lin_alg.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 "lin_alg.h"
 
/*===========================================================================
  === External declarations
  ===========================================================================*/
 
#include <stdexcept>
#include <cmath>
#include "arts.h"
#include "make_vector.h"
#include "array.h"
#include "logic.h"
 
 
 
 
void ludcmp(MatrixView LU,
       ArrayOfIndex& indx, 
       ConstMatrixView A) 
{
  Index imax = 0;
  const Numeric TINY=1.0e-20;
  Numeric big, dum, sum, temp, d;
 
  LU = A;
  const Index dim = A.nrows();
  assert(is_size(A,dim,dim)); //check, if A is quadratic
 
  Vector vv(dim);  //stores implicit scaling of each row
  d = 1.0;
  for (Index i=0; i<dim; i++)
    {
      indx[i]= i;
      big = 0.0;
      for (Index j=0; j<dim; j++)
        {
          if ((temp = abs(LU(i,j))) > big)
            big = temp;
        }
      if (big == 0.)
        throw runtime_error("ludcmp: Matrix is singular");
      vv[i] = 1.0/big; // save scaling
      }
  
  for (Index j=0; j<dim; j++)
    {
      for (Index i=0; i<j; i++)
        {
          sum = LU(i,j);
          for (Index k=0; k<i; k++) 
            sum -= LU(i,k)*LU(k,j);
          LU(i,j) = sum;
        }
      big = 0.0;
      for( Index i=j; i<dim; i++)
        {
          sum = LU(i,j);
          for (Index k=0; k<j; k++)
            sum -= LU(i,k)*LU(k,j);
          LU(i,j) = sum;
          if( (dum = vv[i]*fabs(sum)) >= big) 
            {
              big = dum;
              imax = i;
            }
        }
      if (j!=imax)
        {
          for(Index k=0; k<dim; k++)
            {
              dum = LU(imax,k);
              LU(imax,k) = LU(j,k);
              LU(j,k) = dum;
            }
          d = -d;
          vv[imax] = vv[j];
          indx[j] = imax;
          indx[imax] =j;
         }
     
      if(LU(j,j) == 0.0) LU(j,j) = TINY;
      
      if (j != dim) 
        {
          dum=1.0/LU(j,j);
          for (Index i=j+1; i<dim; i++)
            LU(i,j) *=dum;
        }
    }
}
 
 
 
 
void lubacksub(VectorView x, 
          ConstMatrixView LU, 
          ConstVectorView b, 
          const ArrayOfIndex& indx)
{
  Index dim = LU.nrows(); 
 
  /* Check if the dimensions of the input matrix and vectors agree and if LU is a quadratic matrix.*/
  assert(is_size(LU, dim, dim));
  assert(is_size(b, dim));
  assert(is_size(indx, dim));
 
  
  for(Index i=0; i<dim; i++)
     {
       x[indx[i]] = b[i];
     }
 
  for (Index i=0; i<dim; i++)
    {
     Numeric sum = x[i];
     for (Index j=0; j<=i-1; j++)
       sum -= LU(i,j)*x[j]; 
      x[i] = sum;
     }
 
  for(Index i=dim-1; i>=0; i--)
    {
      Numeric sum = x[i];
      for (Index j=i+1; j<dim; j++)
        sum -= LU(i,j)*x[j];
      x[i] = sum/LU(i,i);
    }
}
 
 
 
 
void matrix_exp(
         MatrixView      F,
         ConstMatrixView A,
   const Index&          q )
{
  const Index n = A.ncols();
 
  /* Check if A and F are a quadratic and of the same dimension. */
  assert( is_size(A,n,n) );
  assert( is_size(F,n,n) );
 
  Numeric A_norm_inf, c;
  Numeric j;
  Matrix D(n,n), N(n,n), X(n,n), cX(n,n,0.0), B(n,n,0.0);
  Vector N_col_vec(n,0.), F_col_vec(n,0.);
 
  A_norm_inf = norm_inf(A);
 
  // This formular is derived in the book by Golub and Van Loan.
  j = 1 +  floor(1./log(2.)*log(A_norm_inf));
  
  if(j<0) j=0.;
  Index j_index = (Index)(j);
 
  // Scale matrix
  F = A;
  F /= pow(2,j);
 
  /* The higher q the more accurate is the computation, 
     see user guide for accuracy */
  //  Index q = 8;
  Numeric q_n = (Numeric)(q);
  id_mat(D);
  id_mat(N);
  id_mat(X);
  c = 1.;
 
  for(Index k=0; k<q; k++)
    {
      Numeric k_n = (Numeric)(k+1);
      c *= (q_n-k_n+1)/((2*q_n-k_n+1)*k_n);
      mult(B,F,X);              // X = F * X
      X = B;
      cX = X;                   
      cX *= c;                  // cX = X*c
      N += cX;                  // N = N + X*c
      cX *= pow(-1,k_n);        // cX = (-1)^k*c*X
      D += cX;                  // D = D + (-1)^k*c*X
    }
 
  /*Solve the equation system DF=N for F using LU decomposition,
   use the backsubstitution routine for columns of N*/ 
 
   /* Now use X  for the LU decomposition matrix of D.*/
  ArrayOfIndex indx(n);
 
  ludcmp(X, indx, D);
 
  for(Index i=0; i<n; i++)
    {
      N_col_vec = N(joker,i);  // extract column vectors of N
      lubacksub(F_col_vec, X, N_col_vec, indx);
      F(joker,i) = F_col_vec;  // construct F matrix  from column vectors 
    }
  
  /* The square of F gives the result. */
  for(Index k=0; k<j_index; k++)
    {
      mult(B,F,F);              // F = F^2
      F = B;
    }
}
 
 
 
Numeric norm_inf(ConstMatrixView A)
{
  Numeric norm_inf = 0;
  
  for(Index j=0; j<A.nrows(); j++)
    {
      Numeric row_sum = 0;
      //Calculate the row sum for all rows
      for(Index i=0; i<A.ncols(); i++)
        row_sum += abs(A(i,j));
      //Pick out the row with the highest row sum
      if( norm_inf < row_sum)
        norm_inf = row_sum;
    }
  return norm_inf;
}
 
 
 
void id_mat(MatrixView I)
{
 
  const Index n = I.ncols();
  assert(n == I.nrows());
  
  I = 0;
  for(Index i=0; i<n; i++)
    I(i,i) = 1.;
}
 
 
 
Numeric det(ConstMatrixView A)
{
    const Index dim = A.nrows();
    assert(dim == A.ncols());
 
    if(dim == 3)
        return A(0,0)*A(1,1)*A(2,2) + A(0,1)*A(1,2)*A(2,0) +
               A(0,2)*A(1,0)*A(2,1) - A(0,2)*A(1,1)*A(2,0) -
               A(0,1)*A(1,0)*A(2,2) - A(0,0)*A(1,2)*A(2,1);
    else if(dim == 2)
        return A(0,0) * A(1,1) - A(0,1) * A(1,0);
    else if(dim == 1)
        return A(0,0);
 
    Numeric ret_val = 0.;
 
    for(Index j = 0; j < dim; j++)
    {
        Matrix temp(dim-1,dim-1);
        for(Index I = 1; I < dim; I++)
            for(Index J = 0; J < dim; J++)
            {
                if(J < j)
                    temp(I-1,J) = A(I,J);
                else if(J > j)
                    temp(I-1,J-1) = A(I,J);
            }
 
        Numeric tempNum = det(temp);
 
        ret_val += ((j%2 == 0)?-1.:1.) * tempNum * A(0,j);
    }
    return ret_val;
}
 
 
 
void linreg(
       Vector&    p,
  ConstVectorView x, 
  ConstVectorView y ) 
{
  const Index n = x.nelem();
  
  assert( y.nelem() == n );
 
  p.resize(2);
 
  // Basic algorithm found at e.g. 
  // http://en.wikipedia.org/wiki/Simple_linear_regression
  // The basic algorithm is as follows:
  /*
  Numeric s1=0, s2=0, s3=0, s4=0;
  for( Index i=0; i<n; i++ )
    {
      s1 += x[i] * y[i];
      s2 += x[i];
      s3 += y[i];
      s4 += x[i] * x[i];
    }
    
  p[1] = ( s1 - (s2*s3)/n ) / ( s4 - s2*s2/n ); 
  p[0] = s3/n - p[1]*s2/n;
  */
 
  // A version abit more numerical stable:
  // Mean value of x is removed before the fit: x' = (x-mean(x)) 
  // This corresponds to that s2 in version above becomes 0
  // y = a + b*x'
  // p[1] = b
  // p[0] = a - p[1]*mean(x)
 
  Numeric s1=0, xm=0, s3=0, s4=0;
 
  for( Index i=0; i<n; i++ )
    { xm += x[i]/Numeric(n); }
 
  for( Index i=0; i<n; i++ )
    {
      const Numeric xv = x[i] - xm;
      s1 += xv * y[i];
      s3 += y[i];
      s4 += xv * xv;
    }
    
  p[1] = s1 / s4; 
  p[0] = s3/Numeric(n) - p[1]*xm;
}