NAG Library Routine Document
f07hrf (zpbtrf) computes the Cholesky factorization of a complex Hermitian positive definite band matrix.
|Integer, Intent (In)||:: ||n, kd, ldab|
|Integer, Intent (Out)||:: ||info|
|Complex (Kind=nag_wp), Intent (Inout)||:: ||ab(ldab,*)|
|Character (1), Intent (In)||:: ||uplo|C Header Interface
f07hrf_ (const char *uplo, const Integer *n, const Integer *kd, Complex ab, const Integer *ldab, Integer *info, const Charlen length_uplo)|
The routine may be called by its
f07hrf (zpbtrf) forms the Cholesky factorization of a complex Hermitian positive definite band matrix either as if or if , where (or ) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as .
Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14
University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn14.pdf
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 1: – Character(1)Input
: specifies whether the upper or lower triangular part of
is stored and how
is to be factorized.
- The upper triangular part of is stored and is factorized as , where is upper triangular.
- The lower triangular part of is stored and is factorized as , where is lower triangular.
- 2: – IntegerInput
On entry: , the order of the matrix .
- 3: – IntegerInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
- 4: – Complex (Kind=nag_wp) arrayInput/Output
the second dimension of the array ab
must be at least
Hermitian positive definite band matrix
The matrix is stored in rows
, more precisely,
- if , the elements of the upper triangle of within the band must be stored with element in ;
- if , the elements of the lower triangle of within the band must be stored with element in
: the upper or lower triangle of
is overwritten by the Cholesky factor
as specified by uplo
, using the same storage format as described above.
- 5: – IntegerInput
: the first dimension of the array ab
as declared in the (sub)program from which f07hrf (zpbtrf)
- 6: – IntegerOutput
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value.
, dynamic memory allocation failed. See Section 3.7
in How to Use the NAG Library and its Documentation for further information. An explanatory message is output, and execution of the program is terminated.
The leading minor of order
is not positive definite and the factorization could not be completed. Hence
itself is not positive definite. This may indicate an error in forming the matrix
. There is no routine specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07brf (zgbtrf)
or as a full Hermitian matrix, by calling f07mrf (zhetrf)
, the computed factor
is the exact factor of a perturbed matrix
is a modest linear function of
is the machine precision
If , a similar statement holds for the computed factor . It follows that .
Parallelism and Performance
f07hrf (zpbtrf) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction
for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note
for your implementation for any additional implementation-specific information.
The total number of real floating-point operations is approximately , assuming .
A call to f07hrf (zpbtrf)
may be followed by calls to the routines:
The real analogue of this routine is f07hdf (dpbtrf)
This example computes the Cholesky factorization of the matrix
Program Text (f07hrfe.f90)
Program Data (f07hrfe.d)
Program Results (f07hrfe.r)