f07hd computes the Cholesky factorization of a real symmetric positive definite band matrix.
- Type: System..::..StringOn entry: specifies whether the upper or lower triangular part of is stored and how is to be factorized.
Constraint: or .
- 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.
- Type: System..::..Int32On entry: , the order of the matrix .Constraint: .
- Type: System..::..Int32On entry: , the number of superdiagonals or subdiagonals of the matrix .Constraint: .
- Type: array<System..::..Double,2>[,](,)[,][,]An array of size [dim1, dim2]Note: dim1 must satisfy the constraint:Note: the second dimension of the array ab must be at least .On entry: the by symmetric positive definite band matrix .The matrix is stored in rows to , more precisely,
On exit: the upper or lower triangle of is overwritten by the Cholesky factor or as specified by uplo, using the same storage format as described above.
- 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
f07hd 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Some error messages may refer to parameters that are dropped from this interface (LDAB) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
- 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 method 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 f07bd or as a full symmetric matrix, by calling f07md.
If , the computed factor is the exact factor of a perturbed matrix , where
is a modest linear function of , and is the machine precision.
If , a similar statement holds for the computed factor . It follows that .
The total number of floating-point operations is approximately , assuming .
A call to f07hd may be followed by calls to the methods:
- f07he to solve ;
- (F07HGF not in this release) to estimate the condition number of .
The complex analogue of this method is (F07HRF not in this release).
This example computes the Cholesky factorization of the matrix , where