The routine may be called by the names f07hff, nagf_lapacklin_dpbequ or its LAPACK name dpbequ.
f07hff computes a diagonal scaling matrix chosen so that
This means that the matrix given by
has diagonal elements equal to unity. This in turn means that the condition number of , , is within a factor of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
1: – Character(1)Input
On entry: indicates whether the upper or lower triangular part of is stored in the array ab, as follows:
The upper triangle of is stored.
The lower triangle of is stored.
2: – IntegerInput
On entry: , the order of the matrix .
3: – IntegerInput
On entry: , the number of superdiagonals of the matrix if , or the number of subdiagonals if .
4: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab
must be at least
On entry: the upper or lower triangle of the symmetric positive definite band matrix whose scaling factors are to be computed.
The matrix is stored in rows to , 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
Only the elements of the array ab corresponding to the diagonal elements of are referenced. (Row of ab when , row of ab when .)
5: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hff is called.
6: – Real (Kind=nag_wp) arrayOutput
On exit: if , s contains the diagonal elements of the scaling matrix .
7: – Real (Kind=nag_wp)Output
On exit: if , scond contains the ratio of the smallest value of s to the largest value of s. If and amax is neither too large nor too small, it is not worth scaling by .
8: – Real (Kind=nag_wp)Output
On exit: . If amax is very close to overflow or underflow, the matrix should be scaled.
9: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The th diagonal element of is not positive
(and hence cannot be positive definite).
The computed scale factors will be close to the exact scale factors.