Developer Reference for Intel® oneAPI Math Kernel Library for C
?getrs
Solves a system of linear equations with an LU-factored square coefficient matrix, with multiple right-hand sides.
Syntax
lapack_intLAPACKE_sgetrs ( intmatrix_layout , chartrans , lapack_intn , lapack_intnrhs , constfloat*a , lapack_intlda , constlapack_int*ipiv , float*b , lapack_intldb );
lapack_intLAPACKE_dgetrs ( intmatrix_layout , chartrans , lapack_intn , lapack_intnrhs , constdouble*a , lapack_intlda , constlapack_int*ipiv , double*b , lapack_intldb );
lapack_intLAPACKE_cgetrs ( intmatrix_layout , chartrans , lapack_intn , lapack_intnrhs , constlapack_complex_float*a , lapack_intlda , constlapack_int*ipiv , lapack_complex_float*b , lapack_intldb );
lapack_intLAPACKE_zgetrs ( intmatrix_layout , chartrans , lapack_intn , lapack_intnrhs , constlapack_complex_double*a , lapack_intlda , constlapack_int*ipiv , lapack_complex_double*b , lapack_intldb );
Include Files
mkl.h
Description
sgetrs dgetrs cgetrs zgetrs getrs
The routine solves for X the following systems of linear equations:
\(AX\) = \(B\) if trans = ‘N’ ,
\(A^{T}X = B\) if trans = ‘T’ ,
\(A^{H}X = B\) if trans = ‘C’ (for complex matrices only).
Before calling this routine, you must call ?getrf (Computes the LU factorization of a general m-by-n matrix.) to compute the LU factorization of A .
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
trans
Must be ‘N’ or ‘T’ or ‘C’ .
Indicates the form of the equations:
If trans = 'N' , then A*X = B is solved for X .
If trans = 'T' , then A^{T}*X = B is solved for X .
If trans = 'C' , then A^{H}*X = B is solved for X .
n
The order of A ; the number of rows in B(n≥ 0) .
nrhs
The number of right-hand sides; nrhs ≥ 0.
a , b
Array of size max(1, lda * n ).
The array a contains LU factorization of matrix A resulting from the call of ?getrf (Computes the LU factorization of a general m-by-n matrix.) .
b
Array of size max(1, ldb * nrhs ) for column major layout, and max(1, ldb * n ) for row major layout.
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
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 .
ipiv
Array, size at least max(1, n) . The ipiv array, as returned by ?getrf (Computes the LU factorization of a general m-by-n matrix.) .
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
\[|E| \le c(n) \varepsilon P|L||U|\]
\(c(n)\) 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 condition number of \(A^{T}\) and \(A^{H}\) might or might not be equal to \(\kappa_\infty (A)\) .
The approximate number of floating-point operations for one right-hand side vector b is \(2n^{2}\) for real flavors and \(8n^{2}\) for complex flavors.
To estimate the condition number \(\kappa_\infty (A)\) , call ?gecon (Estimates the reciprocal of the condition number of a general matrix in the 1-norm or the infinity-norm.) .
To refine the solution and estimate the error, call ?gerfs (Refines the solution of a system of linear equations with a general coefficient matrix and estimates its error.) .