Developer Reference for Intel® oneAPI Math Kernel Library for C
?gttrs
Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by ?gttrf .
Syntax
lapack_int LAPACKE_sgttrs(int matrix_layout, char trans, lapack_int n, lapack_int nrhs,
const float *dl, const float *d, const float *du,
const float *du2, const lapack_int *ipiv, float *b,
lapack_int ldb);
lapack_int LAPACKE_dgttrs(int matrix_layout, char trans, lapack_int n, lapack_int nrhs,
const double *dl, const double *d, const double *du,
const double *du2, const lapack_int *ipiv, double *b,
lapack_int ldb);
lapack_int LAPACKE_cgttrs(int matrix_layout, char trans, lapack_int n, lapack_int nrhs,
const lapack_complex_float *dl, const lapack_complex_float *d,
const lapack_complex_float *du, const lapack_complex_float *du2,
const lapack_int *ipiv, lapack_complex_float *b,
lapack_int ldb);
lapack_int LAPACKE_zgttrs(int matrix_layout, char trans, lapack_int n, lapack_int nrhs,
const lapack_complex_double *dl, const lapack_complex_double *d,
const lapack_complex_double *du,
const lapack_complex_double *du2, const lapack_int *ipiv,
lapack_complex_double *b, lapack_int ldb);Include Files
mkl.h
Description
The routine solves for \(X\) in the following systems of linear equations with multiple right hand sides:
\(AX = B\) 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 ?gttrf (Computes the LU factorization of a tridiagonal matrix.) to compute the LU factorization of A .
Input Parameters
matrix_layout
Specifies whether matrix storage layout for array b 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 ; n ≥ 0.
nrhs
The number of right-hand sides, that is, the number of columns in B; nrhs≥ 0 .
dl , d , du , du2 , b
Arrays: dl(n -1) , d(n) , du(n -1) , du2(n -2) .
The array dl contains the (n - 1) multipliers that define the matrix L from the LU factorization of A .
The array d contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A .
The array du contains the ( n - 1) elements of the first superdiagonal of U .
The array du2 contains the ( n - 2) elements of the second superdiagonal of U .
b
Array of size max(1, ldb * nrhs ) for column major layout and max(1, n * ldb ) for row major layout. Contains the matrix B whose columns are the right-hand sides for the systems of equations.
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 ( n ). The ipiv array, as returned by ?gttrf (Computes the LU factorization of a tridiagonal 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:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(kl + ku + 1) \operatorname{cond}(A, x) \varepsilon\]
where
\[\operatorname{cond(A,x)} = || |A^{-1}| |A| |x|_\infty || / ||x||_\infty \leq ||A^{-1}||_\infty ||A||_\infty = \kappa_\infty(A).\]
Note that \(cond(A,x)\) 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 \(7n\) (including \(n\) divisions) for real flavors and \(34n\) (including \(2n\) divisions) for complex flavors.
To estimate the condition number \(\kappa_\infty (A)\) , call ?gtcon (Estimates the reciprocal of the condition number of a tridiagonal matrix.) .
To refine the solution and estimate the error, call ?gtrfs (Refines the solution of a system of linear equations with a tridiagonal coefficient matrix and estimates its error.) .