# gesvdx - Man Page

gesvdx: SVD, bisection

## Synopsis

### Functions

subroutine **cgesvdx** (jobu, jobvt, range, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)

**CGESVDX computes the singular value decomposition (SVD) for GE matrices**

subroutine **dgesvdx** (jobu, jobvt, range, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, lwork, iwork, info)

**DGESVDX computes the singular value decomposition (SVD) for GE matrices**

subroutine **sgesvdx** (jobu, jobvt, range, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, lwork, iwork, info)

**SGESVDX computes the singular value decomposition (SVD) for GE matrices**

subroutine **zgesvdx** (jobu, jobvt, range, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)

**ZGESVDX computes the singular value decomposition (SVD) for GE matrices**

## Detailed Description

## Function Documentation

### subroutine cgesvdx (character jobu, character jobvt, character range, integer m, integer n, complex, dimension( lda, * ) a, integer lda, real vl, real vu, integer il, integer iu, integer ns, real, dimension( * ) s, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldvt, * ) vt, integer ldvt, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

**CGESVDX computes the singular value decomposition (SVD) for GE matrices**

**Purpose:**

CGESVDX computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V) where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. CGESVDX uses an eigenvalue problem for obtaining the SVD, which allows for the computation of a subset of singular values and vectors. See SBDSVDX for details. Note that the routine returns V**T, not V.

**Parameters***JOBU*JOBU is CHARACTER*1 Specifies options for computing all or part of the matrix U: = 'V': the first min(m,n) columns of U (the left singular vectors) or as specified by RANGE are returned in the array U; = 'N': no columns of U (no left singular vectors) are computed.

*JOBVT*JOBVT is CHARACTER*1 Specifies options for computing all or part of the matrix V**T: = 'V': the first min(m,n) rows of V**T (the right singular vectors) or as specified by RANGE are returned in the array VT; = 'N': no rows of V**T (no right singular vectors) are computed.

*RANGE*RANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th singular values will be found.

*M*M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*N is INTEGER The number of columns of the input matrix A. N >= 0.

*A*A is COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the contents of A are destroyed.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*VL*VL is REAL If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*VU is REAL If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL*IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*NS*NS is INTEGER The total number of singular values found, 0 <= NS <= min(M,N). If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

*S*S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*U*U is COMPLEX array, dimension (LDU,UCOL) If JOBU = 'V', U contains columns of U (the left singular vectors, stored columnwise) as specified by RANGE; if JOBU = 'N', U is not referenced. Note: The user must ensure that UCOL >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'V', LDU >= M.

*VT*VT is COMPLEX array, dimension (LDVT,N) If JOBVT = 'V', VT contains the rows of V**T (the right singular vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N', VT is not referenced. Note: The user must ensure that LDVT >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDVT*LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'V', LDVT >= NS (see above).

*WORK*WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

*LWORK*LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see comments inside the code): - PATH 1 (M much larger than N) - PATH 1t (N much larger than M) LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths. For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*RWORK*RWORK is REAL array, dimension (MAX(1,LRWORK)) LRWORK >= MIN(M,N)*(MIN(M,N)*2+15*MIN(M,N)).

*IWORK*IWORK is INTEGER array, dimension (12*MIN(M,N)) If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in SBDSVDX/SSTEVX.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in SBDSVDX/SSTEVX. if INFO = N*2 + 1, an internal error occurred in SBDSVDX

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **267** of file **cgesvdx.f**.

### subroutine dgesvdx (character jobu, character jobvt, character range, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision vl, double precision vu, integer il, integer iu, integer ns, double precision, dimension( * ) s, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

**DGESVDX computes the singular value decomposition (SVD) for GE matrices**

**Purpose:**

DGESVDX computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V) where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. DGESVDX uses an eigenvalue problem for obtaining the SVD, which allows for the computation of a subset of singular values and vectors. See DBDSVDX for details. Note that the routine returns V**T, not V.

**Parameters***JOBU*JOBU is CHARACTER*1 Specifies options for computing all or part of the matrix U: = 'V': the first min(m,n) columns of U (the left singular vectors) or as specified by RANGE are returned in the array U; = 'N': no columns of U (no left singular vectors) are computed.

*JOBVT*JOBVT is CHARACTER*1 Specifies options for computing all or part of the matrix V**T: = 'V': the first min(m,n) rows of V**T (the right singular vectors) or as specified by RANGE are returned in the array VT; = 'N': no rows of V**T (no right singular vectors) are computed.

*RANGE*RANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th singular values will be found.

*M*M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*N is INTEGER The number of columns of the input matrix A. N >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the contents of A are destroyed.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*VL*VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL*IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*NS*NS is INTEGER The total number of singular values found, 0 <= NS <= min(M,N). If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

*S*S is DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*U*U is DOUBLE PRECISION array, dimension (LDU,UCOL) If JOBU = 'V', U contains columns of U (the left singular vectors, stored columnwise) as specified by RANGE; if JOBU = 'N', U is not referenced. Note: The user must ensure that UCOL >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'V', LDU >= M.

*VT*VT is DOUBLE PRECISION array, dimension (LDVT,N) If JOBVT = 'V', VT contains the rows of V**T (the right singular vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N', VT is not referenced. Note: The user must ensure that LDVT >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDVT*LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'V', LDVT >= NS (see above).

*WORK*WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

*LWORK*LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see comments inside the code): - PATH 1 (M much larger than N) - PATH 1t (N much larger than M) LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths. For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*IWORK*IWORK is INTEGER array, dimension (12*MIN(M,N)) If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DBDSVDX/DSTEVX.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in DBDSVDX/DSTEVX. if INFO = N*2 + 1, an internal error occurred in DBDSVDX

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **260** of file **dgesvdx.f**.

### subroutine sgesvdx (character jobu, character jobvt, character range, integer m, integer n, real, dimension( lda, * ) a, integer lda, real vl, real vu, integer il, integer iu, integer ns, real, dimension( * ) s, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

**SGESVDX computes the singular value decomposition (SVD) for GE matrices**

**Purpose:**

SGESVDX computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V) where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. SGESVDX uses an eigenvalue problem for obtaining the SVD, which allows for the computation of a subset of singular values and vectors. See SBDSVDX for details. Note that the routine returns V**T, not V.

**Parameters***JOBU*JOBU is CHARACTER*1 Specifies options for computing all or part of the matrix U: = 'V': the first min(m,n) columns of U (the left singular vectors) or as specified by RANGE are returned in the array U; = 'N': no columns of U (no left singular vectors) are computed.

*JOBVT*JOBVT is CHARACTER*1 Specifies options for computing all or part of the matrix V**T: = 'V': the first min(m,n) rows of V**T (the right singular vectors) or as specified by RANGE are returned in the array VT; = 'N': no rows of V**T (no right singular vectors) are computed.

*RANGE*RANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th singular values will be found.

*M*M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*N is INTEGER The number of columns of the input matrix A. N >= 0.

*A*A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the contents of A are destroyed.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*VL*VL is REAL If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*VU is REAL If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL*IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*NS*NS is INTEGER The total number of singular values found, 0 <= NS <= min(M,N). If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

*S*S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*U*U is REAL array, dimension (LDU,UCOL) If JOBU = 'V', U contains columns of U (the left singular vectors, stored columnwise) as specified by RANGE; if JOBU = 'N', U is not referenced. Note: The user must ensure that UCOL >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'V', LDU >= M.

*VT*VT is REAL array, dimension (LDVT,N) If JOBVT = 'V', VT contains the rows of V**T (the right singular vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N', VT is not referenced. Note: The user must ensure that LDVT >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDVT*LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'V', LDVT >= NS (see above).

*WORK*WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

*LWORK*LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see comments inside the code): - PATH 1 (M much larger than N) - PATH 1t (N much larger than M) LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths. For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*IWORK*IWORK is INTEGER array, dimension (12*MIN(M,N)) If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in SBDSVDX/SSTEVX.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in SBDSVDX/SSTEVX. if INFO = N*2 + 1, an internal error occurred in SBDSVDX

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **260** of file **sgesvdx.f**.

### subroutine zgesvdx (character jobu, character jobvt, character range, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision vl, double precision vu, integer il, integer iu, integer ns, double precision, dimension( * ) s, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldvt, * ) vt, integer ldvt, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

**ZGESVDX computes the singular value decomposition (SVD) for GE matrices**

**Purpose:**

ZGESVDX computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V) where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A. ZGESVDX uses an eigenvalue problem for obtaining the SVD, which allows for the computation of a subset of singular values and vectors. See DBDSVDX for details. Note that the routine returns V**T, not V.

**Parameters***JOBU**JOBVT**RANGE**M*M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*N is INTEGER The number of columns of the input matrix A. N >= 0.

*A*A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the contents of A are destroyed.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*VL*VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL**IU**NS**S*S is DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*U*U is COMPLEX*16 array, dimension (LDU,UCOL) If JOBU = 'V', U contains columns of U (the left singular vectors, stored columnwise) as specified by RANGE; if JOBU = 'N', U is not referenced. Note: The user must ensure that UCOL >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'V', LDU >= M.

*VT*VT is COMPLEX*16 array, dimension (LDVT,N) If JOBVT = 'V', VT contains the rows of V**T (the right singular vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N', VT is not referenced. Note: The user must ensure that LDVT >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDVT**WORK*WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

*LWORK**RWORK*RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) LRWORK >= MIN(M,N)*(MIN(M,N)*2+15*MIN(M,N)).

*IWORK*IWORK is INTEGER array, dimension (12*MIN(M,N)) If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DBDSVDX/DSTEVX.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in DBDSVDX/DSTEVX. if INFO = N*2 + 1, an internal error occurred in DBDSVDX

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **267** of file **zgesvdx.f**.

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