DecompositionOperations<TReal, TComplex>.SingularValueDecompose Method

Definition

Namespace: Extreme.Mathematics.LinearAlgebra.Implementation
Assembly: Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.23

Overload List

SingularValueDecompose(Char, Int32, Int32, Array2D<TReal>, Array1D<TReal>, Array2D<TReal>, Array2D<TReal>, Int32)

Computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors.

SingularValueDecompose(Char, Int32, Int32, Array2D<TComplex>, Array1D<TReal>, Array2D<TComplex>, Array2D<TComplex>, Int32)

Computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors, by using divide-and-conquer method.

SingularValueDecompose(Char, Int32, Int32, Array2D<TReal>, Array1D<TReal>, Array2D<TReal>, Array2D<TReal>, Int32)

Computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors.

C# 
public abstract void SingularValueDecompose(
	char jobz,
	int m,
	int n,
	Array2D<TReal> a,
	Array1D<TReal> s,
	Array2D<TReal> u,
	Array2D<TReal> vt,
	out int info
)

Parameters

jobz  Char
            Specifies options for computing all or part of the matrix U:
            = 'A':  all M columns of U and all N rows of VT are
                    returned in the arrays U and VT;
            = 'S':  the first min(M,N) columns of U and the first
                    min(M,N) rows of VT are returned in the arrays U
                    and VT;
            = 'O':  If M >= N, the first N columns of U are overwritten
                    on the array A and all rows of VT are returned in
                    the array VT;
                    otherwise, all columns of U are returned in the
                    array U and the first M rows of VT are overwritten
                    in the array A;
            = 'N':  no columns of U or rows of VT are computed.
m  Int32
            The number of rows of the input matrix A.  M >= 0.
n  Int32
            The number of columns of the input matrix A.  N >= 0.
a  Array2D<TReal>
            A is TReal array, dimension (LDA,N)
            On entry, the M-by-N matrix A.
            On exit,
            if JOBZ = 'O',  A is overwritten with the first N columns
                            of U (the left singular vectors, stored
                            columnwise) if M >= N;
                            A is overwritten with the first M rows
                            of VT (the right singular vectors, stored
                            rowwise) otherwise.
            if JOBZ .ne. 'O', the contents of A are destroyed.
            
            The leading dimension of the array A.  LDA >= max(1,M).
s  Array1D<TReal>
            S is TReal array, dimension (min(M,N))
            The singular values of A, sorted so that S(i) >= S(i+1).
u  Array2D<TReal>
            U is TReal array, dimension (LDU,UCOL)
            UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
            UCOL = min(M,N) if JOBZ = 'S'.
            If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
            orthogonal matrix U;
            if JOBZ = 'S', U contains the first min(M,N) columns of U
            (the left singular vectors, stored columnwise);
            if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
            
            The leading dimension of the array U.  LDU >= 1; if
            JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
vt  Array2D<TReal>
            VT is TReal array, dimension (LDVT,N)
            If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
            N-by-N orthogonal matrix VT;
            if JOBZ = 'S', VT contains the first min(M,N) rows of
            VT (the right singular vectors, stored rowwise);
            if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
            
            The leading dimension of the array VT.  LDVT >= 1; if
            JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
            if JOBZ = 'S', LDVT >= min(M,N).
info  Int32
            = 0:  successful exit.
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  DBDSDC did not converge, updating process failed.

Remarks

            If singular vectors are desired, it uses a
            divide-and-conquer algorithm.
            The SVD is written
                 A = U * SIGMA * transpose(V)
            where SIGMA is an M-by-N matrix which is zero except for its
            min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
            V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
            are the singular values of A; they are real and non-negative, and
            are returned in descending order.  The first min(m,n) columns of
            U and V are the left and right singular vectors of A.
            Note that the routine returns VT = VT, not V.
            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

SingularValueDecompose(Char, Int32, Int32, Array2D<TComplex>, Array1D<TReal>, Array2D<TComplex>, Array2D<TComplex>, Int32)

Computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors, by using divide-and-conquer method.

C# 
public abstract void SingularValueDecompose(
	char jobz,
	int m,
	int n,
	Array2D<TComplex> a,
	Array1D<TReal> s,
	Array2D<TComplex> u,
	Array2D<TComplex> vt,
	out int info
)

Parameters

jobz  Char
            Specifies options for computing all or part of the matrix U:
            = 'A':  all M columns of U and all N rows of VH are
                    returned in the arrays U and VT;
            = 'S':  the first min(M,N) columns of U and the first
                    min(M,N) rows of VH are returned in the arrays U
                    and VT;
            = 'O':  If M >= N, the first N columns of U are overwritten
                    in the array A and all rows of VH are returned in
                    the array VT;
                    otherwise, all columns of U are returned in the
                    array U and the first M rows of VH are overwritten
                    in the array A;
            = 'N':  no columns of U or rows of VH are computed.
m  Int32
            The number of rows of the input matrix A.  M >= 0.
n  Int32
            The number of columns of the input matrix A.  N >= 0.
a  Array2D<TComplex>
            A is TComplex array, dimension (LDA,N)
            On entry, the M-by-N matrix A.
            On exit,
            if JOBZ = 'O',  A is overwritten with the first N columns
                            of U (the left singular vectors, stored
                            columnwise) if M >= N;
                            A is overwritten with the first M rows
                            of VH (the right singular vectors, stored
                            rowwise) otherwise.
            if JOBZ .ne. 'O', the contents of A are destroyed.
            
            The leading dimension of the array A.  LDA >= max(1,M).
s  Array1D<TReal>
            S is TReal array, dimension (min(M,N))
            The singular values of A, sorted so that S(i) >= S(i+1).
u  Array2D<TComplex>
            U is TComplex array, dimension (LDU,UCOL)
            UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
            UCOL = min(M,N) if JOBZ = 'S'.
            If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
            unitary matrix U;
            if JOBZ = 'S', U contains the first min(M,N) columns of U
            (the left singular vectors, stored columnwise);
            if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
            
            The leading dimension of the array U.  LDU >= 1; if
            JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
vt  Array2D<TComplex>
            VT is TComplex array, dimension (LDVT,N)
            If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
            N-by-N unitary matrix VH;
            if JOBZ = 'S', VT contains the first min(M,N) rows of
            VH (the right singular vectors, stored rowwise);
            if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
            
            The leading dimension of the array VT.  LDVT >= 1; if
            JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
            if JOBZ = 'S', LDVT >= min(M,N).
info  Int32
            = 0:  successful exit.
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  The updating process of DBDSDC did not converge.

Remarks

            The SVD is written
                 A = U * SIGMA * conjugate-transpose(V)
            where SIGMA is an M-by-N matrix which is zero except for its
            min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
            V is an N-by-N unitary matrix.  The diagonal elements of SIGMA
            are the singular values of A; they are real and non-negative, and
            are returned in descending order.  The first min(m,n) columns of
            U and V are the left and right singular vectors of A.
            Note that the routine returns VT = VH, not V.
            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

See Also