Developer Reference for Intel® oneAPI Math Kernel Library for C
?hptrs
Solves a system of linear equations with a UDU- or LDL-factored Hermitian coefficient matrix using packed storage.
Syntax
lapack_intLAPACKE_chptrs ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , constlapack_complex_float*ap , constlapack_int*ipiv , lapack_complex_float*b , lapack_intldb );
lapack_intLAPACKE_zhptrs ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , constlapack_complex_double*ap , constlapack_int*ipiv , lapack_complex_double*b , lapack_intldb );
Include Files
mkl.h
Description
chptrs zhptrs hptrs
The routine solves for X the system of linear equations A*X = B with a Hermitian matrix A , given the Bunch-Kaufman factorization of A :
if uplo='U' ,
\(A = UDU^{H}\)
if uplo='L' ,
\(A = LDL^{H}\) ,
where U and L are upper and lower packed 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 arrays ap (containing U or L )and ipiv in the form returned by the factorization routine ?hptrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix using packed storage.) .
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 ap stores the packed factor U of the factorization A = U*D*U^{H} . If uplo = 'L' , the array ap stores the packed factor L of the factorization A = L*D*L^{H} .
n
The order of matrix A ; n ≥ 0.
nrhs
The number of right-hand sides; nrhs ≥ 0.
ipiv
Array, size at least max(1, n) . The ipiv array, as returned by ?hptrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix using packed storage.) .
ap
The dimension of array 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(ldb,*) contains the matrix B whose columns are the right-hand sides for the system of equations. The size of b is max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout.
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 \(8n^{2}\) for complex flavors.
To estimate the condition number \(\kappa_{infty}(A)\) , call ?hpcon (Estimates the reciprocal of the condition number of a packed Hermitian matrix.) .
To refine the solution and estimate the error, call ?hprfs (Refines the solution of a system of linear equations with a packed complex Hermitian coefficient matrix and estimates the solution error.) .