DLALN2(l)								 )								 DLALN2(l)

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 perturbation 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 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 imagi-
       nary 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.)

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 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 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.

LAPACK version 3.0						   15 June 2000 							 DLALN2(l)