MAGMA
2.3.0
Matrix Algebra for GPU and Multicore Architectures

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...  
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 Atranspose.)
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 infinitynorm 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.)
[in]  trans  LOGICAL = true (1): A**T will be used. = false (0): A will be used (not transposed.) 
[in]  na  INTEGER The size of the matrix A. It may (only) be 1 or 2. 
[in]  nw  INTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2. 
[in]  smin  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.) 
[in]  ca  DOUBLE PRECISION The coefficient c, which A is multiplied by. 
[in]  A  DOUBLE PRECISION array, dimension (LDA,NA) The NA x NA matrix A. 
[in]  lda  INTEGER The leading dimension of A. It must be at least NA. 
[in]  d1  DOUBLE PRECISION The 1,1 element in the diagonal matrix D. 
[in]  d2  DOUBLE PRECISION The 2,2 element in the diagonal matrix D. Not used if NW=1. 
[in]  B  DOUBLE PRECISION array, dimension (LDB,NW) The NA x NW matrix B (righthand side). If NW=2 ("w" is complex), column 1 contains the real part of B and column 2 contains the imaginary part. 
[in]  ldb  INTEGER The leading dimension of B. It must be at least NA. 
[in]  wr  DOUBLE PRECISION The real part of the scalar "w". 
[in]  wi  DOUBLE PRECISION The imaginary part of the scalar "w". Not used if NW=1. 
[out]  X  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. 
[in]  ldx  INTEGER The leading dimension of X. It must be at least NA. 
[out]  scale  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. 
[out]  xnorm  DOUBLE PRECISION The infinitynorm of X, when X is regarded as an NA x NW real matrix. 
[out]  info  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:

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 Atranspose.)
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 infinitynorm 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.)
[in]  trans  LOGICAL = true (1): A**T will be used. = false (0): A will be used (not transposed.) 
[in]  na  INTEGER The size of the matrix A. It may (only) be 1 or 2. 
[in]  nw  INTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2. 
[in]  smin  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.) 
[in]  ca  REAL The coefficient c, which A is multiplied by. 
[in]  A  REAL array, dimension (LDA,NA) The NA x NA matrix A. 
[in]  lda  INTEGER The leading dimension of A. It must be at least NA. 
[in]  d1  REAL The 1,1 element in the diagonal matrix D. 
[in]  d2  REAL The 2,2 element in the diagonal matrix D. Not used if NW=1. 
[in]  B  REAL array, dimension (LDB,NW) The NA x NW matrix B (righthand side). If NW=2 ("w" is complex), column 1 contains the real part of B and column 2 contains the imaginary part. 
[in]  ldb  INTEGER The leading dimension of B. It must be at least NA. 
[in]  wr  REAL The real part of the scalar "w". 
[in]  wi  REAL The imaginary part of the scalar "w". Not used if NW=1. 
[out]  X  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. 
[in]  ldx  INTEGER The leading dimension of X. It must be at least NA. 
[out]  scale  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. 
[out]  xnorm  REAL The infinitynorm of X, when X is regarded as an NA x NW real matrix. 
[out]  info  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:
