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
    static double[][]
    exp​(double[][] matrix)
    Calculate the "matrix exponential" (expm).
    static 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[][]
    getCholeskyDecomposition​(double[][] symmetricMatrix)
    Create a Cholesky decomposition of a symmetric matrix.
    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[][]
    matrixExp​(double[][] matrix)
    Matrix exponential.
    static double[][]
    matrixLog​(double[][] matrix)
    Matrix logarithm.
    static double[][]
    matrixPow​(double[][] matrix, double exponent)
    Matrix power.
    static double[][]
    multMatrices​(double[][] left, double[][] right)
    Multiplication of two matrices.
    static double[]
    multMatrixVector​(double[][] matrix, double[] vector)
    Multiplication of matrix and vector.
    static double[][]
    pow​(double[][] matrix, double exponent)
    Calculate the power of a matrix Note: The function currently requires jblas.
    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 m x n - matrix given as double[m][n] 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 m x n - matrix given as double[m][n] 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 m x n - matrix given as double[m][n] 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 m x n - matrix given as double[m][n] 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 m x n - matrix given as double[m][n] 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 m x n - matrix given as double[m][n] 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 Details

    • LinearAlgebra

      public LinearAlgebra()

  • Method Details

    • getCholeskyDecomposition

      public static double[][] getCholeskyDecomposition​(double[][] symmetricMatrix)
      Create a Cholesky decomposition of a symmetric matrix.
      Parameters:
      symmetricMatrix - The input matrix.
      Returns:
      A lower triangle matrix representing the CholeskyDecomposition.

    • 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 m x n - matrix given as double[m][n]
      • 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 m x n - matrix given as double[m][n]
      • 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 m x n - matrix given as double[m][n]
      • 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 m x n - matrix given as double[m][n]
      • 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 m x n - matrix given as double[m][n]
      • 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 m x n - matrix given as double[m][n]
      • 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 static 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).

    • pow

      public static double[][] pow​(double[][] matrix, double exponent)
      Calculate the power of a matrix 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.
      exponent - The exponent
      Returns:
      The pow(matrix, exponent).

    • exp

      public static 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)

    • multMatrixVector

      public static double[] multMatrixVector​(double[][] matrix, double[] vector)
      Multiplication of matrix and vector. The vector array is interpreted as column vector.
      Parameters:
      matrix - The matrix A.
      vector - The vector v
      Returns:
      product The matrix product of A*v (if suitable)

    • matrixPow

      public static double[][] matrixPow​(double[][] matrix, double exponent)
      Matrix power. Tries to calculate a matrix A such that M^{exponent} = A.
      Parameters:
      matrix - The matrix M of which we like to have the power.
      exponent - The exponent.
      Returns:
      The exponent-th power of M

    • matrixExp

      public static double[][] matrixExp​(double[][] matrix)
      Matrix exponential. Tries to calculate the matrix A such that exp(M) = A.
      Parameters:
      matrix - The matrix M
      Returns:
      exp(M)

    • matrixLog

      public static double[][] matrixLog​(double[][] matrix)
      Matrix logarithm. Tries to calculate the matrix A such that log(M) = A.
      Parameters:
      matrix - The matrix M
      Returns:
      log(M)