# ?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

FORTRAN 77:

call sgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)

call dgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)

call cgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)

call zgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)

Fortran 95:

call gelsd(a, b [,rank] [,s] [,rcond] [,info])

## Description

This 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:

1. Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS).

2. Solve the BLS using a divide and conquer approach.

3. 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.

The routine uses auxiliary routines ?lals0 and ?lalsa.

## Input Parameters

m

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

n

  INTEGER. The number of columns of the matrix A

  (`n ≥ 0`).

nrhs

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

a, b, work

  REAL for sgelsd

  DOUBLE PRECISION for dgelsd

  COMPLEX for cgelsd

  DOUBLE COMPLEX for zgelsd.

  Arrays:

  a(lda,*) contains the m-by-n matrix A.

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

  b(ldb,*) contains the m-by-nrhs right hand side matrix B.

  The second dimension of b must be at least max(1, nrhs).

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of a; at least max(1, m).

ldb

  INTEGER. The first dimension of b; must be at least max(1, m, n).

rcond

  REAL for single-precision flavors

  DOUBLE PRECISION for double-precision flavors.

  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.

lwork

  INTEGER. The size of the work array; `lwork ≥ 1`.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the array work and the minimum sizes of the arrays rwork and iwork, and returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork is issued by xerbla.

  See Application Notes for the suggested value of lwork.

iwork

  INTEGER. Workspace array. See Application Notes for the suggested dimension of iwork.

rwork

  REAL for cgelsd

  DOUBLE PRECISION for zgelsd.

  Workspace array, used in complex flavors only. See Application Notes for the suggested dimension of rwork.

## 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

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  Array, DIMENSION 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

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

work`(1)`

  If `info = 0`, on exit, `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

rwork`(1)`

  If `info = 0`, on exit, `rwork(1)` returns the minimum size of the workspace array iwork required for optimum performance.

iwork`(1)`

  If `info = 0`, on exit, `iwork(1)` returns the minimum size of the workspace array iwork required for optimum performance.

info

  INTEGER.

  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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine gelsd interface are the following:

a

  Holds the matrix A of size (m,n).

b

  Holds the matrix of size max(m,n)-by-nrhs. On entry, contains the m-by-nrhs right hand side matrix B, On exit, overwritten by the n-by-nrhs solution matrix X.

s

  Holds the vector of length min(m,n).

rcond

  Default value for this element is `rcond = 100*EPSILON(1.0_WP)`.

## Application Notes

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

The exact minimum amount of workspace needed depends on m, n and nrhs. The size lwork of the workspace array work must be as given below.

For real flavors:

If `m ≥ n`,

`lwork ≥ 12n + 2n*smlsiz + 8n*nlvl + n*nrhs + (smlsiz+1)2`;

If `m < n`,

`lwork ≥ 12m + 2m*smlsiz + 8m*nlvl + m*nrhs + (smlsiz+1)2`;

For complex flavors:

If `m ≥ n`,

`lwork< 2n + n*nrhs`;

If `m < n`,

`lwork ≥ 2m + m*nrhs`;

where smlsiz is returned by ilaenv and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and

`nlvl = INT( log2( min( m, n )/(smlsiz+1)) ) + 1`.

For good performance, lwork should generally be larger.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The dimension of the workspace array iwork must be at least

`3*min( m, n )*nlvl + 11*min( m, n )`.

The dimension of the workspace array iwork (for complex flavors) must be at least `max(1, lrwork)`.

`lrwork ≥ 10n + 2n*smlsiz + 8n*nlvl + 3*smlsiz*nrhs + (smlsiz+1)2` if `m ≥ n`, and

`lrwork ≥ 10m + 2m*smlsiz + 8m*nlvl + 3*smlsiz*nrhs + (smlsiz+1)2` if m < n.

Parent topic: Linear Least Squares (LLS) Problems