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# dlaln2

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NAME
DLALN2 - solve a system of the form (ca A - w D ) X = s B or
(ca A' - w D) X = s B with possible scaling ("s") and per-
turbation of A

SYNOPSIS
SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2,
B, LDB, WR, WI, X, LDX, SCALE, XNORM,
INFO )

LOGICAL        LTRANS

INTEGER        INFO, LDA, LDB, LDX, NA, NW

DOUBLE         PRECISION CA, D1, D2, SCALE, SMIN, WI,
WR, XNORM

DOUBLE         PRECISION A( LDA, * ), B( LDB, * ), X(
LDX, * )

PURPOSE
DLALN2 solves a system of the form  (ca A - w D ) X = s B or
(ca A' - w D) X = s B   with possible scaling ("s") and per-
turbation of A.  (A' 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 com-
puted 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.)

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

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

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

SMIN    (input) 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 preci-
sion) and  (machine precision * overflow ).  (See
BIGNUM and ULP.)

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

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

LDA     (input) INTEGER
The leading dimension of A.  It must be at least NA.

D1      (input) DOUBLE PRECISION
The 1,1 entry in the diagonal matrix D.

D2      (input) DOUBLE PRECISION
The 2,2 entry in the diagonal matrix D.  Not used if
NW=1.

B       (input) 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     (input) INTEGER
The leading dimension of B.  It must be at least NA.

WR      (input) DOUBLE PRECISION
The real part of the scalar "w".

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

X       (output) 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     (input) INTEGER
The leading dimension of X.  It must be at least NA.

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

INFO    (output) 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.
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