f01ff computes the matrix function, , of a complex Hermitian by matrix . must also be a complex Hermitian matrix.
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 _ )
public: static void f01ff( String^ uplo, int n, array<Complex,2>^ a, F01..::..F01FF_F^ f, [OutAttribute] int% iflag, [OutAttribute] int% ifail )
- Type: System..::..StringOn entry: if , the upper triangle of the matrix is stored.If , the lower triangle of the matrix is stored.Constraint: or .
- Type: System..::..Int32On entry: , the order of the matrix .Constraint: .
- Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]An array of size [dim1, dim2]Note: dim1 must satisfy the constraint:Note: the second dimension of the array a must be at least .On entry: the by Hermitian matrix .
On exit: if , the upper or lower triangular part of the by matrix function, .
- 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.
- Type: NagLibrary..::..F01..::..F01FF_FThe method f evaluates at a number of points .
A delegate of type F01FF_F.
- Type: System..::..Int32%
is computed using a spectral factorization of
where is the real diagonal matrix whose diagonal elements, , are the eigenvalues of , is a unitary matrix whose columns are the eigenvectors of and is the conjugate transpose of . is then given by
where 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
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.
- If , the th argument had an illegal value.
- iflag has been set nonzero by the user.
- Internal memory allocation failed.
- The algorithm to compute the spectral factorization failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see f08fn).Note: this failure is unlikely to occur.
Provided that 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 . If is the th computed eigenvalue of , then the user-supplied method f will be asked to evaluate the function at , for .
For further information on matrix functions, see Higham (2008).
f01ef can be used to find the matrix function for a real symmetric matrix .
This example finds the matrix cosine, , of the Hermitian matrix