test_linalg.cc

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/* Copyright (C) 2002-2012 Claudia Emde <claudia.emde@dlr.de>
 
   This program is free software; you can redistribute it and/or modify it
   under the terms of the GNU General Public License as published by the
   Free Software Foundation; either version 2, or (at your option) any
   later version.
 
   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.
 
   You should have received a copy of the GNU General Public License
   along with this program; if not, write to the Free Software
   Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
   USA.
 
*/
 
#include <stdlib.h>
#include <time.h>
#include <cmath>
#include <iostream>
#include <random>
#include "array.h"
#include "lin_alg.h"
#include "test_utils.h"
 
using std::abs;
using std::cout;
using std::endl;
using std::setw;
 
 
void test_lusolve1D(void) {
  Matrix a(1, 1);
  ArrayOfIndex indx(1);
  Matrix orig(1, 1);
  Matrix b(1, 1);
 
  /* Assign test-matrix element. */
  a(0, 0) = 3;
 
  /* ------------------------------------------------------------------------
     Test the function ludcmp.
     ----------------------------------------------------------------------- */
  cout << "\n LU decomposition test \n";
  cout << "initial matrix: \n";
 
  cout << " " << a(0, 0) << endl;
 
  /* input: Test-matrix a,
      output: Decomposed matrix b (includes upper and lower triangle, cp.
      Numerical Recipies)
      and index which gives information about pivoting. */
  ludcmp(b, indx, a);
 
  cout << "\n after decomposition: ";
  cout << b(0, 0) << endl;
 
  /* Seperate b into the two triangular matrices. */
  Matrix l(1, 1, 0.0);
  Matrix u(1, 1, 0.0);
  Matrix lu(1, 1, 0.0);
 
  l(0, 0) = 1.0;
  u(0, 0) = b(0, 0);
 
  /*-------------------------------------------------------------------
    end of ludcmp test
     ------------------------------------------------------------------*/
  /*--------------------------------------------------------------------
    test backsubstitution routine lubacksub
    -------------------------------------------------------------------*/
 
  Vector c(1);
  c[0] = 6;
  cout << indx[0] << "  " << c[0] << endl;
 
  Vector x(1);
  lubacksub(x, b, c, indx);
 
  cout << "\n solution vector x: ";
  cout << x[0] << endl;
}
 
 
void test_lusolve4D(void) {
  Matrix a(4, 4);
  ArrayOfIndex indx(4);
  Matrix orig(4, 4);
  Matrix b(4, 4);
 
  /* Assign test-matrix elements. */
 
  a(0, 0) = 1;
  a(0, 1) = 3;
  a(0, 2) = 5;
  a(0, 3) = 6;
  a(1, 0) = 2;
  a(1, 1) = 3;
  a(1, 2) = 4;
  a(1, 3) = 4;
  a(2, 0) = 1;
  a(2, 1) = 2;
  a(2, 2) = 5;
  a(2, 3) = 1;
  a(3, 0) = 7;
  a(3, 1) = 2;
  a(3, 2) = 4;
  a(3, 3) = 3;
 
  /* ------------------------------------------------------------------------
     Test the function ludcmp.
     ----------------------------------------------------------------------- */
 
  cout << "\n LU decomposition test \n";
  cout << "initial matrix: \n";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    for (Index j = 0; j < 4; j++) cout << " " << a(i, j);
  }
  cout << "\n";
 
  /* input: Test-matrix a,
      output: Decomposed matrix b (includes upper and lower triangle, cp.
      Numerical Recipies)
      and index which gives information about pivoting. */
  ludcmp(b, indx, a);
 
  cout << "\n after decomposition";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    for (Index j = 0; j < 4; j++) cout << " " << b(i, j);
  }
  cout << "\n";
 
  /* Seperate b into the two triangular matrices. */
  Matrix l(4, 4, 0.0);
  Matrix u(4, 4, 0.0);
  Matrix lu(4, 4, 0.0);
 
  for (Index i = 0; i < 4; i++) l(i, i) = 1.0;
  l(1, 0) = b(1, 0);
  l(2, Range(0, 2)) = b(2, Range(0, 2));
  l(3, Range(0, 3)) = b(3, Range(0, 3));
 
  cout << "\n Matrix L";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    for (Index j = 0; j < 4; j++) cout << " " << l(i, j);
  }
  cout << "\n";
 
  u(0, Range(0, 4)) = b(0, Range(0, 4));
  u(1, Range(1, 3)) = b(1, Range(1, 3));
  u(2, Range(2, 2)) = b(2, Range(2, 2));
  u(3, Range(3, 1)) = b(3, Range(3, 1));
 
  cout << "\n Matrix U";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    for (Index j = 0; j < 4; j++) cout << " " << u(i, j);
  }
  cout << "\n";
 
  /* Test, if LU = a. */
  mult(lu, l, u);
 
  cout << "\n product L*U";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    for (Index j = 0; j < 4; j++) cout << " " << lu(i, j);
  }
  cout << "\n";
 
  /*-------------------------------------------------------------------
     end of ludcmp test
     ------------------------------------------------------------------*/
 
  /*--------------------------------------------------------------------
     test backsubstitution routine lubacksub
     -------------------------------------------------------------------*/
 
  Vector c(4);
  c[0] = 2;
  c[1] = 5;
  c[2] = 6;
  c[3] = 7;
 
  cout << "\n  vector indx";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    cout << indx[i] << "  " << c[i];
  }
  cout << endl;
 
  Vector x(4);
  lubacksub(x, b, c, indx);
 
  cout << "\n solution vector x" << endl;
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    cout << x[i];
  }
  cout << endl;
 
  cout << "\n test solution LU*x";
  Vector y(4);
  mult(y, a, x);
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    cout << y[i];
  }
  cout << "\n";
}
 
 
void test_solve_linear_system(Index ntests, Index dim, bool verbose) {
  Matrix A(dim, dim);
  Matrix LU(dim, dim);
  Vector x0(dim);
  Vector x(dim);
  Vector b(dim);
  ArrayOfIndex indx(dim);
 
  // initialize random seed
  srand((unsigned int)time(0));
 
  cout << endl << endl << "Testing linear system solution: n = " << dim;
  cout << ", ntests = " << ntests << endl;
  cout << endl << setw(10) << "Test no." << setw(20) << "lubacksub(...)";
  cout << setw(20) << "solve(...)" << endl << endl;
 
  for (Index i = 0; i < ntests; i++) {
    // Generate linear system, make sure the determinant
    // is non-zero.
    random_fill_matrix_pos_def(A, 10, false);
    random_fill_vector(x0, 10, false);
    mult(b, A, x0);
 
    // Test ludcmp/lubacksub.
    ludcmp(LU, indx, A);
    lubacksub(x, LU, b, indx);
 
    Numeric err = 0.0;
    err = get_maximum_error(x, x0, true);
 
    cout << setw(10) << i << setw(20) << err;
 
    // Test solve.
    solve(x, A, b);
 
    err = get_maximum_error(x, x0, true);
 
    cout << setw(20) << err << endl;
 
    if (verbose) {
      cout << endl;
      cout << "A:" << endl << A << endl << endl;
      cout << "x0:" << endl << x0 << endl << endl;
      cout << "x:" << endl << x << endl << endl;
      cout << "Permutation Vector:" << endl << indx << endl;
      cout << endl;
    }
  }
}
 
 
void test_inv(Index ntests, Index dim, bool verbose = false) {
  Matrix A(dim, dim);
  Matrix Ainv(dim, dim);
  Matrix I0(dim, dim);
  id_mat(I0);
  Matrix I(dim, dim);
 
  // initialize random seed
  srand((unsigned int)time(0));
 
  cout << endl << endl << "Testing matrix inversion: n = " << dim;
  cout << ", ntests = " << ntests << endl << endl;
  cout << setw(10) << "Test no." << setw(20) << "Max. rel. error" << endl
       << endl;
 
  for (Index i = 0; i < ntests; i++) {
    // Generate random matrix, make sure the determinant
    // is non-zero.
    random_fill_matrix_pos_def(A, 10, false);
 
    inv(Ainv, A);
    mult(I, Ainv, A);
 
    Numeric err = get_maximum_error(I, I0, false);
    // Print results.
    cout << setw(10) << i << setw(20) << err << endl;
 
    if (verbose) {
      cout << endl;
      cout << "A:" << endl << A << endl << endl;
      cout << "Ainv:" << endl << Ainv << endl << endl;
      cout << "A*Ainv:" << endl << I << endl << endl;
    }
  }
}
 
 
void test_matrix_exp4D(void) {
  Matrix A(4, 4);
  Matrix F(4, 4);
  A(0, 0) = 1;
  A(0, 1) = 3;
  A(0, 2) = 5;
  A(0, 3) = 6;
  A(1, 0) = 2;
  A(1, 1) = 3;
  A(1, 2) = 4;
  A(1, 3) = 4;
  A(2, 0) = 1;
  A(2, 1) = 2;
  A(2, 2) = 5;
  A(2, 3) = 1;
  A(3, 0) = 7;
  A(3, 1) = 2;
  A(3, 2) = 4;
  A(3, 3) = 3;
 
  /* set parameter for accuracy */
  Index q = 8;
 
  /*execute matrix exponential function*/
  matrix_exp(F, A, q);
 
  cout << "\n Exponential of Matrix K";
  for (Index i = 0; i < 4; i++) {
    cout << "\n";
    for (Index j = 0; j < 4; j++) cout << " " << F(i, j);
  }
  cout << "\n";
}
 
 
void test_matrix_exp1D(void) {
  Matrix A(1, 1);
  Matrix F(1, 1);
  A(0, 0) = 5;
 
  /* set parameter for accuracy */
  Index q = 8;
 
  /*execute matrix exponential function*/
  matrix_exp(F, A, q);
 
  cout << "\n Exponential of Matrix A:\n";
  cout << F(0, 0);
  cout << "\n";
}
 
 
void test_matrix_exp3D(void) {
  Matrix A(3, 3);
  Matrix F(3, 3);
  A(0, 0) = 1;
  A(0, 1) = 3;
  A(0, 2) = 5;
  A(1, 0) = 2;
  A(1, 1) = 3;
  A(1, 2) = 4;
  A(2, 0) = 1;
  A(2, 1) = 2;
  A(2, 2) = 5;
 
  /* set parameter for accuracy */
  Index q = 8;
 
  /*execute matrix exponential function*/
  matrix_exp(F, A, q);
 
  cout << "\n Exponential of Matrix A";
  for (Index i = 0; i < 3; i++) {
    cout << "\n";
    for (Index j = 0; j < 3; j++) cout << " " << F(i, j);
  }
  cout << "\n";
}
 
void test_real_diagonalize(Index ntests, Index dim) {
  Matrix A(dim, dim), F1(dim, dim), F2(dim, dim), tmp1(dim, dim),
      tmp2(dim, dim), P(dim, dim);
  Vector Wr(dim), Wi(dim);
 
  const Matrix ZEROES(dim, dim, 0);
 
  // initialize random seed
  srand((unsigned int)time(0));
 
  cout << endl << endl << "Testing diagonalize: n = " << dim;
  cout << ", ntests = " << ntests << endl;
  cout << setw(10) << "Test no." << setw(25) << "Max. rel. expm error";
  cout << setw(25) << "Max. abs. P^-1*A*P-W" << endl << endl;
 
  for (Index i = 0; i < ntests; i++) {
    // Generate a matrix that does not have complex answers...
    random_fill_matrix_symmetric(A, 10, false);
 
    // Use the two expm methods to test that diagonalize works
    matrix_exp(F1, A, 10);
    matrix_exp2(F2, A);
 
    Numeric err1 = 0.0, err2 = 0.0;
    err1 = get_maximum_error(F1, F2, true);
 
    // diagonalize directly to test that P^-1*A*P gives a diagonal matrix
    diagonalize(P, Wr, Wi, A);
 
    // P^-1*A*P
    inv(tmp1, P);
    mult(tmp2, tmp1, A);
    mult(tmp1, tmp2, P);
 
    // Minus W as diagonal matrix
    for (Index j = 0; j < dim; j++) {
      tmp1(j, j) -= Wr[j];
    }
 
    err2 = get_maximum_error(ZEROES, tmp1, false);
 
    cout << setw(10) << i << setw(25) << err1 << setw(25) << err2 << endl;
  }
}
 
void test_complex_diagonalize(Index ntests, Index dim) {
  ComplexMatrix A(dim, dim), tmp1(dim, dim), tmp2(dim, dim), P(dim, dim);
  ComplexVector W(dim);
 
  const ComplexMatrix ZEROES(dim, dim, 0);
 
  // initialize random seed
  srand((unsigned int)time(0));
 
  cout << endl << endl << "Testing diagonalize: n = " << dim;
  cout << ", ntests = " << ntests << endl;
  cout << setw(10) << "Test no.";
  cout << setw(25) << "Max. abs. P^-1*A*P-W" << endl << endl;
 
  for (Index i = 0; i < ntests; i++) {
    // Generate a matrix that does not have complex answers...
    random_fill_matrix_symmetric(A, 10, false);
 
    Numeric err = 0.0;
 
    // diagonalize directly to test that P^-1*A*P gives a diagonal matrix
    diagonalize(P, W, A);
 
    // P^-1*A*P
    inv(tmp1, P);
    mult(tmp2, tmp1, A);
    mult(tmp1, tmp2, P);
 
    // Minus W as diagonal matrix
    for (Index j = 0; j < dim; j++) {
      tmp1(j, j) -= W[j];
    }
 
    err = get_maximum_error(ZEROES, tmp1, false);
 
    cout << setw(10) << i << setw(25) << err << endl;
  }
}
 
void test_matrix_exp_propmat(Index nruns, Index ndiffs) {
  for (Index i = 0; i < nruns; i++) {
    Numeric a = ((Numeric)rand() / RAND_MAX), b = ((Numeric)rand() / RAND_MAX),
            c = ((Numeric)rand() / RAND_MAX), d = ((Numeric)rand() / RAND_MAX),
            u = ((Numeric)rand() / RAND_MAX), v = ((Numeric)rand() / RAND_MAX),
            w = ((Numeric)rand() / RAND_MAX);
 
    Matrix A(4, 4), F0(4, 4), F1(4, 4), F2(4, 4);
 
    A(0, 0) = A(1, 1) = A(2, 2) = A(3, 3) = a;
    A(0, 1) = A(1, 0) = b;
    A(0, 2) = A(2, 0) = c;
    A(0, 3) = A(3, 0) = d;
    A(1, 2) = u;
    A(2, 1) = -A(1, 2);
    A(1, 3) = v;
    A(3, 1) = -A(1, 3);
    A(2, 3) = w;
    A(3, 2) = -A(2, 3);
 
    A *= -1.0;
 
    Tensor3 dF1(ndiffs, 4, 4), dF2(ndiffs, 4, 4), dF3(ndiffs, 4, 4),
        dF4(ndiffs, 4, 4), dA1(ndiffs, 4, 4), dA2(ndiffs, 4, 4);
 
    Numeric da1 = ((Numeric)rand() / RAND_MAX),
            db1 = ((Numeric)rand() / RAND_MAX),
            dc1 = ((Numeric)rand() / RAND_MAX),
            dd1 = ((Numeric)rand() / RAND_MAX),
            du1 = ((Numeric)rand() / RAND_MAX),
            dv1 = ((Numeric)rand() / RAND_MAX),
            dw1 = ((Numeric)rand() / RAND_MAX);
 
    Numeric da2 = ((Numeric)rand() / RAND_MAX),
            db2 = ((Numeric)rand() / RAND_MAX),
            dc2 = ((Numeric)rand() / RAND_MAX),
            dd2 = ((Numeric)rand() / RAND_MAX),
            du2 = ((Numeric)rand() / RAND_MAX),
            dv2 = ((Numeric)rand() / RAND_MAX),
            dw2 = ((Numeric)rand() / RAND_MAX);
 
    dA1(joker, 0, 0) = dA1(joker, 1, 1) = dA1(joker, 2, 2) = dA1(joker, 3, 3) =
        da1;
    dA1(joker, 0, 1) = dA1(joker, 1, 0) = db1;
    dA1(joker, 0, 2) = dA1(joker, 2, 0) = dc1;
    dA1(joker, 0, 3) = dA1(joker, 3, 0) = dd1;
    dA1(joker, 1, 2) = du1;
    dA1(joker, 2, 1) = -du1;
    dA1(joker, 1, 3) = dv1;
    dA1(joker, 3, 1) = -dv1;
    dA1(joker, 2, 3) = dw1;
    dA1(joker, 3, 2) = -dw1;
 
    dA1 *= -1.0;
 
    dA2(joker, 0, 0) = dA2(joker, 1, 1) = dA2(joker, 2, 2) = dA2(joker, 3, 3) =
        da2;
    dA2(joker, 0, 1) = dA2(joker, 1, 0) = db2;
    dA2(joker, 0, 2) = dA2(joker, 2, 0) = dc2;
    dA2(joker, 0, 3) = dA2(joker, 3, 0) = dd2;
    dA2(joker, 1, 2) = du2;
    dA2(joker, 2, 1) = -du2;
    dA2(joker, 1, 3) = dv2;
    dA2(joker, 3, 1) = -dv2;
    dA2(joker, 2, 3) = dw2;
    dA2(joker, 3, 2) = -dw2;
 
    dA2 *= -1.0;
 
    special_matrix_exp_and_dmatrix_exp_dx_for_rt(F1, dF1, dF2, A, dA1, dA2, 10);
 
    cayley_hamilton_fitted_method_4x4_propmat_to_transmat__explicit(
        F1, dF1, dF2, A, dA1, dA2);
    cayley_hamilton_fitted_method_4x4_propmat_to_transmat__eigen(
        F2, dF3, dF4, A, dA1, dA2);
 
    matrix_exp(F0, A);
    cayley_hamilton_fitted_method_4x4_propmat_to_transmat__eigen(F1, A);
    cayley_hamilton_fitted_method_4x4_propmat_to_transmat__explicit(F2, A);
 
    std::cout << MapToEigen(F1) << "\n\n";
    std::cout << MapToEigen(F2) << "\n\n";
    std::cout << MapToEigen(F1) - MapToEigen(F2) << "\n\n";
    std::cout << MapToEigen(dF1(0, joker, joker)) << "\n\n";
    std::cout << MapToEigen(dF3(0, joker, joker)) << "\n\n";
    std::cout << MapToEigen(dF3(0, joker, joker)) -
                     MapToEigen(dF1(0, joker, joker))
              << "\n\n";
    std::cout << MapToEigen(dF2(0, joker, joker)) << "\n\n";
    std::cout << MapToEigen(dF4(0, joker, joker)) << "\n\n";
    std::cout << MapToEigen(dF4(0, joker, joker)) -
                     MapToEigen(dF2(0, joker, joker))
              << "\n\n";
  }
}
 
int main(void) {
  // test_lusolve4D();
  // test_inv( 20, 1000 );
  // test_solve_linear_system( 20, 1000, false );
  // test_matrix_exp1D();
  //  test_real_diagonalize(20,100);
  //  test_complex_diagonalize(20,100);
  test_matrix_exp_propmat(100000, 10);
  return (0);
}