f01ff computes the matrix function, fA, of a complex Hermitian n by n matrix A. fA must also be a complex Hermitian matrix.

Syntax

C#
public static void f01ff(
	string uplo,
	int n,
	Complex[,] a,
	F01..::..F01FF_F f,
	out int iflag,
	out int ifail
)
Visual Basic
Public Shared Sub f01ff ( _
	uplo As String, _
	n As Integer, _
	a As Complex(,), _
	f As F01..::..F01FF_F, _
	<OutAttribute> ByRef iflag As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void f01ff(
	String^ uplo, 
	int n, 
	array<Complex,2>^ a, 
	F01..::..F01FF_F^ f, 
	[OutAttribute] int% iflag, 
	[OutAttribute] int% ifail
)
F#
static member f01ff : 
        uplo : string * 
        n : int * 
        a : Complex[,] * 
        f : F01..::..F01FF_F * 
        iflag : int byref * 
        ifail : int byref -> unit 

Parameters

uplo
Type: System..::..String
On entry: if uplo="U", the upper triangle of the matrix A is stored.
If uplo="L", the lower triangle of the matrix A is stored.
Constraint: uplo="U" or "L".
n
Type: System..::..Int32
On entry: n, the order of the matrix A.
Constraint: n0.
a
Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: dim1max1,n
Note: the second dimension of the array a must be at least n.
On entry: the n by n Hermitian matrix A.
  • If uplo="U", the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo="L", the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ifail=0, the upper or lower triangular part of the n by n matrix function, fA.
f
Type: NagLibrary..::..F01..::..F01FF_F
The method f evaluates fzi at a number of points zi.

A delegate of type F01FF_F.

iflag
Type: System..::..Int32%
On exit: iflag=0, unless you have set iflag nonzero inside f, in which case iflag will be the value you set and ifail will be set to ifail=-6.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

fA is computed using a spectral factorization of A 
A=QDQH,
where D is the real diagonal matrix whose diagonal elements, di, are the eigenvalues of A, Q is a unitary matrix whose columns are the eigenvectors of A and QH is the conjugate transpose of Q. fA is then given by
fA=QfDQH,
where fD is the diagonal matrix whose ith diagonal element is fdi. See for example Section 4.5 of Higham (2008). fdi is assumed to be real.

References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Error Indicators and Warnings

Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (LDA) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
ifail<0andifail-999 or -6
If ifail=-i, the ith argument had an illegal value.
ifail=-6
iflag has been set nonzero by the user.
ifail=-999
Internal memory allocation failed.
The integer allocatable memory required is n, the real allocatable memory required is 4×n-2 and the complex allocatable memory required is approximately n+nb+1×n, where nb is the block size required by f08fn.
ifail=iandifail>0
The algorithm to compute the spectral factorization failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see f08fn).
Note:  this failure is unlikely to occur.
ifail=-9000
An error occured, see message report.
ifail=-6000
Invalid Parameters value
ifail=-4000
Invalid dimension for array value
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value

Accuracy

Provided that fD 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.

Parallelism and Performance

None.

Further Comments

The cost of the algorithm is On3 plus the cost of evaluating fD. If λ^i is the ith computed eigenvalue of A, then the user-supplied method f will be asked to evaluate the function f at fλ^i, for i=1,2,,n.
For further information on matrix functions, see Higham (2008).
f01ef can be used to find the matrix function fA for a real symmetric matrix A.

Example

This example finds the matrix cosine, cosA, of the Hermitian matrix
A=12+i3+2i4+3i2-i12+i3+2i3-2i2-i12+i4-3i3-2i2-i1.

Example program (C#): f01ffe.cs

Example program data: f01ffe.d

Example program results: f01ffe.r

See Also