Decomposition
Definition
Assembly: Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.23
public abstract class DecompositionOperations<T> : DecompositionOperations<T, Complex<T>>
- Inheritance
- Object → DecompositionOperations<T, Complex<T>> → DecompositionOperations<T>
- Derived
Type Parameters
- T
Remarks
The Linear Algebra PACKage (LAPACK) is the de facto standard for more complex linear algebra problems. It complements the Basic Linear Algebra Subroutines (BLAS), which defines standards for low level vector and matrix operations.
Constructors
| Decomposition | Initializes a new instance of the DecompositionOperations<T> class |
Properties
| Has |
Indicates whether the degree of parallelism is a property that is shared
across instances.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Implemented |
Gets the base type of the implementation.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Max |
Gets or sets the maximum degree of parallelism enabled by the instance.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Name |
Gets the name of the implementation.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Platform |
Gets the processor architecture supported by the implementation.
(Inherited from DecompositionOperations<TReal, TComplex>) |
Methods
| Band | Computes the Cholesky factorization of a real symmetric positive definite band matrix A. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite band matrix using the Cholesky factorization A = UT*U or A = L*LT computed by DPBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite band matrix using the Cholesky factorization A = UH*U or A = L*LH computed by ZPBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Solves a system of linear equations A*X = B with a symmetric positive definite band matrix A using the Cholesky factorization A = UT*U or A = L*LT computed by DPBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Solves a system of linear equations A*X = B with a Hermitian positive definite band matrix A using the Cholesky factorization A = UH *U or A = L*LH computed by ZPBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Estimates the reciprocal of the condition number of a real general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by DGBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Estimates the reciprocal of the condition number of a complex general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by ZGBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Solves a system of linear equations A * X = B or AT * X = B with a general band matrix A using the LU factorization computed by DGBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Solves a system of linear equations A * X = B, AT * X = B, or AH * X = B with a general band matrix A using the LU factorization computed by ZGBTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Band | Solves a triangular system of the form A * X = B or AT * X = B, where A is a triangular band matrix of order N, and B is an N-by NRHS matrix. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky | Computes the Cholesky factorization of a real symmetric positive definite matrix A. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky |
Factors a symmetric positive definite matrix.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky | Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite matrix using the Cholesky factorization A = UT*U or A = L*LT computed by DPOTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky |
Estimates the reciprocal of the condition number of a factored hermitian matrix.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky | Computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = UT*U or A = L*LT computed by DPOTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky |
Computes the inverse of a factored hermitian matrix.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky | Solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = UT*U or A = L*LT computed by DPOTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Cholesky |
Solves a hermitian system of equations.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Eigenvalue | Computes for an N-by-N complex non-symmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Eigenvalue | Computes for an N-by-N real non-symmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Equals | Determines whether the specified object is equal to the current object. (Inherited from Object) |
| Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object) |
| Generalized | Computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Generalized | Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Generalized | Computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B: U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) where U, V and Q are orthogonal matrices. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Generalized | Computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B: U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R ) where U, V and Q are unitary matrices. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Get | Serves as the default hash function. (Inherited from Object) |
| Get | Gets the Type of the current instance. (Inherited from Object) |
| Hermitian | Computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Hermitian | Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Hermitian | Estimates the reciprocal of the condition number of a complex Hermitian matrix A using the factorization A = U*D*UH or A = L*D*LH computed by ZHETRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Hermitian | Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Hermitian | Computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = U*D*UH or A = L*D*LH computed by ZHETRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Hermitian | Solves a system of linear equations A*X = B with a complex Hermitian matrix A using the factorization A = U*D*UH or A = L*D*LH computed by ZHETRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| LQDecompose( | Computes an LQ factorization of a real M-by-N matrix A: A = L * Q. (Inherited from DecompositionOperations<TReal, TComplex>) |
| LQDecompose( | Computes an LQ factorization of a complex M-by-N matrix A: A = L * Q. (Inherited from DecompositionOperations<TReal, TComplex>) |
| LQOrthogonal | Overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q**T * C C * Q**T where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(k) . (Inherited from DecompositionOperations<TReal, TComplex>) |
| LQUnitary | Overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(k)**H . (Inherited from DecompositionOperations<TReal, TComplex>) |
| LUDecompose( |
Computes an LU decomposition of a general M-by-N matrix A
using partial pivoting with row interchanges.
The decomposition has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). This is the right-looking Level 3 BLAS version of the algorithm. (Inherited from DecompositionOperations<TReal, TComplex>) |
| LUDecompose( |
ZGETRF computes an LU decomposition of a general M-by-N matrix A
using partial pivoting with row interchanges.
The decomposition has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). This is the right-looking Level 3 BLAS version of the algorithm. (Inherited from DecompositionOperations<TReal, TComplex>) |
| LUEstimate | Estimates the reciprocal of the condition number of a general real matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by DGETRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| LUEstimate |
ZGECON estimates the reciprocal of the condition number of a general
real matrix A, inthis. either the 1-norm or the infinity-norm, using
the LU decomposition computed by ZGETRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
Arguments
=========
NORM (input) CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
N (input) INTEGER
The elementOrder of the matrix A. N >= 0.
A (input) ZOUBLE PRECISION array, dimension (LDA,N)
The factors L and U from the decomposition A = P*L*U
as computed by ZGETRF.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= Max(1,N).
ANORM (input) ZOUBLE PRECISION
If NORM = '1' or 'O', the 1-norm of the original matrix A.
If NORM = 'I', the infinity-norm of the original matrix A.
RCOND (output) ZOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK (workspace) ZOUBLE PRECISION array, dimension (4*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
(Inherited from DecompositionOperations<TReal, TComplex>) |
| LUInvert( |
Computes the inverse of a matrix using the LU decomposition
computed by LUDecompose(Int32, Int32, Array2D<TReal>, Array1D<Int32>, Int32).
(Inherited from DecompositionOperations<TReal, TComplex>) |
| LUInvert( |
ZGETRI computes the inverse of a matrix using the LU decomposition
computed by ZGETRF.
This method inverts U and then computes inv(A) by solving the system
inv(A)*L = inv(U) for inv(A).
Arguments
=========
N (input) INTEGER
The elementOrder of the matrix A. N >= 0.
A (input/output) ZOUBLE PRECISION array, dimension (LDA,N)
On entry, the factors L and U from the decomposition
A = P*L*U as computed by ZGETRF.
On exit, if INFO = 0, the inverse of the original matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= Max(1,N).
IPIV (input) INTEGER array, dimension (N)
The pivot indexes from ZGETRF; for 1< =i< =N, row i of the
matrix was interchanged with row IPIVi.
WORK (workspace/output) ZOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO =0, then WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= Max(1,N).
For optimal performance LWORK >= N*NB, where NB is
the optimal blocksize returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero; the matrix is
singular and its inverse could not be computed.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| LUSolve( |
Solves a system of linear equations
with a general N-by-N matrix A using the LU decomposition computed
by LUDecompose(Int32, Int32, Array2D<TReal>, Array1D<Int32>, Int32).
(Inherited from DecompositionOperations<TReal, TComplex>) |
| LUSolve( |
ZGETRS solves a system of linear equations
A * X = B or A' * X = B
with a general N-by-N matrix A using the LU decomposition computed
by ZGETRF.
Arguments
=========
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= TransposeOperation.Transpose: A'* X = B (Transpose)
= 'C': A'* X = B (Conjugate transpose = Transpose)
N (input) INTEGER
The elementOrder of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A (input) ZOUBLE PRECISION array, dimension (LDA,N)
The factors L and U from the decomposition A = P*L*U
as computed by ZGETRF.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= Max(1,N).
IPIV (input) INTEGER array, dimension (N)
The pivot indexes from ZGETRF; for 1< =i< =N, row i of the
matrix was interchanged with row IPIVi.
B (input/output) ZOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= Max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
=====================================================================
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Memberwise | Creates a shallow copy of the current Object. (Inherited from Object) |
| QLDecompose( | Computes a QL factorization of a real M-by-N matrix A: A = Q * L. (Inherited from DecompositionOperations<TReal, TComplex>) |
| QLDecompose( | Computes a QL factorization of a complex M-by-N matrix A: A = Q * L. (Inherited from DecompositionOperations<TReal, TComplex>) |
| QLOrthogonal | Overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q**T * C C * Q**T where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(k) . (Inherited from DecompositionOperations<TReal, TComplex>) |
| QLUnitary | Overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(k) . (Inherited from DecompositionOperations<TReal, TComplex>) |
| QRDecompose( | Computes a QR factorization of a real M-by-N matrix A: A = Q * R. (Inherited from DecompositionOperations<TReal, TComplex>) |
| QRDecompose( |
ZGEQRF computes a QR decomposition of a real M-by-N matrix A:
A = Q * R.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) ZOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Zetails).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU (output) ZOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Zetails).
WORK (workspace/output) ZOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Zetails
===============
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit inthis. A(i+1:m,i),
and tau inthis. TAU(i).
(Inherited from DecompositionOperations<TReal, TComplex>) |
| QRDecompose( | Computes a QR factorization with column pivoting of a matrix A: A*P = Q*R using Level 3 BLAS. (Inherited from DecompositionOperations<TReal, TComplex>) |
| QROrthogonal | Overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': QT * C C * QT where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(1) H(2) . (Inherited from DecompositionOperations<TReal, TComplex>) |
| QRUnitary | Overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': QH * C C * QH where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1) H(2) . (Inherited from DecompositionOperations<TReal, TComplex>) |
| RQDecompose( | Computes an RQ factorization of a real M-by-N matrix A: A = R * Q. (Inherited from DecompositionOperations<TReal, TComplex>) |
| RQDecompose( | Computes an RQ factorization of a complex M-by-N matrix A: A = R * Q. (Inherited from DecompositionOperations<TReal, TComplex>) |
| RQOrthogonal | Overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q**T * C C * Q**T where Q is a real orthogonal matrix defined as the product of k elementary reflectors Q = H(1) H(2) . (Inherited from DecompositionOperations<TReal, TComplex>) |
| RQUnitary | Overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1)**H H(2)**H . (Inherited from DecompositionOperations<TReal, TComplex>) |
| Singular | Computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Singular | 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. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Symmetric | Computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Symmetric | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Symmetric | Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*UT or A = L*D*LT computed by DSYTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Symmetric | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Symmetric | Computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*UT or A = L*D*LT computed by DSYTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Symmetric | Solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*UT or A = L*D*LT computed by DSYTRF. (Inherited from DecompositionOperations<TReal, TComplex>) |
| ToString | Returns a string that represents the current object. (Inherited from Object) |
| Triangular | Estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Triangular |
Approximates the reciprocal of the condition number of a complex triangular matrix.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Triangular | Computes the inverse of a real upper or lower triangular matrix A. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Triangular |
Computes the inverse of a complex triangular matrix.
(Inherited from DecompositionOperations<TReal, TComplex>) |
| Triangular | Solves a triangular system of the form A * X = B or AT * X = B, where A is a triangular matrix of order N, and B is an N-by-NRHS matrix. (Inherited from DecompositionOperations<TReal, TComplex>) |
| Triangular |
Solves a complex triangular system of equations.
(Inherited from DecompositionOperations<TReal, TComplex>) |