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Numerical singular value decomposition of a matrix
This functionality does not run in MATLAB.
numeric::svd(A, options)
numeric::svd(A) and the equivalent call numeric::singularvectors(A) return numerical singular values and singular vectors of the matrix A.
All entries of A must be numerical. Numerical expressions such as etc. are accepted and converted to floats. Non-numerical symbolic entries lead to an error.
Cat::Matrix objects, i.e., matrices A of a matrix domain such as Dom::Matrix(…) or Dom::SquareMatrix(…) are internally converted to arrays over expressions via expr(A).
The list [U, d, V, res_{U}, res_{V}] returned by numeric::svd corresponds to the singular data of an m×n matrix A as described below.
Let V^{H} denote the Hermitian transpose of the matrix V, i.e., the complex conjugate of the transpose. The singular value decomposition of an m×n matrix A is a factorization A = U D V^{H}. D is an m×n "diagonal" matrix with real nonnegative entries D_{ii} = d_{i}, i = 1, …, p where p = min(m, n):
or
,
respectively. The list d = [d_{1}, …, d_{p}] returned by numeric::svd are the "singular values" of A. They are sorted by numeric::sort, i.e., d_{1} ≥ … ≥ d_{p} ≥ 0.0.
U is a unitary m×m matrix. Its i-th column is an eigenvector of A A^{H} associated with the eigenvalue d_{i}^{2} (d_{i} = 0 for i > p). These are the "left singular vectors" of A. They are returned by numeric::svd as a matrix of floating-point numbers.
V is a unitary n×n matrix. Its i-th column is an eigenvector of A^{H} A associated with the eigenvalue d_{i}^{2} (d_{i} = 0 for i > p). These are the "right singular vectors" of A. They are returned by numeric::svd as an array of floating-point numbers. The matrix V is normalized such that, in each column, the first entry of absolute size larger than is real and positive.
If no return type is specified via the option ReturnType = t, the domain type of the singular vectors U and V depends on the type of the input matrix A:
The singular vectors of an hfarray are returned as hfarrays.
The singular vectors of a dense matrix of type Dom::DenseMatrix() are returned as dense matrices of type Dom::DenseMatrix() over the ring of MuPAD^{®} expressions.
For all other matrices of category Cat::Matrix, the singular vectors 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.
res_{U} = [resU_{1}, …, resU_{m}] is a list of float residues associated with the left singular vectors:
.
Here, u_{i} is the (normalized) i-th column of U, is the usual complex Euclidean scalar product and d_{i} = 0 for p < i ≤ m.
res_{V} = [resV_{1}, …, resV_{n}] is a list of float residues associated with the right singular vectors:
.
Here, v_{i} is the (normalized) i-th column of V, d_{i} = 0 for p < i ≤ n.
The residues res_{U}, res_{V} vanish for exact singular data U, d, V. Their sizes indicate the quality of the numerical data U, d, V.
Note: Singular values are approximated with an absolute precision of , where r is the largest singular value of A. Consequently, large singular values should be computed correctly to DIGITS decimal places. The numerical approximations of small singular values are less accurate. |
The singular values computed by numeric::svd are identical to those computed by numeric::singularvalues.
Singular data may also be computed via [d2, U, resU] := numeric::eigenvectors(A*A^H) or [d2, V, resV] := numeric::eigenvectors(A^H*A), respectively. The list d2 is related to the singular values by
.
The use of numeric::svd avoids the costs of the matrix multiplication. Further, the eigenvector routine requires about twice as many DIGITS to compute the data associated with small singular values with the same precision as numeric::singularvectors. Also note that the normalization of U and V may be different.
The function is sensitive to the environment variable DIGITS, which determines the numerical working precision.
Numerical expressions are converted to floats:
DIGITS := 5: A := array(1..3, 1..2, [[1, PI], [2, 3], [3, exp(sqrt(2))]]): [U, d, V, resU, resV] := numeric::singularvectors(A):
The singular data are:
U, d, V
The small residues indicate that these results are not severely affected by roundoff:
resU, resV
delete DIGITS, A, U, d, V, resU, resV:
We demonstrate how to reconstruct a matrix from its singular data. With the specified ReturnType, the singular vectors are returned as matrices of type Dom::Matrix() and can be handled with the overloaded arithmetic:
DIGITS := 3: A := array(1..2, 1..3, [[1.0, I, PI], [2, 3, I]]): [U, d, V, resU, resV] := numeric::singularvectors(A, NoResidues, ReturnType = Dom::Matrix())
A "diagonal" matrix is built from the singular values:
d := matrix(2, 3, d, Diagonal)
We use the methods conjugate and transpose of the matrix domain to compute the Hermitian transpose of V and reconstruct A. Numerical roundoff is eliminated via numeric::complexRound:
VH := V::dom::conjugate(V::dom::transpose(V)): map(U*d*VH, numeric::complexRound)
delete DIGITS, A, U, d, V, resU, resV, VH:
We demonstrate the use of hardware floats. The following matrix A is degenerate: it has rank 1. For the double eigenvalue 0 of the matrix A^{H} A, different base vectors of the corresponding eigenspace are returned with HardwareFloats and SoftwareFloats, respectively:
A := array(1..2, 1..3, [[1, 2, 3], [30, 60, 90]]): [U1, d1, V1, resU1, resV1] := numeric::singularvectors(A, HardwareFloats): [U2, d2, V2, resU2, resV2] := numeric::singularvectors(A, SoftwareFloats): V1, V2
delete A, U1, d1, V1, resU1, resV1, U2, d2, V2, resU2, resV2:
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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NoLeftVectors |
Suppresses the computation of left singular vectors If only right singular vectors are required, this option may be used to suppress the computation of U and the corresponding residues res_{U}. The return values for these data are NIL. Depending on the size of U, this option may speed up the computation considerably. | |
NoRightVectors |
Suppresses the computation of right singular vectors If only left singular vectors are required, this option may be used to suppress the computation of V and the corresponding residues res_{V}. The return values for these data are NIL. Depending on the size of V, this option may speed up the computation considerably. | |
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_{U} and res_{V}. 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 left and right singular vectors as matrices of domain type t. The following return types t are available: DOM_ARRAY, or DOM_HFARRAY, or Dom::Matrix(), or Dom::DenseMatrix(). This option determines the domain type of the matrices containing the singular vectors. | |
NoWarning |
Suppresses warnings |
List [U, d, V, res_{U}, res_{V}]. U is a unitary square float matrix whose columns are left singular vectors. The list d contains the singular values. V is a unitary square float matrix whose columns are right singular vectors. The lists of float residues res_{U} and res_{V} provide error estimates for the numerical data.