?gelsd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?gelsd

Computes the minimum-norm solution to a linear least squares problem using the singular value decomposition of A and a divide and conquer method.

Syntax

lapack_int LAPACKE_sgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int nrhs, float* a, lapack_int lda, float* b, lapack_int ldb, float* s, float rcond, lapack_int* rank );

lapack_int LAPACKE_dgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int nrhs, double* a, lapack_int lda, double* b, lapack_int ldb, double* s, double rcond, lapack_int* rank );

lapack_int LAPACKE_cgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb, float* s, float rcond, lapack_int* rank );

lapack_int LAPACKE_zgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb, double* s, double rcond, lapack_int* rank );

Include Files

mkl.h

Description

The routine computes the minimum-norm solution to a real linear least squares problem:

minimize ||b - A*x||₂

using the singular value decomposition (SVD) of A. A is an m-by-n matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the m-by-nrhs right hand side matrix B and the n-by-nrhs solution matrix X.

The problem is solved in three steps:

Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS).
Solve the BLS using a divide and conquer approach.
Apply back all the Householder transformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than rcond times the largest singular value.

Input Parameters

matrix_layout

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

m

The number of rows of the matrix A (m≥ 0).

n

The number of columns of the matrix A

(n≥ 0).

nrhs

The number of right-hand sides; the number of columns in B (nrhs≥ 0).

a, b

Arrays:

a(size max(1, lda*n) for column major layout and max(1, lda*m) for row major layout) contains the m-by-n matrix A.

b(size max(1, ldb*nrhs) for column major layout and max(1, ldb*max(m, n)) for row major layout) contains the m-by-nrhs right hand side matrix B.

lda

The leading dimension of a; at least max(1, m)for column major layout and max(1, n) for row major layout.

ldb

The leading dimension of b; must be at least max(1, m, n) for column major layout and at least max(1, nrhs) for row major layout.

rcond

rcond is used to determine the effective rank of A. Singular values s(i) ≤rcond *s(1) are treated as zero. If rcond≤ 0, machine precision is used instead.

Output Parameters

a

On exit, A has been overwritten.

b

Overwritten by the n-by-nrhs solution matrix X.

If m≥n and rank = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of modulus of elements n+1:m in that column.

s

Array, size at least max(1, min(m, n)). The singular values of A in decreasing order. The condition number of A in the 2-norm is

k₂(A) = s(1)/ s(min(m, n)).

rank

The effective rank of A, that is, the number of singular values which are greater than rcond *s(1).

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 = i, then the algorithm for computing the SVD failed to converge; i indicates the number of off-diagonal elements of an intermediate bidiagonal form that did not converge to zero.

Parent topic: Linear Least Squares (LLS) Problems: LAPACK Driver Routines

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