MAGMA  2.3.0
Matrix Algebra for GPU and Multicore Architectures
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laln2: Solve 2x2 system; used by trevc

Functions

magma_int_t magma_dlaln2 (magma_int_t trans, magma_int_t na, magma_int_t nw, double smin, double ca, const double *A, magma_int_t lda, double d1, double d2, const double *B, magma_int_t ldb, double wr, double wi, double *X, magma_int_t ldx, double *scale, double *xnorm, magma_int_t *info)
 DLALN2 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. More...
 
magma_int_t magma_slaln2 (magma_int_t trans, magma_int_t na, magma_int_t nw, float smin, float ca, const float *A, magma_int_t lda, float d1, float d2, const float *B, magma_int_t ldb, float wr, float wi, float *X, magma_int_t ldx, float *scale, float *xnorm, magma_int_t *info)
 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. More...
 

Detailed Description

Function Documentation

magma_int_t magma_dlaln2 ( magma_int_t  trans,
magma_int_t  na,
magma_int_t  nw,
double  smin,
double  ca,
const double *  A,
magma_int_t  lda,
double  d1,
double  d2,
const double *  B,
magma_int_t  ldb,
double  wr,
double  wi,
double *  X,
magma_int_t  ldx,
double *  scale,
double *  xnorm,
magma_int_t *  info 
)

DLALN2 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 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.)

Parameters
[in]transLOGICAL = true (1): A**T will be used. = false (0): A will be used (not transposed.)
[in]naINTEGER The size of the matrix A. It may (only) be 1 or 2.
[in]nwINTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2.
[in]sminDOUBLE 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.)
[in]caDOUBLE PRECISION The coefficient c, which A is multiplied by.
[in]ADOUBLE PRECISION array, dimension (LDA,NA) The NA x NA matrix A.
[in]ldaINTEGER The leading dimension of A. It must be at least NA.
[in]d1DOUBLE PRECISION The 1,1 element in the diagonal matrix D.
[in]d2DOUBLE PRECISION The 2,2 element in the diagonal matrix D. Not used if NW=1.
[in]BDOUBLE 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.
[in]ldbINTEGER The leading dimension of B. It must be at least NA.
[in]wrDOUBLE PRECISION The real part of the scalar "w".
[in]wiDOUBLE PRECISION The imaginary part of the scalar "w". Not used if NW=1.
[out]XDOUBLE 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.
[in]ldxINTEGER The leading dimension of X. It must be at least NA.
[out]scaleDOUBLE 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.
[out]xnormDOUBLE PRECISION The infinity-norm of X, when X is regarded as an NA x NW real matrix.
[out]infoINTEGER 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.
magma_int_t magma_slaln2 ( magma_int_t  trans,
magma_int_t  na,
magma_int_t  nw,
float  smin,
float  ca,
const float *  A,
magma_int_t  lda,
float  d1,
float  d2,
const float *  B,
magma_int_t  ldb,
float  wr,
float  wi,
float *  X,
magma_int_t  ldx,
float *  scale,
float *  xnorm,
magma_int_t *  info 
)

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.)

Parameters
[in]transLOGICAL = true (1): A**T will be used. = false (0): A will be used (not transposed.)
[in]naINTEGER The size of the matrix A. It may (only) be 1 or 2.
[in]nwINTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2.
[in]sminREAL 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.)
[in]caREAL The coefficient c, which A is multiplied by.
[in]AREAL array, dimension (LDA,NA) The NA x NA matrix A.
[in]ldaINTEGER The leading dimension of A. It must be at least NA.
[in]d1REAL The 1,1 element in the diagonal matrix D.
[in]d2REAL The 2,2 element in the diagonal matrix D. Not used if NW=1.
[in]BREAL 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.
[in]ldbINTEGER The leading dimension of B. It must be at least NA.
[in]wrREAL The real part of the scalar "w".
[in]wiREAL The imaginary part of the scalar "w". Not used if NW=1.
[out]XREAL 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.
[in]ldxINTEGER The leading dimension of X. It must be at least NA.
[out]scaleREAL 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.
[out]xnormREAL The infinity-norm of X, when X is regarded as an NA x NW real matrix.
[out]infoINTEGER 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.