Developer Reference for Intel® oneAPI Math Kernel Library for C
?pptrs
Solves a system of linear equations with a packed Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix.
Syntax
lapack_int LAPACKE_spptrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const float *ap, float *b, lapack_int ldb);
lapack_int LAPACKE_dpptrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const double *ap, double *b, lapack_int ldb);
lapack_int LAPACKE_cpptrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const lapack_complex_float *ap, lapack_complex_float *b,
lapack_int ldb);
lapack_int LAPACKE_zpptrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,
const lapack_complex_double *ap, 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 packed symmetric positive-definite or, for complex data, Hermitian positive-definite matrix \(A\) , given the Cholesky factorization of \(A\) :
\(A = U^{T}*U\) for real data, \(A = U^{H}*U\) for complex data |
if uplo='U' |
\(A = L*L^{T}\) for real data, \(A = L*L^{H}\) for complex data |
if uplo='L' |
where \(L\) is a lower triangular matrix and \(U\) is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix \(B\) .
Before calling this routine, you must call ?pptrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.) to compute the Cholesky factorization of \(A\) .
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' , U is stored, where \(A = U^T * U\) for real data, \(A = U^H * U\) for complex data.
If uplo = 'L' , L is stored, where \(A = L * L^T\) for real data, \(A = L * L^H\) for complex data
- 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).
ap , b
The size of ap must be at least max(1, n(n+1)/2).
The array ap contains the factor U or L , as specified by uplo , in packed storage (see Matrix Storage Schemes ).
b
The array b of size max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) 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 .
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
\[|E| \leq c(n) \varepsilon |U| |U^{H}|\]
\(c(n)\) 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 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 ?ppcon (Estimates the reciprocal of the condition number of a packed symmetric (Hermitian) positive-definite matrix.) .
To refine the solution and estimate the error, call ?pprfs (Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite coefficient matrix stored in a packed format and estimates its error.) .