..
DOUBLE PRECISION ZERO, HALF
PARAMETER ( ZERO = 0.0D0, HALF = 0.5 )
* ..
* .. Local Scalars ..
INTEGER J4, J4P2
DOUBLE PRECISION D, EMIN, TEMP, DTHRESH
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( N0-I0-1 ).LE.0 )
$ RETURN
*
DTHRESH = EPS*(SIGMA+TAU)
IF( TAU.LT.DTHRESH*HALF ) TAU = ZERO
IF( TAU.NE.ZERO ) THEN
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 ) - TAU
DMIN = D
DMIN1 = -Z( J4 )
*
IF( IEEE ) THEN
*
* Code for IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 10 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
TEMP = Z( J4+1 ) / Z( J4-2 )
D = D*TEMP - TAU
DMIN = MIN( DMIN, D )
Z( J4 ) = Z( J4-1 )*TEMP
EMIN = MIN( Z( J4 ), EMIN )
10 CONTINUE
ELSE
DO 20 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
TEMP = Z( J4+2 ) / Z( J4-3 )
D = D*TEMP - TAU
DMIN = MIN( DMIN, D )
Z( J4-1 ) = Z( J4 )*TEMP
EMIN = MIN( Z( J4-1 ), EMIN )
20 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DN )
*
ELSE
*
* Code for non IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 30 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
30 CONTINUE
ELSE
DO 40 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
40 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( DNM2.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( DNM1.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DN )
*
END IF
ELSE
* This is the version that sets d's to zero if they are small enough
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 ) - TAU
DMIN = D
DMIN1 = -Z( J4 )
IF( IEEE ) THEN
*
* Code for IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 50 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
TEMP = Z( J4+1 ) / Z( J4-2 )
D = D*TEMP - TAU
IF( D.LT.DTHRESH ) D = ZERO
DMIN = MIN( DMIN, D )
Z( J4 ) = Z( J4-1 )*TEMP
EMIN = MIN( Z( J4 ), EMIN )
50 CONTINUE
ELSE
DO 60 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
TEMP = Z( J4+2 ) / Z( J4-3 )
D = D*TEMP - TAU
IF( D.LT.DTHRESH ) D = ZERO
DMIN = MIN( DMIN, D )
Z( J4-1 ) = Z( J4 )*TEMP
EMIN = MIN( Z( J4-1 ), EMIN )
60 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
DMIN = MIN( DMIN, DN )
*
ELSE
*
* Code for non IEEE arithmetic.
*
IF( PP.EQ.0 ) THEN
DO 70 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU
END IF
IF( D.LT.DTHRESH) D = ZERO
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
70 CONTINUE
ELSE
DO 80 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( D.LT.ZERO ) THEN
RETURN
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU
END IF
IF( D.LT.DTHRESH) D = ZERO
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
80 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( DNM2.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( DNM1.LT.ZERO ) THEN
RETURN
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU
END IF
DMIN = MIN( DMIN, DN )
*
END IF
END IF
*
Z( J4+2 ) = DN
Z( 4*N0-PP ) = EMIN
RETURN
*
* End of DLASQ5
*
END
*> \brief \b DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ6 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
* DNM1, DNM2 )
*
* .. Scalar Arguments ..
* INTEGER I0, N0, PP
* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASQ6 computes one dqd (shift equal to zero) transform in
*> ping-pong form, with protection against underflow and overflow.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] I0
*> \verbatim
*> I0 is INTEGER
*> First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*> N0 is INTEGER
*> Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( 4*N )
*> Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
*> an extra argument.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*> PP is INTEGER
*> PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[out] DMIN
*> \verbatim
*> DMIN is DOUBLE PRECISION
*> Minimum value of d.
*> \endverbatim
*>
*> \param[out] DMIN1
*> \verbatim
*> DMIN1 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[out] DMIN2
*> \verbatim
*> DMIN2 is DOUBLE PRECISION
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[out] DN
*> \verbatim
*> DN is DOUBLE PRECISION
*> d(N0), the last value of d.
*> \endverbatim
*>
*> \param[out] DNM1
*> \verbatim
*> DNM1 is DOUBLE PRECISION
*> d(N0-1).
*> \endverbatim
*>
*> \param[out] DNM2
*> \verbatim
*> DNM2 is DOUBLE PRECISION
*> d(N0-2).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq6
*
* =====================================================================
SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN,
$ DNM1, DNM2 )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER I0, N0, PP
DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2
* ..
* .. Array Arguments ..
DOUBLE PRECISION Z( * )
* ..
*
* =====================================================================
*
* .. Parameter ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER J4, J4P2
DOUBLE PRECISION D, EMIN, SAFMIN, TEMP
* ..
* .. External Function ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( ( N0-I0-1 ).LE.0 )
$ RETURN
*
SAFMIN = DLAMCH( 'Safe minimum' )
J4 = 4*I0 + PP - 3
EMIN = Z( J4+4 )
D = Z( J4 )
DMIN = D
*
IF( PP.EQ.0 ) THEN
DO 10 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-2 ) = D + Z( J4-1 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
D = Z( J4+1 )
DMIN = D
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4+1 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4+1 ) ) THEN
TEMP = Z( J4+1 ) / Z( J4-2 )
Z( J4 ) = Z( J4-1 )*TEMP
D = D*TEMP
ELSE
Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) )
D = Z( J4+1 )*( D / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4 ) )
10 CONTINUE
ELSE
DO 20 J4 = 4*I0, 4*( N0-3 ), 4
Z( J4-3 ) = D + Z( J4 )
IF( Z( J4-3 ).EQ.ZERO ) THEN
Z( J4-1 ) = ZERO
D = Z( J4+2 )
DMIN = D
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4+2 ).LT.Z( J4-3 ) .AND.
$ SAFMIN*Z( J4-3 ).LT.Z( J4+2 ) ) THEN
TEMP = Z( J4+2 ) / Z( J4-3 )
Z( J4-1 ) = Z( J4 )*TEMP
D = D*TEMP
ELSE
Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) )
D = Z( J4+2 )*( D / Z( J4-3 ) )
END IF
DMIN = MIN( DMIN, D )
EMIN = MIN( EMIN, Z( J4-1 ) )
20 CONTINUE
END IF
*
* Unroll last two steps.
*
DNM2 = D
DMIN2 = DMIN
J4 = 4*( N0-2 ) - PP
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM2 + Z( J4P2 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
DNM1 = Z( J4P2+2 )
DMIN = DNM1
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
TEMP = Z( J4P2+2 ) / Z( J4-2 )
Z( J4 ) = Z( J4P2 )*TEMP
DNM1 = DNM2*TEMP
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, DNM1 )
*
DMIN1 = DMIN
J4 = J4 + 4
J4P2 = J4 + 2*PP - 1
Z( J4-2 ) = DNM1 + Z( J4P2 )
IF( Z( J4-2 ).EQ.ZERO ) THEN
Z( J4 ) = ZERO
DN = Z( J4P2+2 )
DMIN = DN
EMIN = ZERO
ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND.
$ SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN
TEMP = Z( J4P2+2 ) / Z( J4-2 )
Z( J4 ) = Z( J4P2 )*TEMP
DN = DNM1*TEMP
ELSE
Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) )
DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) )
END IF
DMIN = MIN( DMIN, DN )
*
Z( J4+2 ) = DN
Z( 4*N0-PP ) = EMIN
RETURN
*
* End of DLASQ6
*
END
*> \brief \b DLASRT sorts numbers in increasing or decreasing order.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASRT + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASRT( ID, N, D, INFO )
*
* .. Scalar Arguments ..
* CHARACTER ID
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Sort the numbers in D in increasing order (if ID = 'I') or
*> in decreasing order (if ID = 'D' ).
*>
*> Use Quick Sort, reverting to Insertion sort on arrays of
*> size <= 20. Dimension of STACK limits N to about 2**32.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ID
*> \verbatim
*> ID is CHARACTER*1
*> = 'I': sort D in increasing order;
*> = 'D': sort D in decreasing order.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of the array D.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the array to be sorted.
*> On exit, D has been sorted into increasing order
*> (D(1) <= ... <= D(N) ) or into decreasing order
*> (D(1) >= ... >= D(N) ), depending on ID.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasrt
*
* =====================================================================
SUBROUTINE DLASRT( ID, N, D, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER ID
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER SELECT
PARAMETER ( SELECT = 20 )
* ..
* .. Local Scalars ..
INTEGER DIR, ENDD, I, J, START, STKPNT
DOUBLE PRECISION D1, D2, D3, DMNMX, TMP
* ..
* .. Local Arrays ..
INTEGER STACK( 2, 32 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
DIR = -1
IF( LSAME( ID, 'D' ) ) THEN
DIR = 0
ELSE IF( LSAME( ID, 'I' ) ) THEN
DIR = 1
END IF
IF( DIR.EQ.-1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASRT', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
STKPNT = 1
STACK( 1, 1 ) = 1
STACK( 2, 1 ) = N
10 CONTINUE
START = STACK( 1, STKPNT )
ENDD = STACK( 2, STKPNT )
STKPNT = STKPNT - 1
IF( ENDD-START.LE.SELECT .AND. ENDD-START.GT.0 ) THEN
*
* Do Insertion sort on D( START:ENDD )
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
DO 30 I = START + 1, ENDD
DO 20 J = I, START + 1, -1
IF( D( J ).GT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
*
ELSE
*
* Sort into increasing order
*
DO 50 I = START + 1, ENDD
DO 40 J = I, START + 1, -1
IF( D( J ).LT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 50
END IF
40 CONTINUE
50 CONTINUE
*
END IF
*
ELSE IF( ENDD-START.GT.SELECT ) THEN
*
* Partition D( START:ENDD ) and stack parts, largest one first
*
* Choose partition entry as median of 3
*
D1 = D( START )
D2 = D( ENDD )
I = ( START+ENDD ) / 2
D3 = D( I )
IF( D1.LT.D2 ) THEN
IF( D3.LT.D1 ) THEN
DMNMX = D1
ELSE IF( D3.LT.D2 ) THEN
DMNMX = D3
ELSE
DMNMX = D2
END IF
ELSE
IF( D3.LT.D2 ) THEN
DMNMX = D2
ELSE IF( D3.LT.D1 ) THEN
DMNMX = D3
ELSE
DMNMX = D1
END IF
END IF
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
I = START - 1
J = ENDD + 1
60 CONTINUE
70 CONTINUE
J = J - 1
IF( D( J ).LT.DMNMX )
$ GO TO 70
80 CONTINUE
I = I + 1
IF( D( I ).GT.DMNMX )
$ GO TO 80
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 60
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
ELSE
*
* Sort into increasing order
*
I = START - 1
J = ENDD + 1
90 CONTINUE
100 CONTINUE
J = J - 1
IF( D( J ).GT.DMNMX )
$ GO TO 100
110 CONTINUE
I = I + 1
IF( D( I ).LT.DMNMX )
$ GO TO 110
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 90
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
END IF
END IF
IF( STKPNT.GT.0 )
$ GO TO 10
RETURN
*
* End of DLASRT
*
END
*> \brief \b DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASV2 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASV2 computes the singular value decomposition of a 2-by-2
*> triangular matrix
*> [ F G ]
*> [ 0 H ].
*> On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
*> smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and
*> right singular vectors for abs(SSMAX), giving the decomposition
*>
*> [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ]
*> [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is DOUBLE PRECISION
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is DOUBLE PRECISION
*> abs(SSMIN) is the smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is DOUBLE PRECISION
*> abs(SSMAX) is the larger singular value.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*> SNL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*> CSL is DOUBLE PRECISION
*> The vector (CSL, SNL) is a unit left singular vector for the
*> singular value abs(SSMAX).
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*> SNR is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*> CSR is DOUBLE PRECISION
*> The vector (CSR, SNR) is a unit right singular vector for the
*> singular value abs(SSMAX).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasv2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Any input parameter may be aliased with any output parameter.
*>
*> Barring over/underflow and assuming a guard digit in subtraction, all
*> output quantities are correct to within a few units in the last
*> place (ulps).
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows or is within a few ulps of overflow.
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION FOUR
PARAMETER ( FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
LOGICAL GASMAL, SWAP
INTEGER PMAX
DOUBLE PRECISION A, CLT, CRT, D, FA, FT, GA, GT, HA, HT, L, M,
$ MM, R, S, SLT, SRT, T, TEMP, TSIGN, TT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Executable Statements ..
*
FT = F
FA = ABS( FT )
HT = H
HA = ABS( H )
*
* PMAX points to the maximum absolute element of matrix
* PMAX = 1 if F largest in absolute values
* PMAX = 2 if G largest in absolute values
* PMAX = 3 if H largest in absolute values
*
PMAX = 1
SWAP = ( HA.GT.FA )
IF( SWAP ) THEN
PMAX = 3
TEMP = FT
FT = HT
HT = TEMP
TEMP = FA
FA = HA
HA = TEMP
*
* Now FA .ge. HA
*
END IF
GT = G
GA = ABS( GT )
IF( GA.EQ.ZERO ) THEN
*
* Diagonal matrix
*
SSMIN = HA
SSMAX = FA
CLT = ONE
CRT = ONE
SLT = ZERO
SRT = ZERO
ELSE
GASMAL = .TRUE.
IF( GA.GT.FA ) THEN
PMAX = 2
IF( ( FA / GA ).LT.DLAMCH( 'EPS' ) ) THEN
*
* Case of very large GA
*
GASMAL = .FALSE.
SSMAX = GA
IF( HA.GT.ONE ) THEN
SSMIN = FA / ( GA / HA )
ELSE
SSMIN = ( FA / GA )*HA
END IF
CLT = ONE
SLT = HT / GT
SRT = ONE
CRT = FT / GT
END IF
END IF
IF( GASMAL ) THEN
*
* Normal case
*
D = FA - HA
IF( D.EQ.FA ) THEN
*
* Copes with infinite F or H
*
L = ONE
ELSE
L = D / FA
END IF
*
* Note that 0 .le. L .le. 1
*
M = GT / FT
*
* Note that abs(M) .le. 1/macheps
*
T = TWO - L
*
* Note that T .ge. 1
*
MM = M*M
TT = T*T
S = SQRT( TT+MM )
*
* Note that 1 .le. S .le. 1 + 1/macheps
*
IF( L.EQ.ZERO ) THEN
R = ABS( M )
ELSE
R = SQRT( L*L+MM )
END IF
*
* Note that 0 .le. R .le. 1 + 1/macheps
*
A = HALF*( S+R )
*
* Note that 1 .le. A .le. 1 + abs(M)
*
SSMIN = HA / A
SSMAX = FA*A
IF( MM.EQ.ZERO ) THEN
*
* Note that M is very tiny
*
IF( L.EQ.ZERO ) THEN
T = SIGN( TWO, FT )*SIGN( ONE, GT )
ELSE
T = GT / SIGN( D, FT ) + M / T
END IF
ELSE
T = ( M / ( S+T )+M / ( R+L ) )*( ONE+A )
END IF
L = SQRT( T*T+FOUR )
CRT = TWO / L
SRT = T / L
CLT = ( CRT+SRT*M ) / A
SLT = ( HT / FT )*SRT / A
END IF
END IF
IF( SWAP ) THEN
CSL = SRT
SNL = CRT
CSR = SLT
SNR = CLT
ELSE
CSL = CLT
SNL = SLT
CSR = CRT
SNR = SRT
END IF
*
* Correct signs of SSMAX and SSMIN
*
IF( PMAX.EQ.1 )
$ TSIGN = SIGN( ONE, CSR )*SIGN( ONE, CSL )*SIGN( ONE, F )
IF( PMAX.EQ.2 )
$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, CSL )*SIGN( ONE, G )
IF( PMAX.EQ.3 )
$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, SNL )*SIGN( ONE, H )
SSMAX = SIGN( SSMAX, TSIGN )
SSMIN = SIGN( SSMIN, TSIGN*SIGN( ONE, F )*SIGN( ONE, H ) )
RETURN
*
* End of DLASV2
*
END
*> \brief \b DPTTRF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTTRF + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTTRF( N, D, E, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTTRF computes the L*D*L**T factorization of a real symmetric
*> positive definite tridiagonal matrix A. The factorization may also
*> be regarded as having the form A = U**T*D*U.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A. On exit, the n diagonal elements of the diagonal matrix
*> D from the L*D*L**T factorization of A.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A. On exit, the (n-1) subdiagonal elements of the
*> unit bidiagonal factor L from the L*D*L**T factorization of A.
*> E can also be regarded as the superdiagonal of the unit
*> bidiagonal factor U from the U**T*D*U factorization of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading principal minor of order k
*> is not positive; if k < N, the factorization could not
*> be completed, while if k = N, the factorization was
*> completed, but D(N) <= 0.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pttrf
*
* =====================================================================
SUBROUTINE DPTTRF( N, D, E, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, I4
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DPTTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
I4 = MOD( N-1, 4 )
DO 10 I = 1, I4
IF( D( I ).LE.ZERO ) THEN
INFO = I
GO TO 30
END IF
EI = E( I )
E( I ) = EI / D( I )
D( I+1 ) = D( I+1 ) - E( I )*EI
10 CONTINUE
*
DO 20 I = I4 + 1, N - 4, 4
*
* Drop out of the loop if d(i) <= 0: the matrix is not positive
* definite.
*
IF( D( I ).LE.ZERO ) THEN
INFO = I
GO TO 30
END IF
*
* Solve for e(i) and d(i+1).
*
EI = E( I )
E( I ) = EI / D( I )
D( I+1 ) = D( I+1 ) - E( I )*EI
*
IF( D( I+1 ).LE.ZERO ) THEN
INFO = I + 1
GO TO 30
END IF
*
* Solve for e(i+1) and d(i+2).
*
EI = E( I+1 )
E( I+1 ) = EI / D( I+1 )
D( I+2 ) = D( I+2 ) - E( I+1 )*EI
*
IF( D( I+2 ).LE.ZERO ) THEN
INFO = I + 2
GO TO 30
END IF
*
* Solve for e(i+2) and d(i+3).
*
EI = E( I+2 )
E( I+2 ) = EI / D( I+2 )
D( I+3 ) = D( I+3 ) - E( I+2 )*EI
*
IF( D( I+3 ).LE.ZERO ) THEN
INFO = I + 3
GO TO 30
END IF
*
* Solve for e(i+3) and d(i+4).
*
EI = E( I+3 )
E( I+3 ) = EI / D( I+3 )
D( I+4 ) = D( I+4 ) - E( I+3 )*EI
20 CONTINUE
*
* Check d(n) for positive definiteness.
*
IF( D( N ).LE.ZERO )
$ INFO = N
*
30 CONTINUE
RETURN
*
* End of DPTTRF
*
END
*> \brief \b IEEECK
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IEEECK + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* .. Scalar Arguments ..
* INTEGER ISPEC
* REAL ONE, ZERO
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> IEEECK is called from the ILAENV to verify that Infinity and
*> possibly NaN arithmetic is safe (i.e. will not trap).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> Specifies whether to test just for infinity arithmetic
*> or whether to test for infinity and NaN arithmetic.
*> = 0: Verify infinity arithmetic only.
*> = 1: Verify infinity and NaN arithmetic.
*> \endverbatim
*>
*> \param[in] ZERO
*> \verbatim
*> ZERO is REAL
*> Must contain the value 0.0
*> This is passed to prevent the compiler from optimizing
*> away this code.
*> \endverbatim
*>
*> \param[in] ONE
*> \verbatim
*> ONE is REAL
*> Must contain the value 1.0
*> This is passed to prevent the compiler from optimizing
*> away this code.
*>
*> RETURN VALUE: INTEGER
*> = 0: Arithmetic failed to produce the correct answers
*> = 1: Arithmetic produced the correct answers
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ieeeck
*
* =====================================================================
INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER ISPEC
REAL ONE, ZERO
* ..
*
* =====================================================================
*
* .. Local Scalars ..
REAL NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF,
$ NEGZRO, NEWZRO, POSINF
* ..
* .. Executable Statements ..
IEEECK = 1
*
POSINF = ONE / ZERO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = -ONE / ZERO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGZRO = ONE / ( NEGINF+ONE )
IF( NEGZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = ONE / NEGZRO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEWZRO = NEGZRO + ZERO
IF( NEWZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = ONE / NEWZRO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = NEGINF*POSINF
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = POSINF*POSINF
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
*
*
*
* Return if we were only asked to check infinity arithmetic
*
IF( ISPEC.EQ.0 )
$ RETURN
*
NAN1 = POSINF + NEGINF
*
NAN2 = POSINF / NEGINF
*
NAN3 = POSINF / POSINF
*
NAN4 = POSINF*ZERO
*
NAN5 = NEGINF*NEGZRO
*
NAN6 = NAN5*ZERO
*
IF( NAN1.EQ.NAN1 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN2.EQ.NAN2 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN3.EQ.NAN3 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN4.EQ.NAN4 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN5.EQ.NAN5 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN6.EQ.NAN6 ) THEN
IEEECK = 0
RETURN
END IF
*
RETURN
END
*> \brief \b ILAENV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ILAENV + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* .. Scalar Arguments ..
* CHARACTER*( * ) NAME, OPTS
* INTEGER ISPEC, N1, N2, N3, N4
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ILAENV is called from the LAPACK routines to choose problem-dependent
*> parameters for the local environment. See ISPEC for a description of
*> the parameters.
*>
*> ILAENV returns an INTEGER
*> if ILAENV >= 0: ILAENV returns the value of the parameter specified by ISPEC
*> if ILAENV < 0: if ILAENV = -k, the k-th argument had an illegal value.
*>
*> This version provides a set of parameters which should give good,
*> but not optimal, performance on many of the currently available
*> computers. Users are encouraged to modify this subroutine to set
*> the tuning parameters for their particular machine using the option
*> and problem size information in the arguments.
*>
*> This routine will not function correctly if it is converted to all
*> lower case. Converting it to all upper case is allowed.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> Specifies the parameter to be returned as the value of
*> ILAENV.
*> = 1: the optimal blocksize; if this value is 1, an unblocked
*> algorithm will give the best performance.
*> = 2: the minimum block size for which the block routine
*> should be used; if the usable block size is less than
*> this value, an unblocked routine should be used.
*> = 3: the crossover point (in a block routine, for N less
*> than this value, an unblocked routine should be used)
*> = 4: the number of shifts, used in the nonsymmetric
*> eigenvalue routines (DEPRECATED)
*> = 5: the minimum column dimension for blocking to be used;
*> rectangular blocks must have dimension at least k by m,
*> where k is given by ILAENV(2,...) and m by ILAENV(5,...)
*> = 6: the crossover point for the SVD (when reducing an m by n
*> matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
*> this value, a QR factorization is used first to reduce
*> the matrix to a triangular form.)
*> = 7: the number of processors
*> = 8: the crossover point for the multishift QR method
*> for nonsymmetric eigenvalue problems (DEPRECATED)
*> = 9: maximum size of the subproblems at the bottom of the
*> computation tree in the divide-and-conquer algorithm
*> (used by xGELSD and xGESDD)
*> =10: ieee infinity and NaN arithmetic can be trusted not to trap
*> =11: infinity arithmetic can be trusted not to trap
*> 12 <= ISPEC <= 17:
*> xHSEQR or related subroutines,
*> see IPARMQ for detailed explanation
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is CHARACTER*(*)
*> The name of the calling subroutine, in either upper case or
*> lower case.
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is CHARACTER*(*)
*> The character options to the subroutine NAME, concatenated
*> into a single character string. For example, UPLO = 'U',
*> TRANS = 'T', and DIAG = 'N' for a triangular routine would
*> be specified as OPTS = 'UTN'.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> \endverbatim
*>
*> \param[in] N3
*> \verbatim
*> N3 is INTEGER
*> \endverbatim
*>
*> \param[in] N4
*> \verbatim
*> N4 is INTEGER
*> Problem dimensions for the subroutine NAME; these may not all
*> be required.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ilaenv
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The following conventions have been used when calling ILAENV from the
*> LAPACK routines:
*> 1) OPTS is a concatenation of all of the character options to
*> subroutine NAME, in the same order that they appear in the
*> argument list for NAME, even if they are not used in determining
*> the value of the parameter specified by ISPEC.
*> 2) The problem dimensions N1, N2, N3, N4 are specified in the order
*> that they appear in the argument list for NAME. N1 is used
*> first, N2 second, and so on, and unused problem dimensions are
*> passed a value of -1.
*> 3) The parameter value returned by ILAENV is checked for validity in
*> the calling subroutine. For example, ILAENV is used to retrieve
*> the optimal blocksize for STRTRI as follows:
*>
*> NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
*> IF( NB.LE.1 ) NB = MAX( 1, N )
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, N1, N2, N3, N4
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IC, IZ, NB, NBMIN, NX
LOGICAL CNAME, SNAME, TWOSTAGE
CHARACTER C1*1, C2*2, C4*2, C3*3, SUBNAM*16
* ..
* .. Intrinsic Functions ..
INTRINSIC CHAR, ICHAR, INT, MIN, REAL
* ..
* .. External Functions ..
INTEGER IEEECK, IPARMQ, IPARAM2STAGE
EXTERNAL IEEECK, IPARMQ, IPARAM2STAGE
* ..
* .. Executable Statements ..
*
GO TO ( 10, 10, 10, 80, 90, 100, 110, 120,
$ 130, 140, 150, 160, 160, 160, 160, 160, 160)ISPEC
*
* Invalid value for ISPEC
*
ILAENV = -1
RETURN
*
10 CONTINUE
*
* Convert NAME to upper case if the first character is lower case.
*
ILAENV = 1
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 20 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
20 CONTINUE
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO 30 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
30 CONTINUE
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 40 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
40 CONTINUE
END IF
END IF
*
C1 = SUBNAM( 1: 1 )
SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
IF( .NOT.( CNAME .OR. SNAME ) )
$ RETURN
C2 = SUBNAM( 2: 3 )
C3 = SUBNAM( 4: 6 )
C4 = C3( 2: 3 )
TWOSTAGE = LEN( SUBNAM ).GE.11
$ .AND. SUBNAM( 11: 11 ).EQ.'2'
*
GO TO ( 50, 60, 70 )ISPEC
*
50 CONTINUE
*
* ISPEC = 1: block size
*
* In these examples, separate code is provided for setting NB for
* real and complex. We assume that NB will take the same value in
* single or double precision.
*
NB = 1
*
IF( SUBNAM(2:6).EQ.'LAORH' ) THEN
*
* This is for *LAORHR_GETRFNP routine
*
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'QR ') THEN
IF( N3 .EQ. 1) THEN
IF( SNAME ) THEN
* M*N
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
ELSE
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
END IF
ELSE
IF( SNAME ) THEN
NB = 1
ELSE
NB = 1
END IF
END IF
ELSE IF( C3.EQ.'LQ ') THEN
IF( N3 .EQ. 2) THEN
IF( SNAME ) THEN
* M*N
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
ELSE
IF ((N1*N2.LE.131072).OR.(N1.LE.8192)) THEN
NB = N1
ELSE
NB = 32768/N2
END IF
END IF
ELSE
IF( SNAME ) THEN
NB = 1
ELSE
NB = 1
END IF
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'PO' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( TWOSTAGE ) THEN
NB = 192
ELSE
NB = 64
END IF
ELSE
IF( TWOSTAGE ) THEN
NB = 192
ELSE
NB = 64
END IF
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( TWOSTAGE ) THEN
NB = 192
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'GB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'PB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'TR' ) THEN
IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF ( C3.EQ.'EVC' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'SYL' ) THEN
* The upper bound is to prevent overly aggressive scaling.
IF( SNAME ) THEN
NB = MIN( MAX( 48, INT( ( MIN( N1, N2 ) * 16 ) / 100) ),
$ 240 )
ELSE
NB = MIN( MAX( 24, INT( ( MIN( N1, N2 ) * 8 ) / 100) ),
$ 80 )
END IF
END IF
ELSE IF( C2.EQ.'LA' ) THEN
IF( C3.EQ.'UUM' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'TRS' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
IF( C3.EQ.'EBZ' ) THEN
NB = 1
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NB = 32
IF( C3.EQ.'HD3' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
END IF
END IF
ILAENV = NB
RETURN
*
60 CONTINUE
*
* ISPEC = 2: minimum block size
*
NBMIN = 2
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NBMIN = 8
ELSE
NBMIN = 8
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NBMIN = 2
IF( C3.EQ.'HD3' ) THEN
NBMIN = 2
END IF
END IF
ILAENV = NBMIN
RETURN
*
70 CONTINUE
*
* ISPEC = 3: crossover point
*
NX = 0
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'GG' ) THEN
NX = 128
IF( C3.EQ.'HD3' ) THEN
NX = 128
END IF
END IF
ILAENV = NX
RETURN
*
80 CONTINUE
*
* ISPEC = 4: number of shifts (used by xHSEQR)
*
ILAENV = 6
RETURN
*
90 CONTINUE
*
* ISPEC = 5: minimum column dimension (not used)
*
ILAENV = 2
RETURN
*
100 CONTINUE
*
* ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD)
*
ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
RETURN
*
110 CONTINUE
*
* ISPEC = 7: number of processors (not used)
*
ILAENV = 1
RETURN
*
120 CONTINUE
*
* ISPEC = 8: crossover point for multishift (used by xHSEQR)
*
ILAENV = 50
RETURN
*
130 CONTINUE
*
* ISPEC = 9: maximum size of the subproblems at the bottom of the
* computation tree in the divide-and-conquer algorithm
* (used by xGELSD and xGESDD)
*
ILAENV = 25
RETURN
*
140 CONTINUE
*
* ISPEC = 10: ieee and infinity NaN arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 1, 0.0, 1.0 )
END IF
RETURN
*
150 CONTINUE
*
* ISPEC = 11: ieee infinity arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 0, 0.0, 1.0 )
END IF
RETURN
*
160 CONTINUE
*
* 12 <= ISPEC <= 17: xHSEQR or related subroutines.
*
ILAENV = IPARMQ( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
RETURN
*
* End of ILAENV
*
END
*> \brief \b IPARMQ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download IPARMQ + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, ISPEC, LWORK, N
* CHARACTER NAME*( * ), OPTS*( * )
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This program sets problem and machine dependent parameters
*> useful for xHSEQR and related subroutines for eigenvalue
*> problems. It is called whenever
*> IPARMQ is called with 12 <= ISPEC <= 16
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ISPEC
*> \verbatim
*> ISPEC is INTEGER
*> ISPEC specifies which tunable parameter IPARMQ should
*> return.
*>
*> ISPEC=12: (INMIN) Matrices of order nmin or less
*> are sent directly to xLAHQR, the implicit
*> double shift QR algorithm. NMIN must be
*> at least 11.
*>
*> ISPEC=13: (INWIN) Size of the deflation window.
*> This is best set greater than or equal to
*> the number of simultaneous shifts NS.
*> Larger matrices benefit from larger deflation
*> windows.
*>
*> ISPEC=14: (INIBL) Determines when to stop nibbling and
*> invest in an (expensive) multi-shift QR sweep.
*> If the aggressive early deflation subroutine
*> finds LD converged eigenvalues from an order
*> NW deflation window and LD > (NW*NIBBLE)/100,
*> then the next QR sweep is skipped and early
*> deflation is applied immediately to the
*> remaining active diagonal block. Setting
*> IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
*> multi-shift QR sweep whenever early deflation
*> finds a converged eigenvalue. Setting
*> IPARMQ(ISPEC=14) greater than or equal to 100
*> prevents TTQRE from skipping a multi-shift
*> QR sweep.
*>
*> ISPEC=15: (NSHFTS) The number of simultaneous shifts in
*> a multi-shift QR iteration.
*>
*> ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
*> following meanings.
*> 0: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are not
*> accumulated when updating the
*> far-from-diagonal matrix entries.
*> 1: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are
*> accumulated, and matrix-matrix
*> multiplication is used to update the
*> far-from-diagonal matrix entries.
*> 2: During the multi-shift QR/QZ sweep,
*> blocked eigenvalue reordering, blocked
*> Hessenberg-triangular reduction,
*> reflections and/or rotations are
*> accumulated, and 2-by-2 block structure
*> is exploited during matrix-matrix
*> multiplies.
*> (If xTRMM is slower than xGEMM, then
*> IPARMQ(ISPEC=16)=1 may be more efficient than
*> IPARMQ(ISPEC=16)=2 despite the greater level of
*> arithmetic work implied by the latter choice.)
*>
*> ISPEC=17: (ICOST) An estimate of the relative cost of flops
*> within the near-the-diagonal shift chase compared
*> to flops within the BLAS calls of a QZ sweep.
*> \endverbatim
*>
*> \param[in] NAME
*> \verbatim
*> NAME is CHARACTER string
*> Name of the calling subroutine
*> \endverbatim
*>
*> \param[in] OPTS
*> \verbatim
*> OPTS is CHARACTER string
*> This is a concatenation of the string arguments to
*> TTQRE.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> N is the order of the Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper triangular
*> in rows and columns 1:ILO-1 and IHI+1:N.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The amount of workspace available.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup iparmq
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Little is known about how best to choose these parameters.
*> It is possible to use different values of the parameters
*> for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.
*>
*> It is probably best to choose different parameters for
*> different matrices and different parameters at different
*> times during the iteration, but this has not been
*> implemented --- yet.
*>
*>
*> The best choices of most of the parameters depend
*> in an ill-understood way on the relative execution
*> rate of xLAQR3 and xLAQR5 and on the nature of each
*> particular eigenvalue problem. Experiment may be the
*> only practical way to determine which choices are most
*> effective.
*>
*> Following is a list of default values supplied by IPARMQ.
*> These defaults may be adjusted in order to attain better
*> performance in any particular computational environment.
*>
*> IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
*> Default: 75. (Must be at least 11.)
*>
*> IPARMQ(ISPEC=13) Recommended deflation window size.
*> This depends on ILO, IHI and NS, the
*> number of simultaneous shifts returned
*> by IPARMQ(ISPEC=15). The default for
*> (IHI-ILO+1) <= 500 is NS. The default
*> for (IHI-ILO+1) > 500 is 3*NS/2.
*>
*> IPARMQ(ISPEC=14) Nibble crossover point. Default: 14.
*>
*> IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
*> a multi-shift QR iteration.
*>
*> If IHI-ILO+1 is ...
*>
*> greater than ...but less ... the
*> or equal to ... than default is
*>
*> 0 30 NS = 2+
*> 30 60 NS = 4+
*> 60 150 NS = 10
*> 150 590 NS = **
*> 590 3000 NS = 64
*> 3000 6000 NS = 128
*> 6000 infinity NS = 256
*>
*> (+) By default matrices of this order are
*> passed to the implicit double shift routine
*> xLAHQR. See IPARMQ(ISPEC=12) above. These
*> values of NS are used only in case of a rare
*> xLAHQR failure.
*>
*> (**) The asterisks (**) indicate an ad-hoc
*> function increasing from 10 to 64.
*>
*> IPARMQ(ISPEC=16) Select structured matrix multiply.
*> (See ISPEC=16 above for details.)
*> Default: 3.
*>
*> IPARMQ(ISPEC=17) Relative cost heuristic for blocksize selection.
*> Expressed as a percentage.
*> Default: 10.
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI,
$ LWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, ISPEC, LWORK, N
CHARACTER NAME*( * ), OPTS*( * )
*
* ================================================================
* .. Parameters ..
INTEGER INMIN, INWIN, INIBL, ISHFTS, IACC22, ICOST
PARAMETER ( INMIN = 12, INWIN = 13, INIBL = 14,
$ ISHFTS = 15, IACC22 = 16, ICOST = 17 )
INTEGER NMIN, K22MIN, KACMIN, NIBBLE, KNWSWP, RCOST
PARAMETER ( NMIN = 75, K22MIN = 14, KACMIN = 14,
$ NIBBLE = 14, KNWSWP = 500, RCOST = 10 )
REAL TWO
PARAMETER ( TWO = 2.0 )
* ..
* .. Local Scalars ..
INTEGER NH, NS
INTEGER I, IC, IZ
CHARACTER SUBNAM*6
* ..
* .. Intrinsic Functions ..
INTRINSIC LOG, MAX, MOD, NINT, REAL
* ..
* .. Executable Statements ..
IF( ( ISPEC.EQ.ISHFTS ) .OR. ( ISPEC.EQ.INWIN ) .OR.
$ ( ISPEC.EQ.IACC22 ) ) THEN
*
* ==== Set the number simultaneous shifts ====
*
NH = IHI - ILO + 1
NS = 2
IF( NH.GE.30 )
$ NS = 4
IF( NH.GE.60 )
$ NS = 10
IF( NH.GE.150 )
$ NS = MAX( 10, NH / NINT( LOG( REAL( NH ) ) / LOG( TWO ) ) )
IF( NH.GE.590 )
$ NS = 64
IF( NH.GE.3000 )
$ NS = 128
IF( NH.GE.6000 )
$ NS = 256
NS = MAX( 2, NS-MOD( NS, 2 ) )
END IF
*
IF( ISPEC.EQ.INMIN ) THEN
*
*
* ===== Matrices of order smaller than NMIN get sent
* . to xLAHQR, the classic double shift algorithm.
* . This must be at least 11. ====
*
IPARMQ = NMIN
*
ELSE IF( ISPEC.EQ.INIBL ) THEN
*
* ==== INIBL: skip a multi-shift qr iteration and
* . whenever aggressive early deflation finds
* . at least (NIBBLE*(window size)/100) deflations. ====
*
IPARMQ = NIBBLE
*
ELSE IF( ISPEC.EQ.ISHFTS ) THEN
*
* ==== NSHFTS: The number of simultaneous shifts =====
*
IPARMQ = NS
*
ELSE IF( ISPEC.EQ.INWIN ) THEN
*
* ==== NW: deflation window size. ====
*
IF( NH.LE.KNWSWP ) THEN
IPARMQ = NS
ELSE
IPARMQ = 3*NS / 2
END IF
*
ELSE IF( ISPEC.EQ.IACC22 ) THEN
*
* ==== IACC22: Whether to accumulate reflections
* . before updating the far-from-diagonal elements
* . and whether to use 2-by-2 block structure while
* . doing it. A small amount of work could be saved
* . by making this choice dependent also upon the
* . NH=IHI-ILO+1.
*
*
* Convert NAME to upper case if the first character is lower case.
*
IPARMQ = 0
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
END DO
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
END DO
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
END DO
END IF
END IF
*
IF( SUBNAM( 2:6 ).EQ.'GGHRD' .OR.
$ SUBNAM( 2:6 ).EQ.'GGHD3' ) THEN
IPARMQ = 1
IF( NH.GE.K22MIN )
$ IPARMQ = 2
ELSE IF ( SUBNAM( 4:6 ).EQ.'EXC' ) THEN
IF( NH.GE.KACMIN )
$ IPARMQ = 1
IF( NH.GE.K22MIN )
$ IPARMQ = 2
ELSE IF ( SUBNAM( 2:6 ).EQ.'HSEQR' .OR.
$ SUBNAM( 2:5 ).EQ.'LAQR' ) THEN
IF( NS.GE.KACMIN )
$ IPARMQ = 1
IF( NS.GE.K22MIN )
$ IPARMQ = 2
END IF
*
ELSE IF( ISPEC.EQ.ICOST ) THEN
*
* === Relative cost of near-the-diagonal chase vs
* BLAS updates ===
*
IPARMQ = RCOST
ELSE
* ===== invalid value of ispec =====
IPARMQ = -1
*
END IF
*
* ==== End of IPARMQ ====
*
END
*> \brief \b LSAME
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* LOGICAL FUNCTION LSAME(CA,CB)
*
* .. Scalar Arguments ..
* CHARACTER CA,CB
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> LSAME returns .TRUE. if CA is the same letter as CB regardless of
*> case.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] CA
*> \verbatim
*> CA is CHARACTER*1
*> \endverbatim
*>
*> \param[in] CB
*> \verbatim
*> CB is CHARACTER*1
*> CA and CB specify the single characters to be compared.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lsame
*
* =====================================================================
LOGICAL FUNCTION LSAME(CA,CB)
*
* -- Reference BLAS level1 routine --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER CA,CB
* ..
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC ICHAR
* ..
* .. Local Scalars ..
INTEGER INTA,INTB,ZCODE
* ..
*
* Test if the characters are equal
*
LSAME = CA .EQ. CB
IF (LSAME) RETURN
*
* Now test for equivalence if both characters are alphabetic.
*
ZCODE = ICHAR('Z')
*
* Use 'Z' rather than 'A' so that ASCII can be detected on Prime
* machines, on which ICHAR returns a value with bit 8 set.
* ICHAR('A') on Prime machines returns 193 which is the same as
* ICHAR('A') on an EBCDIC machine.
*
INTA = ICHAR(CA)
INTB = ICHAR(CB)
*
IF (ZCODE.EQ.90 .OR. ZCODE.EQ.122) THEN
*
* ASCII is assumed - ZCODE is the ASCII code of either lower or
* upper case 'Z'.
*
IF (INTA.GE.97 .AND. INTA.LE.122) INTA = INTA - 32
IF (INTB.GE.97 .AND. INTB.LE.122) INTB = INTB - 32
*
ELSE IF (ZCODE.EQ.233 .OR. ZCODE.EQ.169) THEN
*
* EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or
* upper case 'Z'.
*
IF (INTA.GE.129 .AND. INTA.LE.137 .OR.
+ INTA.GE.145 .AND. INTA.LE.153 .OR.
+ INTA.GE.162 .AND. INTA.LE.169) INTA = INTA + 64
IF (INTB.GE.129 .AND. INTB.LE.137 .OR.
+ INTB.GE.145 .AND. INTB.LE.153 .OR.
+ INTB.GE.162 .AND. INTB.LE.169) INTB = INTB + 64
*
ELSE IF (ZCODE.EQ.218 .OR. ZCODE.EQ.250) THEN
*
* ASCII is assumed, on Prime machines - ZCODE is the ASCII code
* plus 128 of either lower or upper case 'Z'.
*
IF (INTA.GE.225 .AND. INTA.LE.250) INTA = INTA - 32
IF (INTB.GE.225 .AND. INTB.LE.250) INTB = INTB - 32
END IF
LSAME = INTA .EQ. INTB
*
* RETURN
*
* End of LSAME
*
END
*> \brief \b XERBLA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE XERBLA( SRNAME, INFO )
*
* .. Scalar Arguments ..
* CHARACTER*(*) SRNAME
* INTEGER INFO
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> XERBLA is an error handler for the LAPACK routines.
*> It is called by an LAPACK routine if an input parameter has an
*> invalid value. A message is printed and execution stops.
*>
*> Installers may consider modifying the STOP statement in order to
*> call system-specific exception-handling facilities.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SRNAME
*> \verbatim
*> SRNAME is CHARACTER*(*)
*> The name of the routine which called XERBLA.
*> \endverbatim
*>
*> \param[in] INFO
*> \verbatim
*> INFO is INTEGER
*> The position of the invalid parameter in the parameter list
*> of the calling routine.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup xerbla
*
* =====================================================================
SUBROUTINE XERBLA( SRNAME, INFO )
*
* -- Reference BLAS level1 routine --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER*(*) SRNAME
INTEGER INFO
* ..
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC LEN_TRIM
* ..
* .. Executable Statements ..
*
WRITE( *, FMT = 9999 )SRNAME( 1:LEN_TRIM( SRNAME ) ), INFO
*
STOP
*
9999 FORMAT( ' ** On entry to ', A, ' parameter number ', I2, ' had ',
$ 'an illegal value' )
*
* End of XERBLA
*
END
*> \brief \b ZBDSQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZBDSQR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
* LDU, C, LDC, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), RWORK( * )
* COMPLEX*16 C( LDC, * ), U( LDU, * ), VT( LDVT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZBDSQR computes the singular values and, optionally, the right and/or
*> left singular vectors from the singular value decomposition (SVD) of
*> a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
*> zero-shift QR algorithm. The SVD of B has the form
*>
*> B = Q * S * P**H
*>
*> where S is the diagonal matrix of singular values, Q is an orthogonal
*> matrix of left singular vectors, and P is an orthogonal matrix of
*> right singular vectors. If left singular vectors are requested, this
*> subroutine actually returns U*Q instead of Q, and, if right singular
*> vectors are requested, this subroutine returns P**H*VT instead of
*> P**H, for given complex input matrices U and VT. When U and VT are
*> the unitary matrices that reduce a general matrix A to bidiagonal
*> form: A = U*B*VT, as computed by ZGEBRD, then
*>
*> A = (U*Q) * S * (P**H*VT)
*>
*> is the SVD of A. Optionally, the subroutine may also compute Q**H*C
*> for a given complex input matrix C.
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
*> no. 5, pp. 873-912, Sept 1990) and
*> "Accurate singular values and differential qd algorithms," by
*> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
*> Department, University of California at Berkeley, July 1992
*> for a detailed description of the algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': B is upper bidiagonal;
*> = 'L': B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B. N >= 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*> NCVT is INTEGER
*> The number of columns of the matrix VT. NCVT >= 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*> NRU is INTEGER
*> The number of rows of the matrix U. NRU >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*> NCC is INTEGER
*> The number of columns of the matrix C. NCC >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the bidiagonal matrix B.
*> On exit, if INFO=0, the singular values of B in decreasing
*> order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the N-1 offdiagonal elements of the bidiagonal
*> matrix B.
*> On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
*> will contain the diagonal and superdiagonal elements of a
*> bidiagonal matrix orthogonally equivalent to the one given
*> as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is COMPLEX*16 array, dimension (LDVT, NCVT)
*> On entry, an N-by-NCVT matrix VT.
*> On exit, VT is overwritten by P**H * VT.
*> Not referenced if NCVT = 0.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT.
*> LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is COMPLEX*16 array, dimension (LDU, N)
*> On entry, an NRU-by-N matrix U.
*> On exit, U is overwritten by U * Q.
*> Not referenced if NRU = 0.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,NRU).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is COMPLEX*16 array, dimension (LDC, NCC)
*> On entry, an N-by-NCC matrix C.
*> On exit, C is overwritten by Q**H * C.
*> Not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C.
*> LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: If INFO = -i, the i-th argument had an illegal value
*> > 0: the algorithm did not converge; D and E contain the
*> elements of a bidiagonal matrix which is orthogonally
*> similar to the input matrix B; if INFO = i, i
*> elements of E have not converged to zero.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
*> TOLMUL controls the convergence criterion of the QR loop.
*> If it is positive, TOLMUL*EPS is the desired relative
*> precision in the computed singular values.
*> If it is negative, abs(TOLMUL*EPS*sigma_max) is the
*> desired absolute accuracy in the computed singular
*> values (corresponds to relative accuracy
*> abs(TOLMUL*EPS) in the largest singular value.
*> abs(TOLMUL) should be between 1 and 1/EPS, and preferably
*> between 10 (for fast convergence) and .1/EPS
*> (for there to be some accuracy in the results).
*> Default is to lose at either one eighth or 2 of the
*> available decimal digits in each computed singular value
*> (whichever is smaller).
*>
*> MAXITR INTEGER, default = 6
*> MAXITR controls the maximum number of passes of the
*> algorithm through its inner loop. The algorithms stops
*> (and so fails to converge) if the number of passes
*> through the inner loop exceeds MAXITR*N**2.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16OTHERcomputational
*
* =====================================================================
SUBROUTINE ZBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
$ LDU, C, LDC, RWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), RWORK( * )
COMPLEX*16 C( LDC, * ), U( LDU, * ), VT( LDVT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION NEGONE
PARAMETER ( NEGONE = -1.0D0 )
DOUBLE PRECISION HNDRTH
PARAMETER ( HNDRTH = 0.01D0 )
DOUBLE PRECISION TEN
PARAMETER ( TEN = 10.0D0 )
DOUBLE PRECISION HNDRD
PARAMETER ( HNDRD = 100.0D0 )
DOUBLE PRECISION MEIGTH
PARAMETER ( MEIGTH = -0.125D0 )
INTEGER MAXITR
PARAMETER ( MAXITR = 6 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, ROTATE
INTEGER I, IDIR, ISUB, ITER, J, LL, LLL, M, MAXIT, NM1,
$ NM12, NM13, OLDLL, OLDM
DOUBLE PRECISION ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU,
$ OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL,
$ SINR, SLL, SMAX, SMIN, SMINL, SMINOA,
$ SN, THRESH, TOL, TOLMUL, UNFL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLAS2, DLASQ1, DLASV2, XERBLA, ZDROT,
$ ZDSCAL, ZLASR, ZSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NCVT.LT.0 ) THEN
INFO = -3
ELSE IF( NRU.LT.0 ) THEN
INFO = -4
ELSE IF( NCC.LT.0 ) THEN
INFO = -5
ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
$ ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
INFO = -9
ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
INFO = -11
ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
$ ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZBDSQR', -INFO )
RETURN
END IF
IF( N.EQ.0 )
$ RETURN
IF( N.EQ.1 )
$ GO TO 160
*
* ROTATE is true if any singular vectors desired, false otherwise
*
ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
*
* If no singular vectors desired, use qd algorithm
*
IF( .NOT.ROTATE ) THEN
CALL DLASQ1( N, D, E, RWORK, INFO )
*
* If INFO equals 2, dqds didn't finish, try to finish
*
IF( INFO .NE. 2 ) RETURN
INFO = 0
END IF
*
NM1 = N - 1
NM12 = NM1 + NM1
NM13 = NM12 + NM1
IDIR = 0
*
* Get machine constants
*
EPS = DLAMCH( 'Epsilon' )
UNFL = DLAMCH( 'Safe minimum' )
*
* If matrix lower bidiagonal, rotate to be upper bidiagonal
* by applying Givens rotations on the left
*
IF( LOWER ) THEN
DO 10 I = 1, N - 1
CALL DLARTG( D( I ), E( I ), CS, SN, R )
D( I ) = R
E( I ) = SN*D( I+1 )
D( I+1 ) = CS*D( I+1 )
RWORK( I ) = CS
RWORK( NM1+I ) = SN
10 CONTINUE
*
* Update singular vectors if desired
*
IF( NRU.GT.0 )
$ CALL ZLASR( 'R', 'V', 'F', NRU, N, RWORK( 1 ), RWORK( N ),
$ U, LDU )
IF( NCC.GT.0 )
$ CALL ZLASR( 'L', 'V', 'F', N, NCC, RWORK( 1 ), RWORK( N ),
$ C, LDC )
END IF
*
* Compute singular values to relative accuracy TOL
* (By setting TOL to be negative, algorithm will compute
* singular values to absolute accuracy ABS(TOL)*norm(input matrix))
*
TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) )
TOL = TOLMUL*EPS
*
* Compute approximate maximum, minimum singular values
*
SMAX = ZERO
DO 20 I = 1, N
SMAX = MAX( SMAX, ABS( D( I ) ) )
20 CONTINUE
DO 30 I = 1, N - 1
SMAX = MAX( SMAX, ABS( E( I ) ) )
30 CONTINUE
SMINL = ZERO
IF( TOL.GE.ZERO ) THEN
*
* Relative accuracy desired
*
SMINOA = ABS( D( 1 ) )
IF( SMINOA.EQ.ZERO )
$ GO TO 50
MU = SMINOA
DO 40 I = 2, N
MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) )
SMINOA = MIN( SMINOA, MU )
IF( SMINOA.EQ.ZERO )
$ GO TO 50
40 CONTINUE
50 CONTINUE
SMINOA = SMINOA / SQRT( DBLE( N ) )
THRESH = MAX( TOL*SMINOA, MAXITR*N*N*UNFL )
ELSE
*
* Absolute accuracy desired
*
THRESH = MAX( ABS( TOL )*SMAX, MAXITR*N*N*UNFL )
END IF
*
* Prepare for main iteration loop for the singular values
* (MAXIT is the maximum number of passes through the inner
* loop permitted before nonconvergence signalled.)
*
MAXIT = MAXITR*N*N
ITER = 0
OLDLL = -1
OLDM = -1
*
* M points to last element of unconverged part of matrix
*
M = N
*
* Begin main iteration loop
*
60 CONTINUE
*
* Check for convergence or exceeding iteration count
*
IF( M.LE.1 )
$ GO TO 160
IF( ITER.GT.MAXIT )
$ GO TO 200
*
* Find diagonal block of matrix to work on
*
IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH )
$ D( M ) = ZERO
SMAX = ABS( D( M ) )
SMIN = SMAX
DO 70 LLL = 1, M - 1
LL = M - LLL
ABSS = ABS( D( LL ) )
ABSE = ABS( E( LL ) )
IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH )
$ D( LL ) = ZERO
IF( ABSE.LE.THRESH )
$ GO TO 80
SMIN = MIN( SMIN, ABSS )
SMAX = MAX( SMAX, ABSS, ABSE )
70 CONTINUE
LL = 0
GO TO 90
80 CONTINUE
E( LL ) = ZERO
*
* Matrix splits since E(LL) = 0
*
IF( LL.EQ.M-1 ) THEN
*
* Convergence of bottom singular value, return to top of loop
*
M = M - 1
GO TO 60
END IF
90 CONTINUE
LL = LL + 1
*
* E(LL) through E(M-1) are nonzero, E(LL-1) is zero
*
IF( LL.EQ.M-1 ) THEN
*
* 2 by 2 block, handle separately
*
CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR,
$ COSR, SINL, COSL )
D( M-1 ) = SIGMX
E( M-1 ) = ZERO
D( M ) = SIGMN
*
* Compute singular vectors, if desired
*
IF( NCVT.GT.0 )
$ CALL ZDROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT,
$ COSR, SINR )
IF( NRU.GT.0 )
$ CALL ZDROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL, SINL )
IF( NCC.GT.0 )
$ CALL ZDROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL,
$ SINL )
M = M - 2
GO TO 60
END IF
*
* If working on new submatrix, choose shift direction
* (from larger end diagonal element towards smaller)
*
IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN
IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN
*
* Chase bulge from top (big end) to bottom (small end)
*
IDIR = 1
ELSE
*
* Chase bulge from bottom (big end) to top (small end)
*
IDIR = 2
END IF
END IF
*
* Apply convergence tests
*
IF( IDIR.EQ.1 ) THEN
*
* Run convergence test in forward direction
* First apply standard test to bottom of matrix
*
IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR.
$ ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN
E( M-1 ) = ZERO
GO TO 60
END IF
*
IF( TOL.GE.ZERO ) THEN
*
* If relative accuracy desired,
* apply convergence criterion forward
*
MU = ABS( D( LL ) )
SMINL = MU
DO 100 LLL = LL, M - 1
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
E( LLL ) = ZERO
GO TO 60
END IF
MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
SMINL = MIN( SMINL, MU )
100 CONTINUE
END IF
*
ELSE
*
* Run convergence test in backward direction
* First apply standard test to top of matrix
*
IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR.
$ ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN
E( LL ) = ZERO
GO TO 60
END IF
*
IF( TOL.GE.ZERO ) THEN
*
* If relative accuracy desired,
* apply convergence criterion backward
*
MU = ABS( D( M ) )
SMINL = MU
DO 110 LLL = M - 1, LL, -1
IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
E( LLL ) = ZERO
GO TO 60
END IF
MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
SMINL = MIN( SMINL, MU )
110 CONTINUE
END IF
END IF
OLDLL = LL
OLDM = M
*
* Compute shift. First, test if shifting would ruin relative
* accuracy, and if so set the shift to zero.
*
IF( TOL.GE.ZERO .AND. N*TOL*( SMINL / SMAX ).LE.
$ MAX( EPS, HNDRTH*TOL ) ) THEN
*
* Use a zero shift to avoid loss of relative accuracy
*
SHIFT = ZERO
ELSE
*
* Compute the shift from 2-by-2 block at end of matrix
*
IF( IDIR.EQ.1 ) THEN
SLL = ABS( D( LL ) )
CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R )
ELSE
SLL = ABS( D( M ) )
CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R )
END IF
*
* Test if shift negligible, and if so set to zero
*
IF( SLL.GT.ZERO ) THEN
IF( ( SHIFT / SLL )**2.LT.EPS )
$ SHIFT = ZERO
END IF
END IF
*
* Increment iteration count
*
ITER = ITER + M - LL
*
* If SHIFT = 0, do simplified QR iteration
*
IF( SHIFT.EQ.ZERO ) THEN
IF( IDIR.EQ.1 ) THEN
*
* Chase bulge from top to bottom
* Save cosines and sines for later singular vector updates
*
CS = ONE
OLDCS = ONE
DO 120 I = LL, M - 1
CALL DLARTG( D( I )*CS, E( I ), CS, SN, R )
IF( I.GT.LL )
$ E( I-1 ) = OLDSN*R
CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN, D( I ) )
RWORK( I-LL+1 ) = CS
RWORK( I-LL+1+NM1 ) = SN
RWORK( I-LL+1+NM12 ) = OLDCS
RWORK( I-LL+1+NM13 ) = OLDSN
120 CONTINUE
H = D( M )*CS
D( M ) = H*OLDCS
E( M-1 ) = H*OLDSN
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCVT, RWORK( 1 ),
$ RWORK( N ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL ZLASR( 'R', 'V', 'F', NRU, M-LL+1, RWORK( NM12+1 ),
$ RWORK( NM13+1 ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCC, RWORK( NM12+1 ),
$ RWORK( NM13+1 ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( M-1 ) ).LE.THRESH )
$ E( M-1 ) = ZERO
*
ELSE
*
* Chase bulge from bottom to top
* Save cosines and sines for later singular vector updates
*
CS = ONE
OLDCS = ONE
DO 130 I = M, LL + 1, -1
CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R )
IF( I.LT.M )
$ E( I ) = OLDSN*R
CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN, D( I ) )
RWORK( I-LL ) = CS
RWORK( I-LL+NM1 ) = -SN
RWORK( I-LL+NM12 ) = OLDCS
RWORK( I-LL+NM13 ) = -OLDSN
130 CONTINUE
H = D( LL )*CS
D( LL ) = H*OLDCS
E( LL ) = H*OLDSN
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCVT, RWORK( NM12+1 ),
$ RWORK( NM13+1 ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL ZLASR( 'R', 'V', 'B', NRU, M-LL+1, RWORK( 1 ),
$ RWORK( N ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCC, RWORK( 1 ),
$ RWORK( N ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( LL ) ).LE.THRESH )
$ E( LL ) = ZERO
END IF
ELSE
*
* Use nonzero shift
*
IF( IDIR.EQ.1 ) THEN
*
* Chase bulge from top to bottom
* Save cosines and sines for later singular vector updates
*
F = ( ABS( D( LL ) )-SHIFT )*
$ ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) )
G = E( LL )
DO 140 I = LL, M - 1
CALL DLARTG( F, G, COSR, SINR, R )
IF( I.GT.LL )
$ E( I-1 ) = R
F = COSR*D( I ) + SINR*E( I )
E( I ) = COSR*E( I ) - SINR*D( I )
G = SINR*D( I+1 )
D( I+1 ) = COSR*D( I+1 )
CALL DLARTG( F, G, COSL, SINL, R )
D( I ) = R
F = COSL*E( I ) + SINL*D( I+1 )
D( I+1 ) = COSL*D( I+1 ) - SINL*E( I )
IF( I.LT.M-1 ) THEN
G = SINL*E( I+1 )
E( I+1 ) = COSL*E( I+1 )
END IF
RWORK( I-LL+1 ) = COSR
RWORK( I-LL+1+NM1 ) = SINR
RWORK( I-LL+1+NM12 ) = COSL
RWORK( I-LL+1+NM13 ) = SINL
140 CONTINUE
E( M-1 ) = F
*
* Update singular vectors
*
IF( NCVT.GT.0 )
$ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCVT, RWORK( 1 ),
$ RWORK( N ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL ZLASR( 'R', 'V', 'F', NRU, M-LL+1, RWORK( NM12+1 ),
$ RWORK( NM13+1 ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCC, RWORK( NM12+1 ),
$ RWORK( NM13+1 ), C( LL, 1 ), LDC )
*
* Test convergence
*
IF( ABS( E( M-1 ) ).LE.THRESH )
$ E( M-1 ) = ZERO
*
ELSE
*
* Chase bulge from bottom to top
* Save cosines and sines for later singular vector updates
*
F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT /
$ D( M ) )
G = E( M-1 )
DO 150 I = M, LL + 1, -1
CALL DLARTG( F, G, COSR, SINR, R )
IF( I.LT.M )
$ E( I ) = R
F = COSR*D( I ) + SINR*E( I-1 )
E( I-1 ) = COSR*E( I-1 ) - SINR*D( I )
G = SINR*D( I-1 )
D( I-1 ) = COSR*D( I-1 )
CALL DLARTG( F, G, COSL, SINL, R )
D( I ) = R
F = COSL*E( I-1 ) + SINL*D( I-1 )
D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 )
IF( I.GT.LL+1 ) THEN
G = SINL*E( I-2 )
E( I-2 ) = COSL*E( I-2 )
END IF
RWORK( I-LL ) = COSR
RWORK( I-LL+NM1 ) = -SINR
RWORK( I-LL+NM12 ) = COSL
RWORK( I-LL+NM13 ) = -SINL
150 CONTINUE
E( LL ) = F
*
* Test convergence
*
IF( ABS( E( LL ) ).LE.THRESH )
$ E( LL ) = ZERO
*
* Update singular vectors if desired
*
IF( NCVT.GT.0 )
$ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCVT, RWORK( NM12+1 ),
$ RWORK( NM13+1 ), VT( LL, 1 ), LDVT )
IF( NRU.GT.0 )
$ CALL ZLASR( 'R', 'V', 'B', NRU, M-LL+1, RWORK( 1 ),
$ RWORK( N ), U( 1, LL ), LDU )
IF( NCC.GT.0 )
$ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCC, RWORK( 1 ),
$ RWORK( N ), C( LL, 1 ), LDC )
END IF
END IF
*
* QR iteration finished, go back and check convergence
*
GO TO 60
*
* All singular values converged, so make them positive
*
160 CONTINUE
DO 170 I = 1, N
IF( D( I ).LT.ZERO ) THEN
D( I ) = -D( I )
*
* Change sign of singular vectors, if desired
*
IF( NCVT.GT.0 )
$ CALL ZDSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT )
END IF
170 CONTINUE
*
* Sort the singular values into decreasing order (insertion sort on
* singular values, but only one transposition per singular vector)
*
DO 190 I = 1, N - 1
*
* Scan for smallest D(I)
*
ISUB = 1
SMIN = D( 1 )
DO 180 J = 2, N + 1 - I
IF( D( J ).LE.SMIN ) THEN
ISUB = J
SMIN = D( J )
END IF
180 CONTINUE
IF( ISUB.NE.N+1-I ) THEN
*
* Swap singular values and vectors
*
D( ISUB ) = D( N+1-I )
D( N+1-I ) = SMIN
IF( NCVT.GT.0 )
$ CALL ZSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ),
$ LDVT )
IF( NRU.GT.0 )
$ CALL ZSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 )
IF( NCC.GT.0 )
$ CALL ZSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ), LDC )
END IF
190 CONTINUE
GO TO 220
*
* Maximum number of iterations exceeded, failure to converge
*
200 CONTINUE
INFO = 0
DO 210 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
210 CONTINUE
220 CONTINUE
RETURN
*
* End of ZBDSQR
*
END
*> \brief \b ZDROT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZDROT( N, ZX, INCX, ZY, INCY, C, S )
*
* .. Scalar Arguments ..
* INTEGER INCX, INCY, N
* DOUBLE PRECISION C, S
* ..
* .. Array Arguments ..
* COMPLEX*16 ZX( * ), ZY( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Applies a plane rotation, where the cos and sin (c and s) are real
*> and the vectors cx and cy are complex.
*> jack dongarra, linpack, 3/11/78.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of the vectors cx and cy.
*> N must be at least zero.
*> \endverbatim
*>
*> \param[in,out] ZX
*> \verbatim
*> ZX is COMPLEX*16 array, dimension at least
*> ( 1 + ( N - 1 )*abs( INCX ) ).
*> Before entry, the incremented array ZX must contain the n
*> element vector cx. On exit, ZX is overwritten by the updated
*> vector cx.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> On entry, INCX specifies the increment for the elements of
*> ZX. INCX must not be zero.
*> \endverbatim
*>
*> \param[in,out] ZY
*> \verbatim
*> ZY is COMPLEX*16 array, dimension at least
*> ( 1 + ( N - 1 )*abs( INCY ) ).
*> Before entry, the incremented array ZY must contain the n
*> element vector cy. On exit, ZY is overwritten by the updated
*> vector cy.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> On entry, INCY specifies the increment for the elements of
*> ZY. INCY must not be zero.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> On entry, C specifies the cosine, cos.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION
*> On entry, S specifies the sine, sin.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup rot
*
* =====================================================================
SUBROUTINE ZDROT( N, ZX, INCX, ZY, INCY, C, S )
*
* -- Reference BLAS level1 routine --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX, INCY, N
DOUBLE PRECISION C, S
* ..
* .. Array Arguments ..
COMPLEX*16 ZX( * ), ZY( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IX, IY
COMPLEX*16 CTEMP
* ..
* .. Executable Statements ..
*
IF( N.LE.0 )
$ RETURN
IF( INCX.EQ.1 .AND. INCY.EQ.1 ) THEN
*
* code for both increments equal to 1
*
DO I = 1, N
CTEMP = C*ZX( I ) + S*ZY( I )
ZY( I ) = C*ZY( I ) - S*ZX( I )
ZX( I ) = CTEMP
END DO
ELSE
*
* code for unequal increments or equal increments not equal
* to 1
*
IX = 1
IY = 1
IF( INCX.LT.0 )
$ IX = ( -N+1 )*INCX + 1
IF( INCY.LT.0 )
$ IY = ( -N+1 )*INCY + 1
DO I = 1, N
CTEMP = C*ZX( IX ) + S*ZY( IY )
ZY( IY ) = C*ZY( IY ) - S*ZX( IX )
ZX( IX ) = CTEMP
IX = IX + INCX
IY = IY + INCY
END DO
END IF
RETURN
*
* End of ZDROT
*
END
*> \brief \b ZDSCAL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZDSCAL(N,DA,ZX,INCX)
*
* .. Scalar Arguments ..
* DOUBLE PRECISION DA
* INTEGER INCX,N
* ..
* .. Array Arguments ..
* COMPLEX*16 ZX(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZDSCAL scales a vector by a constant.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> number of elements in input vector(s)
*> \endverbatim
*>
*> \param[in] DA
*> \verbatim
*> DA is DOUBLE PRECISION
*> On entry, DA specifies the scalar alpha.
*> \endverbatim
*>
*> \param[in,out] ZX
*> \verbatim
*> ZX is COMPLEX*16 array, dimension ( 1 + ( N - 1 )*abs( INCX ) )
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> storage spacing between elements of ZX
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup scal
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> jack dongarra, 3/11/78.
*> modified 3/93 to return if incx .le. 0.
*> modified 12/3/93, array(1) declarations changed to array(*)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZDSCAL(N,DA,ZX,INCX)
*
* -- Reference BLAS level1 routine --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION DA
INTEGER INCX,N
* ..
* .. Array Arguments ..
COMPLEX*16 ZX(*)
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I,NINCX
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER (ONE=1.0D+0)
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX, DIMAG
* ..
IF (N.LE.0 .OR. INCX.LE.0 .OR. DA.EQ.ONE) RETURN
IF (INCX.EQ.1) THEN
*
* code for increment equal to 1
*
DO I = 1,N
ZX(I) = DCMPLX(DA*DBLE(ZX(I)),DA*DIMAG(ZX(I)))
END DO
ELSE
*
* code for increment not equal to 1
*
NINCX = N*INCX
DO I = 1,NINCX,INCX
ZX(I) = DCMPLX(DA*DBLE(ZX(I)),DA*DIMAG(ZX(I)))
END DO
END IF
RETURN
*
* End of ZDSCAL
*
END
*> \brief \b ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLASET + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, M, N
* COMPLEX*16 ALPHA, BETA
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLASET initializes a 2-D array A to BETA on the diagonal and
*> ALPHA on the offdiagonals.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies the part of the matrix A to be set.
*> = 'U': Upper triangular part is set. The lower triangle
*> is unchanged.
*> = 'L': Lower triangular part is set. The upper triangle
*> is unchanged.
*> Otherwise: All of the matrix A is set.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> On entry, M specifies the number of rows of A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the number of columns of A.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*> ALPHA is COMPLEX*16
*> All the offdiagonal array elements are set to ALPHA.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is COMPLEX*16
*> All the diagonal array elements are set to BETA.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> On entry, the m by n matrix A.
*> On exit, A(i,j) = ALPHA, 1 <= i <= m, 1 <= j <= n, i.ne.j;
*> A(i,i) = BETA , 1 <= i <= min(m,n)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16OTHERauxiliary
*
* =====================================================================
SUBROUTINE ZLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, M, N
COMPLEX*16 ALPHA, BETA
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Set the diagonal to BETA and the strictly upper triangular
* part of the array to ALPHA.
*
DO 20 J = 2, N
DO 10 I = 1, MIN( J-1, M )
A( I, J ) = ALPHA
10 CONTINUE
20 CONTINUE
DO 30 I = 1, MIN( N, M )
A( I, I ) = BETA
30 CONTINUE
*
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
*
* Set the diagonal to BETA and the strictly lower triangular
* part of the array to ALPHA.
*
DO 50 J = 1, MIN( M, N )
DO 40 I = J + 1, M
A( I, J ) = ALPHA
40 CONTINUE
50 CONTINUE
DO 60 I = 1, MIN( N, M )
A( I, I ) = BETA
60 CONTINUE
*
ELSE
*
* Set the array to BETA on the diagonal and ALPHA on the
* offdiagonal.
*
DO 80 J = 1, N
DO 70 I = 1, M
A( I, J ) = ALPHA
70 CONTINUE
80 CONTINUE
DO 90 I = 1, MIN( M, N )
A( I, I ) = BETA
90 CONTINUE
END IF
*
RETURN
*
* End of ZLASET
*
END
*> \brief \b ZLASR applies a sequence of plane rotations to a general rectangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLASR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, PIVOT, SIDE
* INTEGER LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION C( * ), S( * )
* COMPLEX*16 A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLASR applies a sequence of real plane rotations to a complex matrix
*> A, from either the left or the right.
*>
*> When SIDE = 'L', the transformation takes the form
*>
*> A := P*A
*>
*> and when SIDE = 'R', the transformation takes the form
*>
*> A := A*P**T
*>
*> where P is an orthogonal matrix consisting of a sequence of z plane
*> rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
*> and P**T is the transpose of P.
*>
*> When DIRECT = 'F' (Forward sequence), then
*>
*> P = P(z-1) * ... * P(2) * P(1)
*>
*> and when DIRECT = 'B' (Backward sequence), then
*>
*> P = P(1) * P(2) * ... * P(z-1)
*>
*> where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
*>
*> R(k) = ( c(k) s(k) )
*> = ( -s(k) c(k) ).
*>
*> When PIVOT = 'V' (Variable pivot), the rotation is performed
*> for the plane (k,k+1), i.e., P(k) has the form
*>
*> P(k) = ( 1 )
*> ( ... )
*> ( 1 )
*> ( c(k) s(k) )
*> ( -s(k) c(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*>
*> where R(k) appears as a rank-2 modification to the identity matrix in
*> rows and columns k and k+1.
*>
*> When PIVOT = 'T' (Top pivot), the rotation is performed for the
*> plane (1,k+1), so P(k) has the form
*>
*> P(k) = ( c(k) s(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*> ( -s(k) c(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*>
*> where R(k) appears in rows and columns 1 and k+1.
*>
*> Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
*> performed for the plane (k,z), giving P(k) the form
*>
*> P(k) = ( 1 )
*> ( ... )
*> ( 1 )
*> ( c(k) s(k) )
*> ( 1 )
*> ( ... )
*> ( 1 )
*> ( -s(k) c(k) )
*>
*> where R(k) appears in rows and columns k and z. The rotations are
*> performed without ever forming P(k) explicitly.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> Specifies whether the plane rotation matrix P is applied to
*> A on the left or the right.
*> = 'L': Left, compute A := P*A
*> = 'R': Right, compute A:= A*P**T
*> \endverbatim
*>
*> \param[in] PIVOT
*> \verbatim
*> PIVOT is CHARACTER*1
*> Specifies the plane for which P(k) is a plane rotation
*> matrix.
*> = 'V': Variable pivot, the plane (k,k+1)
*> = 'T': Top pivot, the plane (1,k+1)
*> = 'B': Bottom pivot, the plane (k,z)
*> \endverbatim
*>
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies whether P is a forward or backward sequence of
*> plane rotations.
*> = 'F': Forward, P = P(z-1)*...*P(2)*P(1)
*> = 'B': Backward, P = P(1)*P(2)*...*P(z-1)
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. If m <= 1, an immediate
*> return is effected.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. If n <= 1, an
*> immediate return is effected.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> The cosines c(k) of the plane rotations.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension
*> (M-1) if SIDE = 'L'
*> (N-1) if SIDE = 'R'
*> The sines s(k) of the plane rotations. The 2-by-2 plane
*> rotation part of the matrix P(k), R(k), has the form
*> R(k) = ( c(k) s(k) )
*> ( -s(k) c(k) ).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> The M-by-N matrix A. On exit, A is overwritten by P*A if
*> SIDE = 'R' or by A*P**T if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16OTHERauxiliary
*
* =====================================================================
SUBROUTINE ZLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER DIRECT, PIVOT, SIDE
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION C( * ), S( * )
COMPLEX*16 A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J
DOUBLE PRECISION CTEMP, STEMP
COMPLEX*16 TEMP
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( .NOT.( LSAME( SIDE, 'L' ) .OR. LSAME( SIDE, 'R' ) ) ) THEN
INFO = 1
ELSE IF( .NOT.( LSAME( PIVOT, 'V' ) .OR. LSAME( PIVOT,
$ 'T' ) .OR. LSAME( PIVOT, 'B' ) ) ) THEN
INFO = 2
ELSE IF( .NOT.( LSAME( DIRECT, 'F' ) .OR. LSAME( DIRECT, 'B' ) ) )
$ THEN
INFO = 3
ELSE IF( M.LT.0 ) THEN
INFO = 4
ELSE IF( N.LT.0 ) THEN
INFO = 5
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = 9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZLASR ', INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
$ RETURN
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form P * A
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 20 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 10 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
10 CONTINUE
END IF
20 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 40 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 30 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
30 CONTINUE
END IF
40 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 60 J = 2, M
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 50 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
50 CONTINUE
END IF
60 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 80 J = M, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 70 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
70 CONTINUE
END IF
80 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 100 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 90 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
90 CONTINUE
END IF
100 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 120 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 110 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form A * P**T
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 140 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 130 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
130 CONTINUE
END IF
140 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 160 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 150 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
150 CONTINUE
END IF
160 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 180 J = 2, N
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 170 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
170 CONTINUE
END IF
180 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 200 J = N, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 190 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
190 CONTINUE
END IF
200 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 220 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 210 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
210 CONTINUE
END IF
220 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 240 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 230 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
230 CONTINUE
END IF
240 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of ZLASR
*
END
*> \brief \b ZPTEQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZPTEQR + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER COMPZ
* INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * )
* COMPLEX*16 Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric positive definite tridiagonal matrix by first factoring the
*> matrix using DPTTRF and then calling ZBDSQR to compute the singular
*> values of the bidiagonal factor.
*>
*> This routine computes the eigenvalues of the positive definite
*> tridiagonal matrix to high relative accuracy. This means that if the
*> eigenvalues range over many orders of magnitude in size, then the
*> small eigenvalues and corresponding eigenvectors will be computed
*> more accurately than, for example, with the standard QR method.
*>
*> The eigenvectors of a full or band positive definite Hermitian matrix
*> can also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to
*> reduce this matrix to tridiagonal form. (The reduction to
*> tridiagonal form, however, may preclude the possibility of obtaining
*> high relative accuracy in the small eigenvalues of the original
*> matrix, if these eigenvalues range over many orders of magnitude.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Compute eigenvalues only.
*> = 'V': Compute eigenvectors of original Hermitian
*> matrix also. Array Z contains the unitary matrix
*> used to reduce the original matrix to tridiagonal
*> form.
*> = 'I': Compute eigenvectors of tridiagonal matrix also.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix.
*> On normal exit, D contains the eigenvalues, in descending
*> order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is COMPLEX*16 array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', the unitary matrix used in the
*> reduction to tridiagonal form.
*> On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
*> original Hermitian matrix;
*> if COMPZ = 'I', the orthonormal eigenvectors of the
*> tridiagonal matrix.
*> If INFO > 0 on exit, Z contains the eigenvectors associated
*> with only the stored eigenvalues.
*> If COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> COMPZ = 'V' or 'I', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = i, and i is:
*> <= N the Cholesky factorization of the matrix could
*> not be performed because the i-th principal minor
*> was not positive definite.
*> > N the SVD algorithm failed to converge;
*> if INFO = N+i, i off-diagonal elements of the
*> bidiagonal factor did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16PTcomputational
*
* =====================================================================
SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * )
COMPLEX*16 Z( LDZ, * )
* ..
*
* ====================================================================
*
* .. Parameters ..
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPTTRF, XERBLA, ZBDSQR, ZLASET
* ..
* .. Local Arrays ..
COMPLEX*16 C( 1, 1 ), VT( 1, 1 )
* ..
* .. Local Scalars ..
INTEGER I, ICOMPZ, NRU
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
$ N ) ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZPTEQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ICOMPZ.GT.0 )
$ Z( 1, 1 ) = CONE
RETURN
END IF
IF( ICOMPZ.EQ.2 )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
*
* Call DPTTRF to factor the matrix.
*
CALL DPTTRF( N, D, E, INFO )
IF( INFO.NE.0 )
$ RETURN
DO 10 I = 1, N
D( I ) = SQRT( D( I ) )
10 CONTINUE
DO 20 I = 1, N - 1
E( I ) = E( I )*D( I )
20 CONTINUE
*
* Call ZBDSQR to compute the singular values/vectors of the
* bidiagonal factor.
*
IF( ICOMPZ.GT.0 ) THEN
NRU = N
ELSE
NRU = 0
END IF
CALL ZBDSQR( 'Lower', N, 0, NRU, 0, D, E, VT, 1, Z, LDZ, C, 1,
$ WORK, INFO )
*
* Square the singular values.
*
IF( INFO.EQ.0 ) THEN
DO 30 I = 1, N
D( I ) = D( I )*D( I )
30 CONTINUE
ELSE
INFO = N + INFO
END IF
*
RETURN
*
* End of ZPTEQR
*
END
*> \brief \b ZSWAP
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZSWAP(N,ZX,INCX,ZY,INCY)
*
* .. Scalar Arguments ..
* INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
* COMPLEX*16 ZX(*),ZY(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZSWAP interchanges two vectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> number of elements in input vector(s)
*> \endverbatim
*>
*> \param[in,out] ZX
*> \verbatim
*> ZX is COMPLEX*16 array, dimension ( 1 + ( N - 1 )*abs( INCX ) )
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> storage spacing between elements of ZX
*> \endverbatim
*>
*> \param[in,out] ZY
*> \verbatim
*> ZY is COMPLEX*16 array, dimension ( 1 + ( N - 1 )*abs( INCY ) )
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*> INCY is INTEGER
*> storage spacing between elements of ZY
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup swap
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> jack dongarra, 3/11/78.
*> modified 12/3/93, array(1) declarations changed to array(*)
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZSWAP(N,ZX,INCX,ZY,INCY)
*
* -- Reference BLAS level1 routine --
* -- Reference BLAS is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
COMPLEX*16 ZX(*),ZY(*)
* ..
*
* =====================================================================
*
* .. Local Scalars ..
COMPLEX*16 ZTEMP
INTEGER I,IX,IY
* ..
IF (N.LE.0) RETURN
IF (INCX.EQ.1 .AND. INCY.EQ.1) THEN
*
* code for both increments equal to 1
DO I = 1,N
ZTEMP = ZX(I)
ZX(I) = ZY(I)
ZY(I) = ZTEMP
END DO
ELSE
*
* code for unequal increments or equal increments not equal
* to 1
*
IX = 1
IY = 1
IF (INCX.LT.0) IX = (-N+1)*INCX + 1
IF (INCY.LT.0) IY = (-N+1)*INCY + 1
DO I = 1,N
ZTEMP = ZX(IX)
ZX(IX) = ZY(IY)
ZY(IY) = ZTEMP
IX = IX + INCX
IY = IY + INCY
END DO
END IF
RETURN
*
* End of ZSWAP
*
END
*> \brief \b DLAMCH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
* .. Scalar Arguments ..
* CHARACTER CMACH
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAMCH determines double precision machine parameters.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] CMACH
*> \verbatim
*> CMACH is CHARACTER*1
*> Specifies the value to be returned by DLAMCH:
*> = 'E' or 'e', DLAMCH := eps
*> = 'S' or 's , DLAMCH := sfmin
*> = 'B' or 'b', DLAMCH := base
*> = 'P' or 'p', DLAMCH := eps*base
*> = 'N' or 'n', DLAMCH := t
*> = 'R' or 'r', DLAMCH := rnd
*> = 'M' or 'm', DLAMCH := emin
*> = 'U' or 'u', DLAMCH := rmin
*> = 'L' or 'l', DLAMCH := emax
*> = 'O' or 'o', DLAMCH := rmax
*> where
*> eps = relative machine precision
*> sfmin = safe minimum, such that 1/sfmin does not overflow
*> base = base of the machine
*> prec = eps*base
*> t = number of (base) digits in the mantissa
*> rnd = 1.0 when rounding occurs in addition, 0.0 otherwise
*> emin = minimum exponent before (gradual) underflow
*> rmin = underflow threshold - base**(emin-1)
*> emax = largest exponent before overflow
*> rmax = overflow threshold - (base**emax)*(1-eps)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER CMACH
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION RND, EPS, SFMIN, SMALL, RMACH
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC DIGITS, EPSILON, HUGE, MAXEXPONENT,
$ MINEXPONENT, RADIX, TINY
* ..
* .. Executable Statements ..
*
*
* Assume rounding, not chopping. Always.
*
RND = ONE
*
IF( ONE.EQ.RND ) THEN
EPS = EPSILON(ZERO) * 0.5
ELSE
EPS = EPSILON(ZERO)
END IF
*
IF( LSAME( CMACH, 'E' ) ) THEN
RMACH = EPS
ELSE IF( LSAME( CMACH, 'S' ) ) THEN
SFMIN = TINY(ZERO)
SMALL = ONE / HUGE(ZERO)
IF( SMALL.GE.SFMIN ) THEN
*
* Use SMALL plus a bit, to avoid the possibility of rounding
* causing overflow when computing 1/sfmin.
*
SFMIN = SMALL*( ONE+EPS )
END IF
RMACH = SFMIN
ELSE IF( LSAME( CMACH, 'B' ) ) THEN
RMACH = RADIX(ZERO)
ELSE IF( LSAME( CMACH, 'P' ) ) THEN
RMACH = EPS * RADIX(ZERO)
ELSE IF( LSAME( CMACH, 'N' ) ) THEN
RMACH = DIGITS(ZERO)
ELSE IF( LSAME( CMACH, 'R' ) ) THEN
RMACH = RND
ELSE IF( LSAME( CMACH, 'M' ) ) THEN
RMACH = MINEXPONENT(ZERO)
ELSE IF( LSAME( CMACH, 'U' ) ) THEN
RMACH = tiny(zero)
ELSE IF( LSAME( CMACH, 'L' ) ) THEN
RMACH = MAXEXPONENT(ZERO)
ELSE IF( LSAME( CMACH, 'O' ) ) THEN
RMACH = HUGE(ZERO)
ELSE
RMACH = ZERO
END IF
*
DLAMCH = RMACH
RETURN
*
* End of DLAMCH
*
END
************************************************************************
*> \brief \b DLAMC3
*> \details
*> \b Purpose:
*> \verbatim
*> DLAMC3 is intended to force A and B to be stored prior to doing
*> the addition of A and B , for use in situations where optimizers
*> might hold one of these in a register.
*> \endverbatim
*> \author LAPACK is a software package provided by Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
*> \date December 2016
*> \ingroup auxOTHERauxiliary
*>
*> \param[in] A
*> \verbatim
*> A is a DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is a DOUBLE PRECISION
*> The values A and B.
*> \endverbatim
*>
DOUBLE PRECISION FUNCTION DLAMC3( A, B )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2010
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B
* ..
* =====================================================================
*
* .. Executable Statements ..
*
DLAMC3 = A + B
*
RETURN
*
* End of DLAMC3
*
END
*
************************************************************************