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template<typename T>
void laln2(logical *ltrans, integer *na, integer *nw, float *smin, float *ca, float *a, integer *lda, float *d1, float *d2, float *b, integer *ldb, float *wr, float *wi, float *x, integer *ldx, float *scale, float *xnorm, integer *info)# LALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
Purpose:
LALN2 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 (<= 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 fla_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.)
- Parameters:
LTRANS – [in]
LTRANS is LOGICAL
=.TRUE.: A-transpose will be used.
=.FALSE.: A will be used (not transposed.)NA – [in]
NA is INTEGER
The size of the matrix A. It may (only) be 1 or 2.
NW – [in]
NW is INTEGER
1 if “w” is real, 2 if “w” is complex. It may only be 1 or 2.
SMIN – [in]
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 – [in]
CA is REAL
The coefficient c, which A is multiplied by.
A – [in]
A is REAL array, dimension (LDA,NA)
The NA x NA matrix A.
LDA – [in]
LDA is INTEGER
The leading dimension of A. It must be at least NA.
D1 – [in]
D1 is REAL
The 1,1 element in the diagonal matrix D.
D2 – [in]
D2 is REAL
The 2,2 element in the diagonal matrix D. Not used if NA=1.
B – [in]
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 – [in]
LDB is INTEGER
The leading dimension of B. It must be at least NA.
WR – [in]
WR is REAL
The real part of the scalar “w”.
WI – [in]
WI is REAL
The imaginary part of the scalar “w”. Not used if NW=1.
X – [out]
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 – [in]
LDX is INTEGER
The leading dimension of X. It must be at least NA.
SCALE – [out]
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 – [out]
XNORM is REAL
The infinity-norm of X, when X is regarded as an NA x NW real matrix.
INFO – [out]
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.