f01fd computes the matrix exponential, , of a complex Hermitian by matrix .
Public Shared Sub f01fd ( _ uplo As String, _ n As Integer, _ a As Complex(,), _ <OutAttribute> ByRef ifail As Integer _ )
public: static void f01fd( String^ uplo, int n, array<Complex,2>^ a, [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 exponential, .
- 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.
is computed using a spectral factorization of
where is the diagonal matrix whose diagonal elements, , are the eigenvalues of , and is a unitary matrix whose columns are the eigenvectors of . is then given by
where is the diagonal matrix whose th diagonal element is . See for example Section 4.5 of Higham (2008).
Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl. 26(4) 1179–1193
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Rev. 45 3–49
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.
- 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.
For an Hermitian matrix , the matrix , has the relative condition number
which is the minimal possible for the matrix exponential and so the computed matrix exponential is guaranteed to be close to the exact matrix. See Section 10.2 of Higham (2008) for details and further discussion.
The cost of the algorithm is .
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).
This example finds the matrix exponential of the Hermitian matrix