.TH "bdsqr" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
bdsqr \- bdsqr: bidiagonal SVD, QR iteration (dqds)
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcbdsqr\fP (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info)"
.br
.RI "\fBCBDSQR\fP "
.ti -1c
.RI "subroutine \fBdbdsqr\fP (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info)"
.br
.RI "\fBDBDSQR\fP "
.ti -1c
.RI "subroutine \fBsbdsqr\fP (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info)"
.br
.RI "\fBSBDSQR\fP "
.ti -1c
.RI "subroutine \fBzbdsqr\fP (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info)"
.br
.RI "\fBZBDSQR\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldvt, * ) vt, integer ldvt, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldc, * ) c, integer ldc, real, dimension( * ) rwork, integer info)"

.PP
\fBCBDSQR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CBDSQR 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 CGEBRD, 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix B\&.  N >= 0\&.
.fi
.PP
.br
\fINCVT\fP 
.PP
.nf
          NCVT is INTEGER
          The number of columns of the matrix VT\&. NCVT >= 0\&.
.fi
.PP
.br
\fINRU\fP 
.PP
.nf
          NRU is INTEGER
          The number of rows of the matrix U\&. NRU >= 0\&.
.fi
.PP
.br
\fINCC\fP 
.PP
.nf
          NCC is INTEGER
          The number of columns of the matrix C\&. NCC >= 0\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL 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\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is REAL 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\&.
.fi
.PP
.br
\fIVT\fP 
.PP
.nf
          VT is COMPLEX 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\&.
.fi
.PP
.br
\fILDVT\fP 
.PP
.nf
          LDVT is INTEGER
          The leading dimension of the array VT\&.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0\&.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is COMPLEX 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\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&.  LDU >= max(1,NRU)\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is COMPLEX 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\&.
.fi
.PP
.br
\fILDC\fP 
.PP
.nf
          LDC is INTEGER
          The leading dimension of the array C\&.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (4*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  TOLMUL  REAL, 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\&.
.fi
.PP
 
.RE
.PP
\fBNote:\fP
.RS 4

.PP
.nf
  Bug report from Cezary Dendek\&.
  On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919)\&. We instead use MAXITDIVN = MAXITR*N\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB231\fP of file \fBcbdsqr\&.f\fP\&.
.SS "subroutine dbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)"

.PP
\fBDBDSQR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DBDSQR 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**T

 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**T*VT instead of
 P**T, for given real input matrices U and VT\&.  When U and VT are the
 orthogonal matrices that reduce a general matrix A to bidiagonal
 form:  A = U*B*VT, as computed by DGEBRD, then

    A = (U*Q) * S * (P**T*VT)

 is the SVD of A\&.  Optionally, the subroutine may also compute Q**T*C
 for a given real 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix B\&.  N >= 0\&.
.fi
.PP
.br
\fINCVT\fP 
.PP
.nf
          NCVT is INTEGER
          The number of columns of the matrix VT\&. NCVT >= 0\&.
.fi
.PP
.br
\fINRU\fP 
.PP
.nf
          NRU is INTEGER
          The number of rows of the matrix U\&. NRU >= 0\&.
.fi
.PP
.br
\fINCC\fP 
.PP
.nf
          NCC is INTEGER
          The number of columns of the matrix C\&. NCC >= 0\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIVT\fP 
.PP
.nf
          VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT\&.
          On exit, VT is overwritten by P**T * VT\&.
          Not referenced if NCVT = 0\&.
.fi
.PP
.br
\fILDVT\fP 
.PP
.nf
          LDVT is INTEGER
          The leading dimension of the array VT\&.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0\&.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is DOUBLE PRECISION 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\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&.  LDU >= max(1,NRU)\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is DOUBLE PRECISION array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C\&.
          On exit, C is overwritten by Q**T * C\&.
          Not referenced if NCC = 0\&.
.fi
.PP
.br
\fILDC\fP 
.PP
.nf
          LDC is INTEGER
          The leading dimension of the array C\&.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (4*(N-1))
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:
             if NCVT = NRU = NCC = 0,
                = 1, a split was marked by a positive value in E
                = 2, current block of Z not diagonalized after 30*N
                     iterations (in inner while loop)
                = 3, termination criterion of outer while loop not met
                     (program created more than N unreduced blocks)
             else NCVT = NRU = NCC = 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\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  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\&.
.fi
.PP
 
.RE
.PP
\fBNote:\fP
.RS 4

.PP
.nf
  Bug report from Cezary Dendek\&.
  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919)\&. We instead use MAXITDIVN = MAXITR*N\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB239\fP of file \fBdbdsqr\&.f\fP\&.
.SS "subroutine sbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)"

.PP
\fBSBDSQR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SBDSQR 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**T

 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**T*VT instead of
 P**T, for given real input matrices U and VT\&.  When U and VT are the
 orthogonal matrices that reduce a general matrix A to bidiagonal
 form:  A = U*B*VT, as computed by SGEBRD, then

    A = (U*Q) * S * (P**T*VT)

 is the SVD of A\&.  Optionally, the subroutine may also compute Q**T*C
 for a given real 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix B\&.  N >= 0\&.
.fi
.PP
.br
\fINCVT\fP 
.PP
.nf
          NCVT is INTEGER
          The number of columns of the matrix VT\&. NCVT >= 0\&.
.fi
.PP
.br
\fINRU\fP 
.PP
.nf
          NRU is INTEGER
          The number of rows of the matrix U\&. NRU >= 0\&.
.fi
.PP
.br
\fINCC\fP 
.PP
.nf
          NCC is INTEGER
          The number of columns of the matrix C\&. NCC >= 0\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL 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\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is REAL 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\&.
.fi
.PP
.br
\fIVT\fP 
.PP
.nf
          VT is REAL array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT\&.
          On exit, VT is overwritten by P**T * VT\&.
          Not referenced if NCVT = 0\&.
.fi
.PP
.br
\fILDVT\fP 
.PP
.nf
          LDVT is INTEGER
          The leading dimension of the array VT\&.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0\&.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is REAL 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\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&.  LDU >= max(1,NRU)\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is REAL array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C\&.
          On exit, C is overwritten by Q**T * C\&.
          Not referenced if NCC = 0\&.
.fi
.PP
.br
\fILDC\fP 
.PP
.nf
          LDC is INTEGER
          The leading dimension of the array C\&.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (4*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:
             if NCVT = NRU = NCC = 0,
                = 1, a split was marked by a positive value in E
                = 2, current block of Z not diagonalized after 30*N
                     iterations (in inner while loop)
                = 3, termination criterion of outer while loop not met
                     (program created more than N unreduced blocks)
             else NCVT = NRU = NCC = 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\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  TOLMUL  REAL, 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\&.
.fi
.PP
 
.RE
.PP
\fBNote:\fP
.RS 4

.PP
.nf
  Bug report from Cezary Dendek\&.
  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919)\&. We instead use MAXITDIVN = MAXITR*N\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB238\fP of file \fBsbdsqr\&.f\fP\&.
.SS "subroutine zbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldvt, * ) vt, integer ldvt, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) rwork, integer info)"

.PP
\fBZBDSQR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix B\&.  N >= 0\&.
.fi
.PP
.br
\fINCVT\fP 
.PP
.nf
          NCVT is INTEGER
          The number of columns of the matrix VT\&. NCVT >= 0\&.
.fi
.PP
.br
\fINRU\fP 
.PP
.nf
          NRU is INTEGER
          The number of rows of the matrix U\&. NRU >= 0\&.
.fi
.PP
.br
\fINCC\fP 
.PP
.nf
          NCC is INTEGER
          The number of columns of the matrix C\&. NCC >= 0\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIVT\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDVT\fP 
.PP
.nf
          LDVT is INTEGER
          The leading dimension of the array VT\&.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0\&.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&.  LDU >= max(1,NRU)\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDC\fP 
.PP
.nf
          LDC is INTEGER
          The leading dimension of the array C\&.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (4*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  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\&.
.fi
.PP
 
.RE
.PP
\fBNote:\fP
.RS 4

.PP
.nf
  Bug report from Cezary Dendek\&.
  On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919)\&. We instead use MAXITDIVN = MAXITR*N\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB231\fP of file \fBzbdsqr\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.