NAG Library Routine Document
F01BUF performs a decomposition of a real symmetric positive definite band matrix.
||N, M1, K, LDA, IFAIL
The symmetric positive definite matrix , of order and bandwidth , is divided into the leading principal sub-matrix of order and its complement, where . A decomposition of the latter and an decomposition of the former are obtained by means of a sequence of elementary transformations, where is unit upper triangular, is unit lower triangular and is diagonal. Thus if , an decomposition of is obtained.
This routine is specifically designed to precede F01BVF
for the transformation of the symmetric-definite eigenproblem
by the method of Crawford where
are of band form. In this context,
is chosen to be close to
and the decomposition is applied to the matrix
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
- 1: N – INTEGERInput
On entry: , the order of the matrix .
- 2: M1 – INTEGERInput
On entry: , where is the number of nonzero superdiagonals in . Normally .
- 3: K – INTEGERInput
On entry: , the change-over point in the decomposition.
- 4: A(LDA,N) – REAL (KIND=nag_wp) arrayInput/Output
: the upper triangle of the
symmetric band matrix
, with the diagonal of the matrix stored in the
th row of the array, and the
superdiagonals within the band stored in the first
rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if
, the storage scheme is
Elements in the top left corner of the array are not used. The following code assigns the matrix elements within the band to the correct elements of the array:
DO 20 J = 1, N
DO 10 I = MAX(1,J-M1+1), J
A(I-J+M1,J) = matrix(I,J)
On exit: is overwritten by the corresponding elements of , and .
- 5: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F01BUF is called.
- 6: W(M1) – REAL (KIND=nag_wp) arrayWorkspace
- 7: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
is not positive definite, perhaps as a result of rounding errors, giving an element of
which is zero or negative.
when the failure occurs in the leading principal sub-matrix of order K
when it occurs in the complement.
The Cholesky decomposition of a positive definite matrix is known for its remarkable numerical stability (see Wilkinson (1965)
). The computed
satisfy the relation
are related by
is a constant of order unity and
is the machine precision
. In practice, the error is usually appreciably smaller than this.
The time taken by F01BUF is approximately proportional to .
This routine is specifically designed for use as the first stage in the solution of the generalized symmetric eigenproblem by Crawford's method which preserves band form in the transformation to a similar standard problem. In this context, for maximum efficiency, should be chosen as the multiple of nearest to .
The matrix is such that is diagonal in its last rows and columns, is such that and is diagonal. To find , and where requires multiplications and divisions which, is independent of .
This example finds a
decomposition of the real symmetric positive definite matrix
9.1 Program Text
Program Text (f01bufe.f90)
9.2 Program Data
Program Data (f01bufe.d)
9.3 Program Results
Program Results (f01bufe.r)