matrix inverse

moctar (9)
i am trying to write a program to get the inverse of any type of square matrix using strassen based matrix inversion.
but what i don't understand is how to represent the sub-block matrix in the form A^-1, B^-1 etc...

since i have to apply it to different matrixes i can't really use something like
A^-1 = 1/determinant. so how to implement the blockkwise inverse.
a code would really be helpful.
lastchance (6980)
What do you want to do with this matrix inverse?

Note that it's actually surprisingly rare to require a matrix inverse. It's usually easier to QR or LU factorise, because those factorised systems can be inverted quickly as and when necessary.
moctar (9)
i mean not a program but a function, to get the inverse of a matrix using strassen matrix inversion method given a matrix as input. It is explained on this link but i don't know how to implement it.

http://tesla.pmf.ni.ac.rs/people/DeXteR/old/Papers/RapidLU.pdf
lastchance (6980)
You haven't answered my question.

What are you intending to do with the matrix inverse once you've got it?
moctar (9)
i want to multiply the matrix inverse with the matrix using strassen multiplication to get the identity matrix
lastchance (6980)
i want to multiply the matrix inverse with the matrix using strassen multiplication to get the identity matrix

Well, I can think of easier ways of getting the identity matrix.


Anyway, I tried coding from the link that you provided (see the Algorithm 2.1 on page 4, and the equations it refers to.)

It appears to work ...

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#include <iostream>
#include <iomanip>
#include <string>
#include <vector>
#include <algorithm>
#include <cassert>
#include <cstdlib>
#include <cmath>
using namespace std;

const double SMALL = 1.0E-30;          // used to stop divide-by-zero
const double NEARZERO = 1.0e-10;       // helps in printing

using vec    = vector<double>;         // vector
using matrix = vector<vec>;            // matrix

// Function prototypes
void print( const string &title, const matrix &A );
matrix matmul( const matrix &A, const matrix &B );
matrix subtract( const matrix &A, const matrix &B );
matrix oppsign( matrix A );
matrix subMatrix( const matrix &A, int i1, int i2, int j1, int j2 );
matrix assembly( const matrix &A11, const matrix &A12, const matrix &A21, const matrix &A22 );
matrix inverse( const matrix &A );

//======================================================================

void print( const string &title, const matrix &A )
{
   if ( title != "" ) cout << title << '\n';

   for ( auto &row : A )
   {
      for ( auto x : row ) cout << setw( 15 ) << ( abs( x ) < NEARZERO ? 0.0 : x );
      cout << '\n';
   }
}

//======================================================================

matrix matmul( const matrix &A, const matrix &B )          // Matrix times matrix
{
   int rowsA = A.size(),   colsA = A[0].size();
   int rowsB = B.size(),   colsB = B[0].size();
   assert( colsA == rowsB );

   matrix C( rowsA, vec( colsB, 0.0 ) );
   for ( int i = 0; i < rowsA; i++ )
   {
      for ( int j = 0; j < colsB; j++ )
      {
         for ( int k = 0; k < colsA; k++ ) C[i][j] += A[i][k] * B[k][j];
      }
   }
   return C;
}

//======================================================================

matrix subtract( const matrix &A, const matrix &B )        // Subtract matrices
{
   int rows = A.size(),   cols = A[0].size();
   assert( rows == B.size() && cols == B[0].size() );

   matrix result( rows, vec( cols ) );
   for ( int i = 0; i < rows; i++ )
   {
      for ( int j = 0; j < cols; j++ ) result[i][j] = A[i][j] - B[i][j];
   }
   return result;
}

//======================================================================

matrix oppsign( matrix A )                                  // Minus matrix
{
   for ( auto &row : A )
   {
      for ( auto &e : row ) e = -e;
   }
   return A;
}

//======================================================================

matrix subMatrix( const matrix &A, int i1, int i2, int j1, int j2 )
{
   int rows = i2 - i1 + 1, cols = j2 - j1 + 1;
   matrix result( rows, vec( cols ) );
   for ( int i = i1, r = 0; i <= i2; i++, r++ )
   {
      auto it1 = A[i].begin() + j1, it2 = A[i].begin() + j2 + 1;
      copy( it1, it2, result[r].begin() );
   }
   return result;
}

//======================================================================

matrix assembly( const matrix &A11, const matrix &A12, const matrix &A21, const matrix &A22 )
{
   int k = A11.size();           
   int n = k + A22.size();
   matrix result( n, vec( n ) );

   for ( int i = 0; i < k; i++ )
   {
      copy( A11[i].begin(), A11[i].end(), result[i].begin()     );
      copy( A12[i].begin(), A12[i].end(), result[i].begin() + k );
   }

   for ( int i = k; i < n; i++ )
   {
      copy( A21[i-k].begin(), A21[i-k].end(), result[i].begin()     );
      copy( A22[i-k].begin(), A22[i-k].end(), result[i].begin() + k );
   }

   return result;
}

//======================================================================

matrix inverse( const matrix &A )
{
   int n = A.size();
   if ( n == 1 ) 
   {
      double value = A[0][0];
      if ( abs( value ) < SMALL )
      {
         cerr << "Non-invertible. Giving up.\n";
         exit( 0 );
      }
      return matrix( 1, vec( 1, 1.0 / value ) );
   }

   // Partition into four
   int k = n / 2;
   matrix A11 = subMatrix( A, 0, k - 1, 0, k - 1 );
   matrix A12 = subMatrix( A, 0, k - 1, k, n - 1 );
   matrix A21 = subMatrix( A, k, n - 1, 0, k - 1 );
   matrix A22 = subMatrix( A, k, n - 1, k, n - 1 );

   // Strassen steps
   matrix R1  = inverse( A11 );
   matrix R2  = matmul( A21, R1 );
   matrix R3  = matmul( R1, A12 );
   matrix R4  = matmul( A21, R3 );
   matrix R5  = subtract( R4, A22 );
   matrix R6  = inverse( R5 );
   matrix X12 = matmul( R3, R6 );
   matrix X21 = matmul( R6, R2 );
   matrix R7  = matmul( R3, X21 );
   matrix X11 = subtract( R1, R7 );
   matrix X22 = oppsign( R6 );

   return assembly( X11, X12, X21, X22);
}

//====================================================================== 

int main()
{
   // Data
   matrix A = { { 1, 2, 3, 4, 5 }, { 9, 88, 7, 63, 5 }, { 1, 3, 15, 32, 1 }, { 2, 4, 22, 4, 222 }, { 3, 5, 9, 0, 0 } };
   print( "A:", A );
   matrix B = inverse( A );
   print( "\nB:", B );
   print( "\nCheck AB=I:", matmul( A, B ) );
}


A:
              1              2              3              4              5
              9             88              7             63              5
              1              3             15             32              1
              2              4             22              4            222
              3              5              9              0              0

B:
        2.34541     -0.0288739      -0.229938     -0.0511385       -0.25111
       -0.36173      0.0150113       0.014695     0.00774279      0.0654828
      -0.580842     0.00128504      0.0684823      0.0127446       0.158435
       0.231676    -0.00110368     0.00511536    -0.00521612     -0.0721421
      0.0387744   -0.000117808    -0.00507196     0.00365671     -0.0133185

Check AB=I:
              1              0              0              0              0
              0              1              0              0              0
              0              0              1              0              0
              0              0              0              1              0
              0              0              0              0              1


Last edited on
moctar (9)
Thank you! i will have a look into it.
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