NAG Library Routine Document
F07HRF (ZPBTRF) computes the Cholesky factorization of a complex Hermitian positive definite band matrix.
||N, KD, LDAB, INFO
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 1: UPLO – 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: N – INTEGERInput
On entry: , the order of the matrix .
- 3: KD – INTEGERInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
- 4: AB(LDAB,) – 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: LDAB – INTEGERInput
: the first dimension of the array AB
as declared in the (sub)program from which F07HRF (ZPBTRF) is called.
- 6: INFO – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th parameter 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
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
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 .
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
9.1 Program Text
Program Text (f07hrfe.f90)
9.2 Program Data
Program Data (f07hrfe.d)
9.3 Program Results
Program Results (f07hrfe.r)