MAGMA
2.3.0
Matrix Algebra for GPU and Multicore Architectures

Functions  
magma_int_t  magma_clatrsd (magma_uplo_t uplo, magma_trans_t trans, magma_diag_t diag, magma_bool_t normin, magma_int_t n, const magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex lambda, magmaFloatComplex *x, float *scale, float *cnorm, magma_int_t *info) 
CLATRSD solves one of the triangular systems with modified diagonal (A  lambda*I) * x = s*b, (A  lambda*I)**T * x = s*b, or (A  lambda*I)**H * x = s*b, with scaling to prevent overflow. More...  
magma_int_t  magma_zlatrsd (magma_uplo_t uplo, magma_trans_t trans, magma_diag_t diag, magma_bool_t normin, magma_int_t n, const magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex lambda, magmaDoubleComplex *x, double *scale, double *cnorm, magma_int_t *info) 
ZLATRSD solves one of the triangular systems with modified diagonal (A  lambda*I) * x = s*b, (A  lambda*I)**T * x = s*b, or (A  lambda*I)**H * x = s*b, with scaling to prevent overflow. More...  
magma_int_t magma_clatrsd  (  magma_uplo_t  uplo, 
magma_trans_t  trans,  
magma_diag_t  diag,  
magma_bool_t  normin,  
magma_int_t  n,  
const magmaFloatComplex *  A,  
magma_int_t  lda,  
magmaFloatComplex  lambda,  
magmaFloatComplex *  x,  
float *  scale,  
float *  cnorm,  
magma_int_t *  info  
) 
CLATRSD solves one of the triangular systems with modified diagonal (A  lambda*I) * x = s*b, (A  lambda*I)**T * x = s*b, or (A  lambda*I)**H * x = s*b, with scaling to prevent overflow.
Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are nelement vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a nontrivial solution to A*x = 0 is returned.
This version subtracts lambda from the diagonal, for use in ctrevc to compute eigenvectors. It does not modify A during the computation.
[in]  uplo  magma_uplo_t Specifies whether the matrix A is upper or lower triangular.

[in]  trans  magma_trans_t Specifies the operation applied to A.

[in]  diag  magma_diag_t Specifies whether or not the matrix A is unit triangular.

[in]  normin  magma_bool_t Specifies whether CNORM has been set or not.

[in]  n  INTEGER The order of the matrix A. N >= 0. 
[in]  A  COMPLEX array, dimension (LDA,N) The triangular matrix A. If UPLO = MagmaUpper, the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = MagmaUnit, the diagonal elements of A are also not referenced and are assumed to be 1. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max (1,N). 
[in]  lambda  COMPLEX Eigenvalue to subtract from diagonal of A. 
[in,out]  x  COMPLEX array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. 
[out]  scale  REAL The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. 
[in,out]  cnorm  (input or output) REAL array, dimension (N)

[out]  info  INTEGER

A rough bound on x is computed; if that is less than overflow, CTRSV is called, otherwise, specific code is used which checks for possible overflow or dividebyzero at every operation.
A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is
x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n]  x(j) * A[j+1:n,j] end
Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have M(j+1) <= G(j) /  A(j+1,j+1)  G(j+1) <= G(j) + M(j+1) *  A[j+2:n,j+1]  <= G(j) ( 1 + CNORM(j+1) /  A(j+1,j+1)  )
where CNORM(j+1) is greater than or equal to the infinitynorm of column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) /  A(i,i)  ) 1 <= i <= j and
x(j) <= ( G(0) / A(j,j) ) product ( 1 + CNORM(i) / A(i,i) ) 1 <= i < j
Since x(j) <= M(j), we use the Level 2 BLAS routine CTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a nontrivial solution to A*x = 0 is found.
Similarly, a rowwise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for upper triangular A is:
for j = 1, ..., n x(j) := ( b(j)  A[1:j1,j]' * x[1:j1] ) / A(j,j) end
We simultaneously compute two bounds G(j) = bound on ( b(i)  A[1:i1,i]' * x[1:i1] ), 1 <= i <= j M(j) = bound on x(i), 1 <= i <= j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j1) and M(j) >= M(j1) for j >= 1. Then the bound on x(j) is
M(j) <= M(j1) * ( 1 + CNORM(j) ) /  A(j,j)  <= M(0) * product ( ( 1 + CNORM(i) ) / A(i,i) ) 1 <= i <= j
and we can safely call CTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).
magma_int_t magma_zlatrsd  (  magma_uplo_t  uplo, 
magma_trans_t  trans,  
magma_diag_t  diag,  
magma_bool_t  normin,  
magma_int_t  n,  
const magmaDoubleComplex *  A,  
magma_int_t  lda,  
magmaDoubleComplex  lambda,  
magmaDoubleComplex *  x,  
double *  scale,  
double *  cnorm,  
magma_int_t *  info  
) 
ZLATRSD solves one of the triangular systems with modified diagonal (A  lambda*I) * x = s*b, (A  lambda*I)**T * x = s*b, or (A  lambda*I)**H * x = s*b, with scaling to prevent overflow.
Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are nelement vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a nontrivial solution to A*x = 0 is returned.
This version subtracts lambda from the diagonal, for use in ztrevc to compute eigenvectors. It does not modify A during the computation.
[in]  uplo  magma_uplo_t Specifies whether the matrix A is upper or lower triangular.

[in]  trans  magma_trans_t Specifies the operation applied to A.

[in]  diag  magma_diag_t Specifies whether or not the matrix A is unit triangular.

[in]  normin  magma_bool_t Specifies whether CNORM has been set or not.

[in]  n  INTEGER The order of the matrix A. N >= 0. 
[in]  A  COMPLEX_16 array, dimension (LDA,N) The triangular matrix A. If UPLO = MagmaUpper, the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = MagmaUnit, the diagonal elements of A are also not referenced and are assumed to be 1. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max (1,N). 
[in]  lambda  COMPLEX_16 Eigenvalue to subtract from diagonal of A. 
[in,out]  x  COMPLEX_16 array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. 
[out]  scale  DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. 
[in,out]  cnorm  (input or output) DOUBLE PRECISION array, dimension (N)

[out]  info  INTEGER

A rough bound on x is computed; if that is less than overflow, ZTRSV is called, otherwise, specific code is used which checks for possible overflow or dividebyzero at every operation.
A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is
x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n]  x(j) * A[j+1:n,j] end
Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have M(j+1) <= G(j) /  A(j+1,j+1)  G(j+1) <= G(j) + M(j+1) *  A[j+2:n,j+1]  <= G(j) ( 1 + CNORM(j+1) /  A(j+1,j+1)  )
where CNORM(j+1) is greater than or equal to the infinitynorm of column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) /  A(i,i)  ) 1 <= i <= j and
x(j) <= ( G(0) / A(j,j) ) product ( 1 + CNORM(i) / A(i,i) ) 1 <= i < j
Since x(j) <= M(j), we use the Level 2 BLAS routine ZTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a nontrivial solution to A*x = 0 is found.
Similarly, a rowwise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for upper triangular A is:
for j = 1, ..., n x(j) := ( b(j)  A[1:j1,j]' * x[1:j1] ) / A(j,j) end
We simultaneously compute two bounds G(j) = bound on ( b(i)  A[1:i1,i]' * x[1:i1] ), 1 <= i <= j M(j) = bound on x(i), 1 <= i <= j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j1) and M(j) >= M(j1) for j >= 1. Then the bound on x(j) is
M(j) <= M(j1) * ( 1 + CNORM(j) ) /  A(j,j)  <= M(0) * product ( ( 1 + CNORM(i) ) / A(i,i) ) 1 <= i <= j
and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).