dgelsd man page on Scientific

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```DGELSD(1)	      LAPACK driver routine (version 3.2)	     DGELSD(1)

NAME
DGELSD  -  computes  the	 minimum-norm  solution to a real linear least
squares problem

SYNOPSIS
SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S,  RCOND,  RANK,	 WORK,
LWORK, IWORK, INFO )

INTEGER	  INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

DOUBLE	  PRECISION RCOND

INTEGER	  IWORK( * )

DOUBLE	  PRECISION  A( LDA, * ), B( LDB, * ), S( * ), WORK( *
)

PURPOSE
DGELSD 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 tranformations to solve
the original least squares problem.
The effective rank of A is determined by treating as zero those	singu‐
lar values which are less than RCOND times the largest singular value.
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, or on those binary machines without guard digits which
subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It	 could
conceivably  fail on hexadecimal or decimal machines without guard dig‐
its, but we know of none.

ARGUMENTS
M       (input) INTEGER
The number of rows of A. M >= 0.

N       (input) INTEGER
The number of columns of A. N >= 0.

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

A       (input) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.  On exit, A has been destroyed.

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

B       (input/output) DOUBLE PRECISION 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  elements
n+1:m in that column.

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

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

WORK	  (workspace/output)   DOUBLE	PRECISION   array,   dimension
(MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK   (input) 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 12*N + 2*N*SMLSIZ + 8*N*NLVL
+ N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or
12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, 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 sub‐
problems	 at  the bottom of the computation tree (usually about
25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) +
1 ) 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 WORK array, returns this
value as the first entry of the WORK array, and no  error  mes‐
sage related to LWORK is issued by XERBLA.

IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N
).

INFO    (output) 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 bidiag‐
onal form did not converge to zero.

FURTHER DETAILS
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

LAPACK driver routine (version 3November 2008			     DGELSD(1)
```
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