Solves a system of linear equations with a UDU- or LDL-factored Hermitian matrix using packed storage.
Syntax
FORTRAN 77:
call chptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )
call zhptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )
FORTRAN 95:
call hptrs( ap, b, ipiv [,uplo] [,info] )
C:
lapack_int LAPACKE_<?>hptrs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const <datatype>* ap, const lapack_int* ipiv, <datatype>* b, lapack_int ldb );
Description
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 = P*U*D*UH*PT |
if uplo='L', |
A = P*L*D*LH*PT, |
where P is a permutation matrix, 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.
Input Parameters
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
uplo |
CHARACTER*1. 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 = P*U*D*UH*PT. If uplo = 'L', the array ap stores the packed factor L of the factorization A = P*L*D*LH*PT. |
n |
INTEGER. The order of matrix A; n ≥ 0. |
nrhs |
INTEGER. The number of right-hand sides; nrhs ≥ 0. |
ipiv |
INTEGER. Array, DIMENSION at least max(1, n). The ipiv array, as returned by ?hptrf. |
ap, b |
COMPLEX for chptrs DOUBLE COMPLEX for zhptrs. Arrays: ap(*), b(ldb,*). The dimension 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). The array b contains the matrix B whose columns are the right-hand sides for the system of equations. The second dimension of b must be at least max(1, nrhs). |
ldb |
INTEGER. The leading dimension of b; ldb ≥ max(1, n). |
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. |
Fortran 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 Fortran 95 Interface Conventions.
Specific details for the routine hptrs interface are as follows:
ap |
Holds the array A of size (n*(n+1)/2). |
b |
Holds the matrix B of size (n, nrhs). |
ipiv |
Holds the vector of length n. |
uplo |
Must be 'U' or 'L'. The default value is 'U'. |
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| ≤ c(n)ε P|U||D||UH|PT or |E| ≤ c(n)ε P|L||D||LH|PT
c(n) is a modest linear function of n, and ε is the machine precision.
If x0 is the true solution, the computed solution x satisfies this error bound:
where cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞ ≤ ||A-1||∞ ||A||∞ = κ∞(A).
Note that cond(A,x) can be much smaller than κ∞(A).
The total number of floating-point operations for one right-hand side vector is approximately 8n2 for complex flavors.
To estimate the condition number κ∞(A), call ?hpcon.
To refine the solution and estimate the error, call ?hprfs.