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Numerical eigenvalues and eigenvectors of a matrix
This functionality does not run in MATLAB.
numeric::eigenvectors(A, options)
numeric::eigenvectors(A) returns numerical eigenvalues and eigenvectors of the matrix A.
All entries of the matrix must be numerical. Numerical expressions such as etc. are accepted and converted to floats. Non-numerical symbolic entries lead to an error.
The eigenvalues are sorted by numeric::sort.
The matrix X provides the eigenvectors: the i-th column of X is a numerical eigenvector corresponding to the eigenvalue d_{i}. Each column is either zero or normalized to the Euclidean length 1.0.
For matrices with multiple eigenvalues and an insufficient number of eigenvectors, some of the eigenvectors may coincide or may be zero, i.e., X is not necessarily invertible.
The list of residues res = [res_{1}, res_{2}, …] provides some control over the quality of the numerical spectral data. The residues are given by
,
where x_{i} is the normalized eigenvector (the i-th column of X) associated with the numerical eigenvalue d_{i}. For Hermitian matrices, res_{i} provides an upper bound for the absolute error of d_{i}.
With the option NoResidues, the computation of the residues is suppressed, the returned value is NIL.
If no return type is specified via the option ReturnType = t, the domain type of the eigenvector matrix X depends on the type of the input matrix A:
The eigenvectors of a dense matrix of type Dom::DenseMatrix() are returned as a dense matrix of type Dom::DenseMatrix() over the ring of expressions.
For all other matrices of category Cat::Matrix, the eigenvectors are returned as matrices of type Dom::Matrix() over the ring of MuPAD^{®} expressions. This includes input matrices A of type Dom::Matrix(...), Dom::SquareMatrix(...), Dom::MatrixGroup(...) etc.
Note: Matrices A of a matrix domain such as Dom::Matrix(...) or Dom::SquareMatrix(...) are internally converted to arrays over expressions via expr(A). Note that linalg::eigenvectors must be used, when the eigenvalues/vectors are to be computed over the component domain. Cf. Example 3. |
Note: Eigenvalues are approximated with an absolute precision of , where r is the spectral radius of A (i.e., r is the maximal singular value of A). Consequently, large eigenvalues should be computed correctly to DIGITS decimal places. The numerical approximations of the small eigenvalues are less accurate. |
Note: For a numerical algorithm, it is not possible to distinguish between badly separated distinct eigenvalues and multiple eigenvalues. For this reason, numeric::eigenvectors and linalg::eigenvectors use different return formats: the latter can provide information on the multiplicity of eigenvalues due to its internal exact arithmetic. |
Use numeric::eigenvalues if only eigenvalues are to be computed.
The function is sensitive to the environment variable DIGITS, which determines the numerical working precision.
We compute the spectral data of the 2×2 Hilbert matrix:
A := linalg::hilbert(2)
[d, X, res] := numeric::eigenvectors(A):
The eigenvalues:
d
The eigenvectors:
X
Hilbert matrices are Hermitian, i.e., computing the spectral data is a numerically stable process. This is confirmed by the small residues:
res
The routine linalg::hilbert provides the input as a matrix of type Dom::Matrix(). Consequently, the eigenvectors also consist of such a matrix. For further processing, we convert the list of eigenvalues to a diagonal matrix:
d := matrix(2, 2, d, Diagonal):
We reconstruct the matrix from its spectral data:
X*d*X^(-1)
We extract an eigenvector from the matrix X and doublecheck its numerical quality:
eigenvector1 := X::dom::col(X, 1); norm(A*eigenvector1 - d[1, 1]*eigenvector1)
delete A, d, X, res, eigenvector1:
We demonstrate a numerically ill-conditioned case. The following matrix has only one eigenvector and cannot be diagonalized. Numerically, the zero vector is returned as the second column of the eigenvector matrix:
A := array(1..2, 1..2, [[5, -1], [4, 1]]): DIGITS := 6: numeric::eigenvectors(A)
delete A, DIGITS:
The following matrix has domain components:
A := Dom::Matrix(Dom::IntegerMod(7))([[6, -1], [0, 3]])
Note that numeric::eigenvectors computes the spectral data of the following matrix:
expr(A)
numeric::eigenvectors(A, NoResidues)
The routine linalg::eigenvectors should be used if the spectral data are to be computed over the component domain Dom::IntegerMod(7):
linalg::eigenvectors(A)
delete A:
We demonstrate the use of hardware floats. The following matrix is degenerate: it has rank 1. For the double eigenvalue 0, different base vectors of the corresponding eigenspace are returned with HardwareFloats and SoftwareFloats, respectively:
A := array(1..3, 1..3, [[1, 2, 3], [2, 4, 6], [3*10^12, 6*10^12, 9*10^12]]): [d1, X1, res1] := numeric::eigenvectors(A, HardwareFloats): d1, X1
[d2, X2, res2] := numeric::eigenvectors(A, SoftwareFloats): d2, X2
delete A, d1, X1, res1, d2, X2, res2:
A |
A numerical matrix of domain type DOM_ARRAY, or DOM_HFARRAY, or of category Cat::Matrix. |
Hard, HardwareFloats, Soft, SoftwareFloats |
With Hard (or HardwareFloats), computations are done using fast hardware float arithmetic from within a MuPAD session. Hard and HardwareFloats are equivalent. With this option, the input data are converted to hardware floats and processed by compiled C code. The result is reconverted to MuPAD floats and returned to the MuPAD session. With Soft (or SoftwareFloats) computations are dome using software float arithmetic provided by the MuPAD kernel. Soft and SoftwareFloats are equivalent. SoftwareFloats is used by default if the current value of DIGITS is larger than 15 and the input matrix A is not of domain type DOM_HFARRAY. Compared to the SoftwareFloats used by the MuPAD kernel, the computation with HardwareFloats may be many times faster. Note, however, that the precision of hardware arithmetic is limited to about 15 digits. Further, the size of floating-point numbers may not be larger than approximately 10^{308} and not smaller than approximately 10^{- 308}. If no HardwareFloats or SoftwareFloats are requested explicitly, the following strategy is used: If the current value of DIGITS is smaller than 16 or if the matrix A is a hardware float array of domain type DOM_HFARRAY, then hardware arithmetic is tried. If this is successful, the result is returned. If the result cannot be computed with hardware floats, software arithmetic by the MuPAD kernel is tried. If the current value of DIGITS is larger than 15 and the input matrix A is not of domain type DOM_HFARRAY, or if one of the options Soft, SoftwareFloats or Symbolic is specified, MuPAD computes the result with its software arithmetic without trying to use hardware floats first. There may be several reasons for hardware arithmetic to fail:
If neither HardwareFloats nor SoftwareFloats is specified, the user is not informed whether hardware floats or software floats are used. If HardwareFloats are specified but fail due to one of the reasons above, a warning is issued that the (much slower) software floating-point arithmetic of the MuPAD kernel is used. Note that HardwareFloats can only be used if all input data can be converted to floating-point numbers. The trailing digits in floating-point results computed with HardwareFloats and SoftwareFloats may differ.
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NoResidues |
Suppresses the computation of error estimates If no error estimates are required, this option may be used to suppress the computation of the residues res. The return values for these data are NIL. The alternative option name NoErrors used in previous MuPAD versions is still available. | |
ReturnType |
Option, specified as ReturnType = t Return the eigenvectors as a matrix of domain type t. The following return types t are available: DOM_ARRAY, DOM_HFARRAY, Dom::Matrix(), or Dom::DenseMatrix(). | |
NoWarning |
Suppresses warnings |
List [d, X, res]. The list d = [d_{1}, d_{2}, …] contains the numerical eigenvalue. The i-th column of the matrix X is the eigenvector associated with the eigenvalue d_{i}. The list of residues res = [res_{1}, res_{2}, …] provides error estimates for the numerical eigenvalues.
The routine implements standard numerical algorithms from the Handbook of Automatic Computation by Wilkinson and Reinsch.