Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?getrs
Solves a system of linear equations with an LU-factored square coefficient matrix, with multiple right-hand sides.
Syntax
call sgetrs ( trans , n , nrhs , a , lda , ipiv , b , ldb , info )
call dgetrs ( trans , n , nrhs , a , lda , ipiv , b , ldb , info )
call cgetrs ( trans , n , nrhs , a , lda , ipiv , b , ldb , info )
call zgetrs ( trans , n , nrhs , a , lda , ipiv , b , ldb , info )
call getrs ( a , ipiv , b [ , trans ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
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
trans
CHARACTER*1 . 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
INTEGER . The order of A ; the number of rows in B(n≥ 0) .
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
a , b
REAL for sgetrs
DOUBLE PRECISION for dgetrs
COMPLEX for cgetrs
DOUBLE COMPLEX for zgetrs .
Arrays: a (size lda by *), b (size ldb by *).
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.) .
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
The second dimension of a must be at least max(1,n) and the second dimension of b at least max(1,nrhs) .
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
INTEGER . The leading dimension of a ; lda≥ max(1, n) .
ldb
INTEGER . The leading dimension of b ; ldb≥ max(1, n) .
ipiv
INTEGER .
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 .
info
INTEGER .
If info = 0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine getrs interface are as follows:
a
Holds the matrix A of size ( n , n ).
b
Holds the matrix B of size ( n , nrhs ).
ipiv
Holds the vector of length n .
trans
Must be ‘N’ , ‘C’ , or ‘T’ . The default value is ‘N’ .
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.) .