NAG Library Routine Document
F01EFF computes the matrix function, , of a real symmetric by matrix . must also be a real symmetric matrix.
||N, LDA, IUSER(*), IFLAG, IFAIL
is computed using a spectral factorization of
is the diagonal matrix whose diagonal elements,
, are the eigenvalues of
is an orthogonal matrix whose columns are the eigenvectors of
is then given by
is the diagonal matrix whose
th diagonal element is
. See for example Section 4.5 of Higham (2008)
is assumed to be real.
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
- 1: UPLO – CHARACTER(1)Input
, the upper triangle of the matrix
If , the lower triangle of the matrix is stored.
- 2: N – INTEGERInput
On entry: , the order of the matrix .
- 3: A(LDA,) – REAL (KIND=nag_wp) arrayInput/Output
the second dimension of the array A
must be at least
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if , the upper or lower triangular part of the by matrix function, .
- 4: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F01EFF is called.
- 5: F – SUBROUTINE, supplied by the user.External Procedure
The subroutine F
at a number of points
The specification of F
||IFLAG, N, IUSER(*)
||X(N), FX(N), RUSER(*)
- 1: IFLAG – INTEGERInput/Output
will be zero.
should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function
; for instance
may not be defined, or may be complex. If IFLAG
is returned as nonzero then F01EFF will terminate the computation, with
- 2: N – INTEGERInput
On entry: , the number of function values required.
- 3: X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the points at which the function is to be evaluated.
- 4: FX(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the function values.
should return the value , for .
- 5: IUSER() – INTEGER arrayUser Workspace
- 6: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
is called with the parameters IUSER
as supplied to F01EFF. You are free to use the arrays IUSER
to supply information to F
as an alternative to using COMMON global variables.
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F01EFF is called. Parameters denoted as Input
be changed by this procedure.
- 6: IUSER() – INTEGER arrayUser Workspace
- 7: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
are not used by F01EFF, but are passed directly to F
and may be used to pass information to this routine as an alternative to using COMMON global variables.
- 8: IFLAG – INTEGEROutput
, unless you have set IFLAG
nonzero inside F
, in which case IFLAG
will be the value you set and IFAIL
will be set to
- 9: 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:
If , the th argument had an illegal value.
has been set nonzero by the user.
Internal memory allocation failed.
The integer allocatable memory required is N
, and the real allocatable memory required is approximately
, where nb
is the block size required by F08FAF (DSYEV)
The algorithm to compute the spectral factorization failed to converge;
off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see F08FAF (DSYEV)
Note: this failure is unlikely to occur.
can be computed accurately then the computed matrix function will be close to the exact matrix function. See Section 10.2 of Higham (2008)
for details and further discussion.
The cost of the algorithm is
plus the cost of evaluating
th computed eigenvalue of
, then the user-supplied subroutine F
will be asked to evaluate the function
For further information on matrix functions, see Higham (2008)
can be used to find the matrix function
for a complex Hermitian matrix
This example finds the matrix cosine,
, of the symmetric matrix
9.1 Program Text
Program Text (f01effe.f90)
9.2 Program Data
Program Data (f01effe.d)
9.3 Program Results
Program Results (f01effe.r)