Developer Reference for Intel® oneAPI Math Kernel Library for C
?sytrs
Solves a system of linear equations with a UDU T - or LDL T -factored symmetric coefficient matrix.
Syntax
lapack_int LAPACKE_ssytrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const float *a, lapack_int lda, const lapack_int *ipiv,
float *b, lapack_int ldb);
lapack_int LAPACKE_dsytrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const double *a, lapack_int lda, const lapack_int *ipiv,
double *b, lapack_int ldb);
lapack_int LAPACKE_csytrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const lapack_complex_float *a, lapack_int lda,
const lapack_int *ipiv, lapack_complex_float *b,
lapack_int ldb);
lapack_int LAPACKE_zsytrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const lapack_complex_double *a, lapack_int lda,
const lapack_int *ipiv, lapack_complex_double *b,
lapack_int ldb);Include Files
mkl.h
Description
The routine solves for \(X\) the system of linear equations \(A*X = B\) with a symmetric matrix \(A\) , given the Bunch-Kaufman factorization of \(A\) :
if uplo='U' ,
\(A = U*D*U^{T}\)
if uplo='L' ,
\(A = L*D*L^{T}\) ,
where \(U\) and \(L\) are upper and lower triangular matrices with unit diagonal and \(D\) is a symmetric block-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix \(B\) . You must supply to this routine the factor \(U\) (or \(L\) ) and the array ipiv returned by the factorization routine ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
uplo
Must be ‘U’ or ‘L’ .
Indicates how the input matrix A has been factored:
If uplo = 'U' , the array a stores the upper triangular factor \(U\) of the factorization \(A = U*D*U^{T}\) .
If uplo = 'L' , the array a stores the lower triangular factor \(L\) of the factorization \(A = L*D*L^{T}\) .
- n
-
Specifies the order of the matrix A . The value of n must be at least zero.
nrhs
The number of right-hand sides; nrhs ≥ 0.
ipiv
Array, size at least max(1, n) . The ipiv array, as returned by ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
a , b
The array a of size max(1, lda * n ) contains the factor U or L (see uplo ). .
b
The array b contains the matrix B whose columns are the right-hand sides for the system of equations.
The size of b is at least max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout.
lda
The leading dimension of a ; lda≥ max(1, n) .
ldb
The leading dimension of b ; ldb≥ max(1, n)ldb ≥ max(1, n ) for column major layout and ldb ≥ nrhs for row major layout .
Output Parameters
b
Overwritten by the solution matrix X .
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , parameter i had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
For each right-hand side \(b\) , the computed solution is the exact solution of a perturbed system of equations \((A + E)x = b\) , where
\[|E| \leq c(n) \varepsilon P|U||D||U^{H}|P^{T} or |E| \le c(n) \varepsilon P|L||D||L^{H}|P^{T}\]
\(c(n)\) is a modest linear function of \(k\) , and \(\varepsilon\) is the machine precision.
If \(x_{0}\) is the true solution, the computed solution \(x\) satisfies this error bound:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(n) \operatorname{cond}(A, x) \varepsilon\]
where
\[\operatorname{cond}(A,x) = || |A^{-1}| |A| |x|_\infty || / ||x||_\infty \leq ||A^{-1}||_\infty ||A||_\infty = \kappa_\infty(A).\]
Note that \(\operatorname{cond}(A,x)\) can be much smaller than \(\kappa_{\infty}(A)\) .
The total number of floating-point operations for one right-hand side vector is approximately \(2n^{2}\) for real flavors or \(8n^{2}\) for complex flavors.
To estimate the condition number \(\kappa_{\infty}(A)\) , call ?sycon (Estimates the reciprocal of the condition number of a symmetric matrix.) .
To refine the solution and estimate the error, call ?syrfs (Refines the solution of a system of linear equations with a symmetric coefficient matrix and estimates its error.) .