Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?bdsqr

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

Syntax

lapack_int LAPACKE_sbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt, lapack_int nru, lapack_int ncc, float* d, float* e, float* vt, lapack_int ldvt, float* u, lapack_int ldu, float* c, lapack_int ldc );

lapack_int LAPACKE_dbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt, lapack_int nru, lapack_int ncc, double* d, double* e, double* vt, lapack_int ldvt, double* u, lapack_int ldu, double* c, lapack_int ldc );

lapack_int LAPACKE_cbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt, lapack_int nru, lapack_int ncc, float* d, float* e, lapack_complex_float* vt, lapack_int ldvt, lapack_complex_float* u, lapack_int ldu, lapack_complex_float* c, lapack_int ldc );

lapack_int LAPACKE_zbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt, lapack_int nru, lapack_int ncc, double* d, double* e, lapack_complex_double* vt, lapack_int ldvt, lapack_complex_double* u, lapack_int ldu, lapack_complex_double* c, lapack_int ldc );

Include Files

  • mkl.h

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*PH 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 PH *VT instead of PH, 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*(PH*VT)

is the SVD of A. Optionally, the subroutine may also compute QH *C for a given real/complex input matrix C.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

uplo

Must be 'U' or 'L'.

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

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

n

The order of the matrix B (n 0).

ncvt

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

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

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

d, e

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).

vt, u, c

Arrays:

vt, size max(1, ldvt*ncvt) for column major layout and max(1, ldvt*n) for row major layout, contains an n-by-ncvt matrix VT.

vt is not referenced if ncvt = 0.

u, size max(1, ldu*n) for column major layout and max(1, ldu*nru) for row major layout, contains an nru by n matrix U.

u is not referenced if nru = 0.

c, size max(1, ldc*ncc) for column major layout and max(1, ldc*n) for row major layout, contains the n-by-ncc matrix C for computing the product QH*C.

ldvt

The leading dimension of vt. Constraints:

ldvt max(1, n) if ncvt > 0 for column major layout and ldvt max(1, ncvt) for row major layout;

ldvt 1 if ncvt = 0.

ldu

The leading dimension of u. Constraint:

ldu max(1, nru) for column major layout and ldu max(1, n) for row major layout .

ldc

The leading dimension of c. Constraints:

ldc max(1, n) if ncc > 0 for column major layout and ldc max(1, ncc) for row major layout; 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 QH*C.

vt

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

u

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

Return Values

This function returns a value info.

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.

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

|si - σi| p*(m,n)*ε*σi

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

θ(ui, wi) p(m,n)*ε / min ij(|σi - σj|/|σi + σj|).

Here minij(|σi - σj|/|σi + σj|) is the relative gap between σi 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.