NAG Library Routine Document
F07HDF (DPBTRF) computes the Cholesky factorization of a real symmetric positive definite band matrix.
||N, KD, LDAB, INFO
The routine may be called by its
F07HDF (DPBTRF) forms the Cholesky factorization of a real symmetric 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: – REAL (KIND=nag_wp) arrayInput/Output
the second dimension of the array AB
must be at least
symmetric 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 F07HDF (DPBTRF) is called.
- 6: – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
If , argument had an illegal value. 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 symmetric band matrix which is
not positive definite; the matrix must be treated either as a nonsymmetric
band matrix, by calling F07BDF (DGBTRF)
or as a full symmetric matrix, by
calling F07MDF (DSYTRF)
, 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 .
8 Parallelism and Performance
F07HDF (DPBTRF) is not threaded by NAG in any implementation.
F07HDF (DPBTRF) 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 floating-point operations is approximately , assuming .
A call to F07HDF (DPBTRF) may be followed by calls to the routines:
The complex analogue of this routine is F07HRF (ZPBTRF)
This example computes the Cholesky factorization of the matrix
10.1 Program Text
Program Text (f07hdfe.f90)
10.2 Program Data
Program Data (f07hdfe.d)
10.3 Program Results
Program Results (f07hdfe.r)