?bdsqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?bdsqr

Computes the singular value decomposition of a general matrix that has been reduced to bidiagonal form.

Syntax

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

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

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

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

call rbdsqr(d, e [,vt] [,u] [,c] [,uplo] [,info])

call bdsqr(d, e [,vt] [,u] [,c] [,uplo] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine 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 real/complex input matrices U and VT. When U and VT are the orthogonal/unitary matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by ?gebrd, 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 real/complex input matrix C.

See also lasq1, lasq2, lasq3, lasq4, lasq5, lasq6 used by this routine.

Input Parameters

uplo

CHARACTER*1. Must be 'U' or 'L'.

If uplo = 'U', B is an upper bidiagonal matrix.

If uplo = 'L', B is a lower bidiagonal matrix.

n

INTEGER. The order of the matrix B (n≥ 0).

ncvt

INTEGER. The number of columns of the matrix VT, that is, the number of right singular vectors (ncvt≥ 0).

Set ncvt = 0 if no right singular vectors are required.

nru

INTEGER. The number of rows in U, that is, the number of left singular vectors (nru≥ 0).

Set nru = 0 if no left singular vectors are required.

ncc

INTEGER. The number of columns in the matrix C used for computing the product Q^H*C (ncc≥ 0). Set ncc = 0 if no matrix C is supplied.

d, e

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

Arrays:

d(*) contains the diagonal elements of B.

The size of d must be at least max(1, n).

e(*) contains the (n-1) off-diagonal elements of B.

The size of e must be at least max(1, n - 1).

work

REAL for sbdsqr

DOUBLE PRECISION for dbdsqr.

work(*) is a workspace array.

The size of work must be at least max(1, 4*n).

rwork

REAL for cbdsqr

DOUBLE PRECISION for zbdsqr.

rwork(*) is a workspace array.

The size of rwork must be at least max(1, 4*n).

vt, u, c

REAL for sbdsqr

DOUBLE PRECISION for dbdsqr

COMPLEX for cbdsqr

DOUBLE COMPLEX for zbdsqr.

Arrays:

vt(ldvt,*) contains an n-by-ncvt matrix VT.

The second dimension of vt must be at least max(1, ncvt).

vt is not referenced if ncvt = 0.

u(ldu,*) contains an nru by n matrix U.

The second dimension of u must be at least max(1, n).

u is not referenced if nru = 0.

c(ldc,*) contains the n-by-ncc matrix C for computing the product Q^H*C.

The second dimension of c must be at least max(1, ncc). The array is not referenced if ncc = 0.

ldvt

INTEGER. The leading dimension of vt. Constraints:

ldvt≥ max(1, n) if ncvt > 0;

ldvt≥ 1 if ncvt = 0.

ldu

INTEGER. The leading dimension of u. Constraint:

ldu≥ max(1, nru) .

ldc

INTEGER. The leading dimension of c. Constraints:

ldc≥ max(1, n) if ncc > 0; ldc≥ 1 otherwise.

Output Parameters

d

On exit, if info = 0, overwritten by the singular values in decreasing order (see info).

e

On exit, if info = 0, e is destroyed. See also info below.

c

Overwritten by the product Q^H*C.

vt

On exit, this array is overwritten by P^H *VT. Not referenced if ncvt = 0.

u

On exit, this array is overwritten by U *Q. Not referenced if nru = 0.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info > 0,

If ncvt = nru = ncc = 0,

info = 1, a split was marked by a positive value in e
info = 2, the current block of z not diagonalized after 100*n iterations (in the inner while loop)
info = 3, termination criterion of the outer while loop is not met (the program created more than n unreduced blocks).

In all other cases when ncvt, nru, or ncc > 0, the algorithm did not converge; d and e contain the elements of a bidiagonal matrix that is orthogonally similar to the input matrix B; if info = i, i elements of e have not converged to zero.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine bdsqr interface are the following:

d: Holds the vector of length (n).
e: Holds the vector of length (n).
vt: Holds the matrix VT of size (n, ncvt).
u: Holds the matrix U of size (nru,n).
c: Holds the matrix C of size (n,ncc).
uplo: Must be 'U' or 'L'. The default value is 'U'.
ncvt: If argument vt is present, then ncvt is equal to the number of columns in matrix VT; otherwise, ncvt is set to zero.
nru: If argument u is present, then nru is equal to the number of rows in matrix U; otherwise, nru is set to zero.
ncc: If argument c is present, then ncc is equal to the number of columns in matrix C; otherwise, ncc is set to zero.

Note that two variants of Fortran 95 interface for bdsqr routine are needed because of an ambiguous choice between real and complex cases appear when vt, u, and c are omitted. Thus, the name rbdsqr is used in real cases (single or double precision), and the name bdsqr is used in complex cases (single or double precision).

Application Notes

Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the routine) may decrease the relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.

If s_i is an exact singular value of B, and s_i is the corresponding computed value, then

|s_i - σ_i| ≤p*(m,n)*ε*σ_i

where p(m, n) is a modestly increasing function of m and n, and ε is the machine precision.

If only singular values are computed, they are computed more accurately than when some singular vectors are also computed (that is, the function p(m, n) is smaller).

If u_i is the corresponding exact left singular vector of B, and w_i is the corresponding computed left singular vector, then the angle θ(u_i, w_i) between them is bounded as follows:

θ(u_i, w_i) ≤p(m,n)*ε / min _i≠j(|σi - σj|/|σ_i + σj|).

Here min_i≠j(|σ_i - σ_j|/|σ_i + σ_j|) is the relative gap between σ_i and the other singular values. A similar error bound holds for the right singular vectors.

The total number of real floating-point operations is roughly proportional to n² if only the singular values are computed. About 6n²*nru additional operations (12n²*nru for complex flavors) are required to compute the left singular vectors and about 6n²*ncvt operations (12n²*ncvt for complex flavors) to compute the right singular vectors.

Parent topic: Singular Value Decomposition: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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