# bdsqr - Man Page

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

## 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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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 Tennessee

Univ. of California Berkeley

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

```          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*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 Tennessee

Univ. of California Berkeley