SLALN2 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 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.)

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 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

          CA is REAL


          The coefficient c, which A is multiplied by.

A

          A is REAL 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 REAL


          The 1,1 element in the diagonal matrix D.

D2

          D2 is REAL


          The 2,2 element in the diagonal matrix D.  Not used if NA=1.

B

          B is 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

          LDB is INTEGER


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

WR

          WR is REAL


          The real part of the scalar "w".

WI

          WI is REAL


          The imaginary part of the scalar "w".  Not used if NW=1.

X

          X is 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

          LDX is INTEGER


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

SCALE

          SCALE is 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

          XNORM is REAL


          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016