f01ed computes the matrix exponential, , of a real symmetric by matrix .
Public Shared Sub f01ed ( _ uplo As String, _ n As Integer, _ a As Double(,), _ <OutAttribute> ByRef ifail As Integer _ )
public: static void f01ed( String^ uplo, int n, array<double,2>^ a, [OutAttribute] int% ifail )
- Type: System..::..StringOn entry: indicates whether the upper or lower triangular part of is stored.
Constraint: or .
- The upper triangular part of is stored.
- The lower triangular part of is stored.
- Type: System..::..Int32On entry: , the order of the matrix .Constraint: .
- Type: array<System..::..Double,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 symmetric 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 an orthogonal 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 (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.
- On entry, or .
- On entry, .
- An unexpected internal error occurred when computing the spectral factorization. Please contact NAG.
- On entry, .
- The algorithm to compute the spectral factorization failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see f08fa).Note: this failure is unlikely to occur.
For a symmetric matrix , the matrix , has the relative condition number
which is the minimum 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 symmetric matrix