Module net.finmath.lib

Package net.finmath.functions

Class LinearAlgebra

java.lang.Object

net.finmath.functions.LinearAlgebra

public class LinearAlgebra
extends Object

This class implements some methods from linear algebra (e.g. solution of a linear equation, PCA). It is basically a functional wrapper using either the Apache commons math or JBlas

Version:

1.6

Author:

Christian Fries

Constructor Summary

Constructors

Constructor	Description
LinearAlgebra()

Method Summary

Modifier and Type	Method	Description
static double[][]	diag(double[] vector)	Generates a diagonal matrix with the input vector on its diagonal
double[][]	exp(double[][] matrix)	Calculate the "matrix exponential" (expm).
org.apache.commons.math3.linear.RealMatrix	exp(org.apache.commons.math3.linear.RealMatrix matrix)	Calculate the "matrix exponential" (expm).
static double[][]	factorReduction(double[][] correlationMatrix, int numberOfFactors)	Returns a correlation matrix which has rank < n and for which the first n factors agree with the factors of correlationMatrix.
static double[][]	factorReductionUsingCommonsMath(double[][] correlationMatrix, int numberOfFactors)	Returns a correlation matrix which has rank < n and for which the first n factors agree with the factors of correlationMatrix.
static double[][]	getFactorMatrix(double[][] correlationMatrix, int numberOfFactors)	Returns the matrix of the n Eigenvectors corresponding to the first n largest Eigenvalues of a correlation matrix.
static double[][]	invert(double[][] matrix)	Returns the inverse of a given matrix.
static double[][]	multMatrices(double[][] left, double[][] right)	Multiplication of two matrices.
static double[][]	pseudoInverse(double[][] matrix)	Pseudo-Inverse of a matrix calculated in the least square sense.
static double[]	solveLinearEquation(double[][] matrixA, double[] b)	Find a solution of the linear equation A x = b where A is an n x m - matrix given as double[n][m] b is an m - vector given as double[m], x is an n - vector given as double[n],
static double[]	solveLinearEquationLeastSquare(double[][] matrix, double[] vector)	Find a solution of the linear equation A x = b in the least square sense where A is an n x m - matrix given as double[n][m] b is an m - vector given as double[m], x is an n - vector given as double[n],
static double[][]	solveLinearEquationLeastSquare(double[][] matrix, double[][] rhs)	Find a solution of the linear equation A X = B in the least square sense where A is an n x m - matrix given as double[n][m] B is an m x k - matrix given as double[m][k], X is an n x k - matrix given as double[n][k],
static double[]	solveLinearEquationSVD(double[][] matrixA, double[] b)	Find a solution of the linear equation A x = b where A is an n x m - matrix given as double[n][m] b is an m - vector given as double[m], x is an n - vector given as double[n],
static double[]	solveLinearEquationSymmetric(double[][] matrix, double[] vector)	Find a solution of the linear equation A x = b where A is an symmetric n x n - matrix given as double[n][n] b is an n - vector given as double[n], x is an n - vector given as double[n],
static double[]	solveLinearEquationTikonov(double[][] matrixA, double[] b, double lambda)	Find a solution of the linear equation A x = b where A is an n x m - matrix given as double[n][m] b is an m - vector given as double[m], x is an n - vector given as double[n], using a standard Tikhonov regularization, i.e., we solve in the least square sense A* x = b* where A* = (A^T, lambda I)^T and b* = (b^T , 0)^T.
static double[]	solveLinearEquationTikonov(double[][] matrixA, double[] b, double lambda0, double lambda1, double lambda2)	Find a solution of the linear equation A x = b where A is an n x m - matrix given as double[n][m] b is an m - vector given as double[m], x is an n - vector given as double[n], using a Tikhonov regularization, i.e., we solve in the least square sense A* x = b* where A* = (A^T, lambda0 I, lambda1 S, lambda2 C)^T and b* = (b^T , 0 , 0 , 0)^T.
static double[][]	transpose(double[][] matrix)	Transpose a matrix

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

LinearAlgebra

public LinearAlgebra()

Method Detail

solveLinearEquationTikonov

public static double[] solveLinearEquationTikonov(double[][] matrixA,
                                                  double[] b,
                                                  double lambda)

Find a solution of the linear equation A x = b where

• A is an n x m - matrix given as double[n][m]
• b is an m - vector given as double[m],
• x is an n - vector given as double[n],

using a standard Tikhonov regularization, i.e., we solve in the least square sense A* x = b* where A* = (A^T, lambda I)^T and b* = (b^T , 0)^T.

Parameters:

matrixA - The matrix A (left hand side of the linear equation).

b - The vector (right hand of the linear equation).

lambda - The parameter lambda of the Tikhonov regularization. Lambda effectively measures which small numbers are considered zero.

Returns:

A solution x to A x = b.

solveLinearEquationTikonov

public static double[] solveLinearEquationTikonov(double[][] matrixA,
                                                  double[] b,
                                                  double lambda0,
                                                  double lambda1,
                                                  double lambda2)

Find a solution of the linear equation A x = b where

• A is an n x m - matrix given as double[n][m]
• b is an m - vector given as double[m],
• x is an n - vector given as double[n],

using a Tikhonov regularization, i.e., we solve in the least square sense A* x = b* where A* = (A^T, lambda0 I, lambda1 S, lambda2 C)^T and b* = (b^T , 0 , 0 , 0)^T. The matrix I is the identity matrix, effectively reducing the level of the solution vector. The matrix S is the first order central finite difference matrix with -lambda1 on the element [i][i-1] and +lambda1 on the element [i][i+1] The matrix C is the second order central finite difference matrix with -0.5 lambda2 on the element [i][i-1] and [i][i+1] and lambda2 on the element [i][i].

Parameters:

matrixA - The matrix A (left hand side of the linear equation).

b - The vector (right hand of the linear equation).

lambda0 - The parameter lambda0 of the Tikhonov regularization. Reduces the norm of the solution vector.

lambda1 - The parameter lambda1 of the Tikhonov regularization. Reduces the slope of the solution vector.

lambda2 - The parameter lambda1 of the Tikhonov regularization. Reduces the curvature of the solution vector.

Returns:

The solution x of the equation A* x = b*

solveLinearEquation

public static double[] solveLinearEquation(double[][] matrixA,
                                           double[] b)

Find a solution of the linear equation A x = b where

• A is an n x m - matrix given as double[n][m]
• b is an m - vector given as double[m],
• x is an n - vector given as double[n],

Parameters:

matrixA - The matrix A (left hand side of the linear equation).

b - The vector (right hand of the linear equation).

Returns:

A solution x to A x = b.

solveLinearEquationSVD

public static double[] solveLinearEquationSVD(double[][] matrixA,
                                              double[] b)

Find a solution of the linear equation A x = b where

• A is an n x m - matrix given as double[n][m]
• b is an m - vector given as double[m],
• x is an n - vector given as double[n],

Parameters:

matrixA - The matrix A (left hand side of the linear equation).

b - The vector (right hand of the linear equation).

Returns:

A solution x to A x = b.

invert

public static double[][] invert(double[][] matrix)

Returns the inverse of a given matrix.

Parameters:

matrix - A matrix given as double[n][n].

Returns:

The inverse of the given matrix.

solveLinearEquationSymmetric

public static double[] solveLinearEquationSymmetric(double[][] matrix,
                                                    double[] vector)

Find a solution of the linear equation A x = b where

• A is an symmetric n x n - matrix given as double[n][n]
• b is an n - vector given as double[n],
• x is an n - vector given as double[n],

Parameters:

matrix - The matrix A (left hand side of the linear equation).

vector - The vector b (right hand of the linear equation).

Returns:

A solution x to A x = b.

solveLinearEquationLeastSquare

public static double[] solveLinearEquationLeastSquare(double[][] matrix,
                                                      double[] vector)

Find a solution of the linear equation A x = b in the least square sense where

• A is an n x m - matrix given as double[n][m]
• b is an m - vector given as double[m],
• x is an n - vector given as double[n],

Parameters:

matrix - The matrix A (left hand side of the linear equation).

vector - The vector b (right hand of the linear equation).

Returns:

A solution x to A x = b.

solveLinearEquationLeastSquare

public static double[][] solveLinearEquationLeastSquare(double[][] matrix,
                                                        double[][] rhs)

Find a solution of the linear equation A X = B in the least square sense where

• A is an n x m - matrix given as double[n][m]
• B is an m x k - matrix given as double[m][k],
• X is an n x k - matrix given as double[n][k],

Parameters:

matrix - The matrix A (left hand side of the linear equation).

rhs - The matrix B (right hand of the linear equation).

Returns:

A solution X to A X = B.

getFactorMatrix

public static double[][] getFactorMatrix(double[][] correlationMatrix,
                                         int numberOfFactors)

Returns the matrix of the n Eigenvectors corresponding to the first n largest Eigenvalues of a correlation matrix. These Eigenvectors can also be interpreted as "principal components" (i.e., the method implements the PCA).

Parameters:

correlationMatrix - The given correlation matrix.

numberOfFactors - The requested number of factors (eigenvectors).

Returns:

Matrix of n Eigenvectors (columns) (matrix is given as double[n][numberOfFactors], where n is the number of rows of the correlationMatrix.

factorReduction

public static double[][] factorReduction(double[][] correlationMatrix,
                                         int numberOfFactors)

Returns a correlation matrix which has rank < n and for which the first n factors agree with the factors of correlationMatrix.

Parameters:

correlationMatrix - The given correlation matrix.

numberOfFactors - The requested number of factors (Eigenvectors).

Returns:

Factor reduced correlation matrix.

factorReductionUsingCommonsMath

public static double[][] factorReductionUsingCommonsMath(double[][] correlationMatrix,
                                                         int numberOfFactors)

Returns a correlation matrix which has rank < n and for which the first n factors agree with the factors of correlationMatrix.

Parameters:

correlationMatrix - The given correlation matrix.

numberOfFactors - The requested number of factors (Eigenvectors).

Returns:

Factor reduced correlation matrix.

exp

public double[][] exp(double[][] matrix)

Calculate the "matrix exponential" (expm). Note: The function currently requires jblas. If jblas is not availabe on your system, an exception will be thrown. A future version of this function may implement a fall back.

Parameters:

matrix - The given matrix.

Returns:

The exp(matrix).

exp

public org.apache.commons.math3.linear.RealMatrix exp(org.apache.commons.math3.linear.RealMatrix matrix)

Calculate the "matrix exponential" (expm). Note: The function currently requires jblas. If jblas is not availabe on your system, an exception will be thrown. A future version of this function may implement a fall back.

Parameters:

matrix - The given matrix.

Returns:

The exp(matrix).

transpose

public static double[][] transpose(double[][] matrix)

Transpose a matrix

Parameters:

matrix - The given matrix.

Returns:

The transposed matrix.

pseudoInverse

public static double[][] pseudoInverse(double[][] matrix)

Pseudo-Inverse of a matrix calculated in the least square sense.

Parameters:

matrix - The given matrix A.

Returns:

pseudoInverse The pseudo-inverse matrix P, such that A*P*A = A and P*A*P = P

diag

public static double[][] diag(double[] vector)

Generates a diagonal matrix with the input vector on its diagonal

Parameters:

vector - The given matrix A.

Returns:

diagonalMatrix The matrix with the vectors entries on its diagonal

multMatrices

public static double[][] multMatrices(double[][] left,
                                      double[][] right)

Multiplication of two matrices.

Parameters:

left - The matrix A.

right - The matrix B

Returns:

product The matrix product of A*B (if suitable)