The routine may be called by the names f07gtf, nagf_lapacklin_zppequ or its LAPACK name zppequ.
f07gtf 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 ap, as follows:
The upper triangle of is stored.
The lower triangle of is stored.
2: – IntegerInput
On entry: , the order of the matrix .
3: – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array ap
must be at least
On entry: the Hermitian matrix , packed by columns.
if , the upper triangle of must be stored with element in for ;
if , the lower triangle of must be stored with element in for .
Only the elements of ap corresponding to the diagonal elements are referenced.
4: – Real (Kind=nag_wp) arrayOutput
On exit: if , s contains the diagonal elements of the scaling matrix .
5: – 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 .
6: – Real (Kind=nag_wp)Output
On exit: . If amax is very close to overflow or underflow, the matrix should be scaled.
7: – 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.