# zgelsd.f - Man Page

SRC/zgelsd.f

## Synopsis

### Functions/Subroutines

subroutine zgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)
ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

## Function/Subroutine Documentation

### subroutine zgelsd (integer m, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) s, double precision rcond, integer rank, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

``` ZGELSD computes the minimum-norm solution to a real linear least
squares problem:
minimize 2-norm(| b - A*x |)
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.```
Parameters

M

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

N

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

NRHS

```          NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A has been destroyed.```

LDA

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

B

```          B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, B is 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 the modulus of elements n+1:m in that column.```

LDB

```          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,M,N).```

S

```          S is DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A in decreasing order.
The condition number of A in the 2-norm = S(1)/S(min(m,n)).```

RCOND

```          RCOND is DOUBLE PRECISION
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.```

RANK

```          RANK is INTEGER
The effective rank of A, i.e., the number of singular values
which are greater than RCOND*S(1).```

WORK

```          WORK is COMPLEX*16 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 must be at least 1.
The exact minimum amount of workspace needed depends on M,
N and NRHS. As long as LWORK is at least
2*N + N*NRHS
if M is greater than or equal to N or
2*M + M*NRHS
if M is less than N, the code will execute correctly.
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 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.```

RWORK

```          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
LRWORK >=
10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
if M is greater than or equal to N or
10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
if M is less than N, the code will execute correctly.
SMLSIZ is returned by ILAENV and is equal to the maximum
size of the subproblems at the bottom of the computation
NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.```

IWORK

```          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
where MINMN = MIN( M,N ).
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.```

INFO

```          INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0:  the algorithm for computing the SVD failed to converge;
if INFO = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Definition at line 217 of file zgelsd.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zgelsd(3) is an alias of zgelsd.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK