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NAME
     dlaln2 - solve a system of the form (ca A-wD ) X =  s  B  or
     (ca A'-wD) X = s B with possible scaling ("s") and perturba-
     tion 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, * )



     #include <sunperf.h>

     void dlaln2(int ltrans, int na, int nw, double smin,  double
               ca,  double  *da,  int  lda, double d1, double d2,
               double *db, int ldb, double wr, double wi,  double
               *dx,  int  ldx, double *dscale, double *xnorm, int
               *info) ;

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 precision) and  (machine preci-
               sion * 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 element in the diagonal matrix D.

     D2        (input) DOUBLE PRECISION
               The 2,2 element 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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