dlr::linearAlgebra Namespace Reference

This namespace contains functions which implement common linear algebra tasks, such as eigenvector computations, SVD, etc. More...

Functions

void choleskyFactorization (const Array2D< Float64 > &inputArray, Array2D< Float64 > &kArray, bool isUpperTriangular=true)

This function computes the Cholesky factorization of a symmetric, positive definite matrix.

Float64 determinant (const Array2D< Float64 > &A)

This function computes the determinant of a square Array2D<Float64> instance.

Array1D< Float64 > eigenvaluesSymmetric (const Array2D< Float64 > &inputArray)

This function computes the eigenvalues of a symmetric real matrix.

void eigenvectorsSymmetric (const Array2D< Float64 > &inputArray, Array1D< Float64 > &eigenvalues, Array2D< Float64 > &eigenvectors)

This function computes the eigenvalues and eigenvectors of a symmetric real matrix.

Array2D< Float64 > inverse (const Array2D< Float64 > &A)

This function accepts a square Array2D<Float64> instance and returns an Array2D<Float64> instance such that the matrix product of the two is equal to the identity matrix.

std::pair< Float64, Float64 > linearFit (const Array1D< Float64 > &array0, const Array1D< Float64 > &array1)

This function computes the best linear fit between the two input arrays.

Array1D< Float64 > linearLeastSquares (const Array2D< Float64 > &A, const Array1D< Float64 > &b)

This function solves the system of equations A*x = b, where A and b are known Array2D<Float64> instances.

void linearSolveInPlace (Array2D< Float64 > &A, Array1D< Float64 > &b)

This function solves the system of equations A*x = b, where A is a known matrix, and b is a known vector.

void linearSolveInPlace (Array2D< Float64 > &A, Array2D< Float64 > &b)

This function is identical to linearSolveInPlace(Array2D<Float64>, Array1D<Float64>&), except that b (and therefore x) is not constrained to be a vector.

Array1D< Float64 > linearSolveTridiagonal (const Array1D< Float64 > &subDiagonal, const Array1D< Float64 > &centerDiagonal, const Array1D< Float64 > &superDiagonal, const Array1D< Float64 > &bVector)

This function solves the system of equations A*x = b, where A is a known tridiagonal matrix and b is a known vector.

Array2D< Float64 > pseudoinverse (const Array2D< Float64 > &A)

This function accepts an Array2D<Float64> instance having at least as many rows as columns, and returns the Moore-Penrose pseudoinverse.

void singularValueDecomposition (const Array2D< Float64 > &inputArray, Array2D< Float64 > &uArray, Array1D< Float64 > &sigmaArray, Array2D< Float64 > &vTransposeArray)

This function computes the singular value decomposition of a matrix.

void singularValueDecompositionSimple (const Array2D< Float64 > &inputArray, Array2D< Float64 > &uArray, Array1D< Float64 > &sigmaArray, Array2D< Float64 > &vTransposeArray)

Array1D< Float64 > singularValues (const Array2D< Float64 > &inputArray)

This function computes the singular values a matrix without computing the associated U and V matrices.

Detailed Description

This namespace contains functions which implement common linear algebra tasks, such as eigenvector computations, SVD, etc.

Function Documentation

void dlr::linearAlgebra::choleskyFactorization	(	const Array2D< Float64 > &	inputArray,
		Array2D< Float64 > &	kArray,
		bool	isUpperTriangular = `true`
	)

This function computes the Cholesky factorization of a symmetric, positive definite matrix.

That is, for a symmetric, positive definite matrix E, it computes the upper triangular matrix K such that E == K^T * K (if argument isUpperTriangular is true), or the lower triangular matrix such that E = K * K^T (if argument isUpperTriangular is false).

Parameters:

	inputArray	This argument is the matrix to be factored.
	kArray	This argument will be filled in with the upper triangular matrix K.
	isUpperTriangular	This argument specifies whether the result should be returned as an upper triangular or lower triangular matrix.

Definition at line 28 of file linearAlgebra.cpp.

References dpotrf_().

Float64 dlr::linearAlgebra::determinant ( const Array2D< Float64 > & A )

This function computes the determinant of a square Array2D<Float64> instance.

Parameters:

A

This argument represents the matrix from which to compute the determinant.

Returns:: The return value is a Float64 representing the determinant of the input argument.

Definition at line 107 of file linearAlgebra.cpp.

References dgetrf_().

Array1D< Float64 > dlr::linearAlgebra::eigenvaluesSymmetric ( const Array2D< Float64 > & inputArray )

This function computes the eigenvalues of a symmetric real matrix.

Parameters:

inputArray

This argument is and Array2D<Float64> instance representing a symmetric matrix. It must be square and have non-zero size. Only the data in the upper triangular portion of the array (including the diagonal) will be examined. The remaining elements are assumed to be symmetric.

Returns:: The return value is an Array1D<Float64> instance containing the eigenvalues, sorted into descending order.

Definition at line 162 of file linearAlgebra.cpp.

References dsyev_().

void dlr::linearAlgebra::eigenvectorsSymmetric	(	const Array2D< Float64 > &	inputArray,
		Array1D< Float64 > &	eigenvalues,
		Array2D< Float64 > &	eigenvectors
	)

This function computes the eigenvalues and eigenvectors of a symmetric real matrix.

Parameters:

	inputArray	This argument is and Array2D<Float64> instance representing a symmetric matrix. It must be square and have non-zero size. Only the data in the upper triangular portion of the array (including the diagonal) will be examined. The remaining elements are assumed to be symmetric.
	eigenvalues	This argument is used to return an Array1D<Float64> instance containing the eigenvalues, sorted into descending order.
	eigenvectors	This argument is used to return the eigenvectors. On return, the first column contains the eigenvector corresponding to the first eigenvalue, the second column contains the eigenvector corresponding to the second eigenvalue, and so on.

Definition at line 244 of file linearAlgebra.cpp.

References dsyev_().

Array2D< Float64 > dlr::linearAlgebra::inverse ( const Array2D< Float64 > & A )

This function accepts a square Array2D<Float64> instance and returns an Array2D<Float64> instance such that the matrix product of the two is equal to the identity matrix.

It is an error if the argument is not invertible.

Parameters:

A

This argument is the matrix to be inverted.

Returns:: The return value is the inverse of argument A.

Definition at line 360 of file linearAlgebra.cpp.

References linearSolveInPlace().

Referenced by linearFit(), and pseudoinverse().

std::pair< Float64, Float64 > dlr::linearAlgebra::linearFit	(	const Array1D< Float64 > &	array0,
		const Array1D< Float64 > &	array1
	)

This function computes the best linear fit between the two input arrays.

That is, it solves for scalars a and b which minimize the quantity

e = |(a * array0 + b) - array1|^2,

where array0 and array1 are the two input arrays. Put another way, this means we're solving (in the least squares sense) for the variables a and b which most nearly satisfy the following equation:

a * array0 + b = array1

Parameters:

	array0	This argument is the first input array.
	array1	This argument is the second input array.

Returns:: The return value is a pair of Float64s in which the first element is the variable a and the second is the variable b.

Definition at line 381 of file linearAlgebra.cpp.

References inverse().

Array1D< Float64 > dlr::linearAlgebra::linearLeastSquares	(	const Array2D< Float64 > &	A,
		const Array1D< Float64 > &	b
	)

This function solves the system of equations A*x = b, where A and b are known Array2D<Float64> instances.

The contents of the arguments are not modified. If the solution fails, a ValueException will be generated.

Parameters:

	A	This argument specifies the A matrix in the system "Ax = b," and must either be square or have more rows than columns.
	b	This argument specifies the b vector in the system "Ax = b." It must have the same number of elements as argument A has rows.

Returns:: The return value is the vector x which most nearly satisfies the equation. "Nearly" is defined in the least-squares sense.

Definition at line 439 of file linearAlgebra.cpp.

References dgels_().

void dlr::linearAlgebra::linearSolveInPlace	(	Array2D< Float64 > &	A,
		Array2D< Float64 > &	b
	)

This function is identical to linearSolveInPlace(Array2D<Float64>, Array1D<Float64>&), except that b (and therefore x) is not constrained to be a vector.

Parameters:

	A	This argument specifies the A matrix in the system "Ax = b," and must be square. In the current implementation, it will be replaced the LU decomposition of A, however this behavior may change in the future.
	b	This argument specifies the b matrix in the system "Ax = b." It must have the same size as x, and will be replaced with the recovered value of x.

Definition at line 536 of file linearAlgebra.cpp.

References dgesv_().

void dlr::linearAlgebra::linearSolveInPlace	(	Array2D< Float64 > &	A,
		Array1D< Float64 > &	b
	)

This function solves the system of equations A*x = b, where A is a known matrix, and b is a known vector.

The contents of both arguments are modified as part of the process. If the solution fails, a ValueException will be generated.

Parameters:

	A	This argument specifies the A matrix in the system "Ax = b," and must be square.
	b	This argument specifies the b vector in the system "Ax = b." It must have the same size as x, and will be replaced with the recovered value of x.

Definition at line 525 of file linearAlgebra.cpp.

Referenced by inverse().

Array1D< Float64 > dlr::linearAlgebra::linearSolveTridiagonal	(	const Array1D< Float64 > &	subDiagonal,
		const Array1D< Float64 > &	centerDiagonal,
		const Array1D< Float64 > &	superDiagonal,
		const Array1D< Float64 > &	bVector
	)

This function solves the system of equations A*x = b, where A is a known tridiagonal matrix and b is a known vector.

The contents of the arguments are not modified. If the solution fails, a ValueException will be generated.

Parameters:

	subDiagonal	This argument specifies the lower diagonal of the A matrix in the system "A*x = b."
	centerDiagonal	This argument specifies the center diagonal of the A matrix in the system "A*x = b." It's length must be 1 greater than the length of argument subDiagonal.
	superDiagonal	This argument specifies the upper diagonal of the A matrix in the system "A*x = b." It's length must be the same as the length of argument subDiagonal.
	b	This argument specifies the b vector in the system "Ax = b." It must have the same length as argument centerDiagonal.

Returns:: The return value is the vector x which most nearly satisfies the equation. "Nearly" is defined in the least-squares sense.

Definition at line 578 of file linearAlgebra.cpp.

References dgtsv_().

Array2D< Float64 > dlr::linearAlgebra::pseudoinverse ( const Array2D< Float64 > & A )

This function accepts an Array2D<Float64> instance having at least as many rows as columns, and returns the Moore-Penrose pseudoinverse.

That is, for an input matrix A, it returns inverse(matrixMultiply(transpose(A), A)). It is an error if the rank of A is less than A.columns().

Parameters:

A

This argument is the matrix to be psuedoinverted.

Returns:: The return value is the pseudoinverse of argument A.

Definition at line 635 of file linearAlgebra.cpp.

References inverse().

void dlr::linearAlgebra::singularValueDecomposition	(	const Array2D< Float64 > &	inputArray,
		Array2D< Float64 > &	uArray,
		Array1D< Float64 > &	sigmaArray,
		Array2D< Float64 > &	vTransposeArray
	)

This function computes the singular value decomposition of a matrix.

After a successful call, matrixMultiply(matrixMultiply(uArray, sigmaArray), vTransposeArray) == inputArray.

Parameters:

	inputArray	This argument is the matrix to be decomposed.
	uArray	This argument will be filled in with an orthonormal basis spanning the range of the input matrix, one basis vector per column. If inputArray has M rows and N columns, then this array will have M rows and min(M, N) columns when the funtion returns.
	sigmaArray	This argument will be filled in with the singular values of the matrix, in descending order. If inputArray has M rows and N columns, then this array will have min(M, N) elements when the funtion returns.
	vTransposeArray	This argument will be filled in with an orthonormal basis spanning the domain of the input matrix, one basis vector per row. If inputArray has M rows and N columns, then this array will have min(M, N) rows and N columns when the funtion returns.

Definition at line 646 of file linearAlgebra.cpp.

References dgesdd_().

Array1D< Float64 > dlr::linearAlgebra::singularValues ( const Array2D< Float64 > & inputArray )

This function computes the singular values a matrix without computing the associated U and V matrices.

Parameters:

inputArray

This argument is the matrix from which to compute the singular values.

Returns:: The return value is an Array1D of singular values in descending order.

Definition at line 925 of file linearAlgebra.cpp.

References dgesdd_().


Functions
void	choleskyFactorization (const Array2D< Float64 > &inputArray, Array2D< Float64 > &kArray, bool isUpperTriangular=true)
	This function computes the Cholesky factorization of a symmetric, positive definite matrix.
Float64	determinant (const Array2D< Float64 > &A)
	This function computes the determinant of a square Array2D<Float64> instance.
Array1D< Float64 >	eigenvaluesSymmetric (const Array2D< Float64 > &inputArray)
	This function computes the eigenvalues of a symmetric real matrix.
void	eigenvectorsSymmetric (const Array2D< Float64 > &inputArray, Array1D< Float64 > &eigenvalues, Array2D< Float64 > &eigenvectors)
	This function computes the eigenvalues and eigenvectors of a symmetric real matrix.
Array2D< Float64 >	inverse (const Array2D< Float64 > &A)
	This function accepts a square Array2D<Float64> instance and returns an Array2D<Float64> instance such that the matrix product of the two is equal to the identity matrix.
std::pair< Float64, Float64 >	linearFit (const Array1D< Float64 > &array0, const Array1D< Float64 > &array1)
	This function computes the best linear fit between the two input arrays.
Array1D< Float64 >	linearLeastSquares (const Array2D< Float64 > &A, const Array1D< Float64 > &b)
	This function solves the system of equations A*x = b, where A and b are known Array2D<Float64> instances.
void	linearSolveInPlace (Array2D< Float64 > &A, Array1D< Float64 > &b)
	This function solves the system of equations A*x = b, where A is a known matrix, and b is a known vector.
void	linearSolveInPlace (Array2D< Float64 > &A, Array2D< Float64 > &b)
	This function is identical to linearSolveInPlace(Array2D<Float64>, Array1D<Float64>&), except that b (and therefore x) is not constrained to be a vector.
Array1D< Float64 >	linearSolveTridiagonal (const Array1D< Float64 > &subDiagonal, const Array1D< Float64 > &centerDiagonal, const Array1D< Float64 > &superDiagonal, const Array1D< Float64 > &bVector)
	This function solves the system of equations A*x = b, where A is a known tridiagonal matrix and b is a known vector.
Array2D< Float64 >	pseudoinverse (const Array2D< Float64 > &A)
	This function accepts an Array2D<Float64> instance having at least as many rows as columns, and returns the Moore-Penrose pseudoinverse.
void	singularValueDecomposition (const Array2D< Float64 > &inputArray, Array2D< Float64 > &uArray, Array1D< Float64 > &sigmaArray, Array2D< Float64 > &vTransposeArray)
	This function computes the singular value decomposition of a matrix.
void	singularValueDecompositionSimple (const Array2D< Float64 > &inputArray, Array2D< Float64 > &uArray, Array1D< Float64 > &sigmaArray, Array2D< Float64 > &vTransposeArray)
Array1D< Float64 >	singularValues (const Array2D< Float64 > &inputArray)
	This function computes the singular values a matrix without computing the associated U and V matrices.