bdsqr(3) | Library Functions Manual | bdsqr(3) |

# NAME

bdsqr - bdsqr: bidiagonal SVD, QR iteration (dqds)

# SYNOPSIS

## Functions

subroutine **cbdsqr** (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt,
u, ldu, c, ldc, rwork, info)

**CBDSQR** subroutine **dbdsqr** (uplo, n, ncvt, nru, ncc, d, e, vt,
ldvt, u, ldu, c, ldc, work, info)

**DBDSQR** subroutine **sbdsqr** (uplo, n, ncvt, nru, ncc, d, e, vt,
ldvt, u, ldu, c, ldc, work, info)

**SBDSQR** subroutine **zbdsqr** (uplo, n, ncvt, nru, ncc, d, e, vt,
ldvt, u, ldu, c, ldc, rwork, info)

**ZBDSQR**

# Detailed Description

# Function Documentation

## 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)

**CBDSQR**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.

*N*

N is INTEGER The order of the matrix B. N >= 0.

*NCVT*

NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0.

*NRU*

NRU is INTEGER The number of rows of the matrix U. NRU >= 0.

*NCC*

NCC is INTEGER The number of columns of the matrix C. NCC >= 0.

*D*

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.

*E*

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.

*VT*

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.

*LDVT*

LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

*U*

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.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU).

*C*

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.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

*RWORK*

RWORK is REAL array, dimension (4*N)

*INFO*

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.

**Internal Parameters:**

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.

**Note:**

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **231** of file **cbdsqr.f**.

## 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)

**DBDSQR**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.

*N*

N is INTEGER The order of the matrix B. N >= 0.

*NCVT*

NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0.

*NRU*

NRU is INTEGER The number of rows of the matrix U. NRU >= 0.

*NCC*

NCC is INTEGER The number of columns of the matrix C. NCC >= 0.

*D*

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.

*E*

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.

*VT*

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.

*LDVT*

LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

*U*

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.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU).

*C*

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.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

*WORK*

WORK is DOUBLE PRECISION array, dimension (4*(N-1))

*INFO*

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.

**Internal Parameters:**

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.

**Note:**

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **239** of file **dbdsqr.f**.

## 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)

**SBDSQR**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.

*N*

N is INTEGER The order of the matrix B. N >= 0.

*NCVT*

NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0.

*NRU*

NRU is INTEGER The number of rows of the matrix U. NRU >= 0.

*NCC*

NCC is INTEGER The number of columns of the matrix C. NCC >= 0.

*D*

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.

*E*

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.

*VT*

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.

*LDVT*

LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

*U*

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.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU).

*C*

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.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

*WORK*

WORK is REAL array, dimension (4*N)

*INFO*

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.

**Internal Parameters:**

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.

**Note:**

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **238** of file **sbdsqr.f**.

## 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)

**ZBDSQR**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.

*N*

N is INTEGER The order of the matrix B. N >= 0.

*NCVT*

NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0.

*NRU*

NRU is INTEGER The number of rows of the matrix U. NRU >= 0.

*NCC*

NCC is INTEGER The number of columns of the matrix C. NCC >= 0.

*D*

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.

*E*

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.

*VT*

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.

*LDVT*

*U*

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.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU).

*C*

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.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (4*N)

*INFO*

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.

**Internal Parameters:**

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.

**Note:**

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **231** of file **zbdsqr.f**.

# Author

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