# DLALN2 (3) - Linux Manuals

dlaln2.f -

## SYNOPSIS

### Functions/Subroutines

subroutine dlaln2 (LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO)
DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

## Function/Subroutine Documentation

### subroutine dlaln2 (logicalLTRANS, integerNA, integerNW, double precisionSMIN, double precisionCA, double precision, dimension( lda, * )A, integerLDA, double precisionD1, double precisionD2, double precision, dimension( ldb, * )B, integerLDB, double precisionWR, double precisionWI, double precision, dimension( ldx, * )X, integerLDX, double precisionSCALE, double precisionXNORM, integerINFO)

DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

Purpose:

DLALN2 solves a system of the form  (ca A - w D ) X = s B
or (ca A**T - w D) X = s B   with possible scaling ("s") and
perturbation of A.  (A**T means A-transpose.)

A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
real diagonal matrix, w is a real or complex value, and X and B are
NA x 1 matrices -- real if w is real, complex if w is complex.  NA
may be 1 or 2.

If w is complex, X and B are represented as NA x 2 matrices,
the first column of each being the real part and the second
being the imaginary part.

"s" is a scaling factor (.LE. 1), computed by DLALN2, which is
so chosen that X can be computed without overflow.  X is further
scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
than overflow.

If both singular values of (ca A - w D) are less than SMIN,
SMIN*identity will be used instead of (ca A - w D).  If only one
singular value is less than SMIN, one element of (ca A - w D) will be
perturbed enough to make the smallest singular value roughly SMIN.
If both singular values are at least SMIN, (ca A - w D) will not be
perturbed.  In any case, the perturbation will be at most some small
multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
are computed by infinity-norm approximations, and thus will only be
correct to a factor of 2 or so.

Note: all input quantities are assumed to be smaller than overflow
by a reasonable factor.  (See BIGNUM.)

Parameters:

LTRANS

LTRANS is LOGICAL
=.TRUE.:  A-transpose will be used.
=.FALSE.: A will be used (not transposed.)

NA

NA is INTEGER
The size of the matrix A.  It may (only) be 1 or 2.

NW

NW is INTEGER
1 if "w" is real, 2 if "w" is complex.  It may only be 1
or 2.

SMIN

SMIN is DOUBLE PRECISION
The desired lower bound on the singular values of A.  This
should be a safe distance away from underflow or overflow,
say, between (underflow/machine precision) and  (machine
precision * overflow ).  (See BIGNUM and ULP.)

CA

CA is DOUBLE PRECISION
The coefficient c, which A is multiplied by.

A

A is DOUBLE PRECISION array, dimension (LDA,NA)
The NA x NA matrix A.

LDA

LDA is INTEGER
The leading dimension of A.  It must be at least NA.

D1

D1 is DOUBLE PRECISION
The 1,1 element in the diagonal matrix D.

D2

D2 is DOUBLE PRECISION
The 2,2 element in the diagonal matrix D.  Not used if NW=1.

B

B is DOUBLE PRECISION array, dimension (LDB,NW)
The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
complex), column 1 contains the real part of B and column 2
contains the imaginary part.

LDB

LDB is INTEGER
The leading dimension of B.  It must be at least NA.

WR

WR is DOUBLE PRECISION
The real part of the scalar "w".

WI

WI is DOUBLE PRECISION
The imaginary part of the scalar "w".  Not used if NW=1.

X

X is DOUBLE PRECISION array, dimension (LDX,NW)
The NA x NW matrix X (unknowns), as computed by DLALN2.
If NW=2 ("w" is complex), on exit, column 1 will contain
the real part of X and column 2 will contain the imaginary
part.

LDX

LDX is INTEGER
The leading dimension of X.  It must be at least NA.

SCALE

SCALE is DOUBLE PRECISION
The scale factor that B must be multiplied by to insure
that overflow does not occur when computing X.  Thus,
(ca A - w D) X  will be SCALE*B, not B (ignoring
perturbations of A.)  It will be at most 1.

XNORM

XNORM is DOUBLE PRECISION
The infinity-norm of X, when X is regarded as an NA x NW
real matrix.

INFO

INFO is INTEGER
An error flag.  It will be set to zero if no error occurs,
a negative number if an argument is in error, or a positive
number if  ca A - w D  had to be perturbed.
The possible values are:
= 0: No error occurred, and (ca A - w D) did not have to be
perturbed.
= 1: (ca A - w D) had to be perturbed to make its smallest
(or only) singular value greater than SMIN.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.

Author:

Univ. of Tennessee

Univ. of California Berkeley