slaln2(3S)

NAME
     SLALN2 - 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 perturbation of A

SYNOPSIS
     SUBROUTINE SLALN2( 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

         REAL           CA, D1, D2, SCALE, SMIN, WI, WR, XNORM

         REAL           A( LDA, * ), B( LDB, * ), X( LDX, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SLALN2 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 perturbation 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 SLALN2, 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.)


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) REAL
             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      (input) REAL
             The coefficient c, which A is multiplied by.

     A       (input) REAL 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) REAL
             The 1,1 element in the diagonal matrix D.

     D2      (input) REAL
             The 2,2 element in the diagonal matrix D.  Not used if NW=1.

     B       (input) REAL 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) REAL
             The real part of the scalar "w".

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

     X       (output) REAL array, dimension (LDX,NW)
             The NA x NW matrix X (unknowns), as computed by SLALN2.  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) REAL
             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) REAL
             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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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