- Many numerical methods involves
matrix-vector multiplications
- A matrix is typically
represented
by a
two-dimensional array
- Example:
#define N ... double A[N][N];
- A vector is
represented
by a
one-dimensional array
- Example:
#define N ... double v[N];
- Matrix-vector multiplication algorithm (with ordinary matrices)
double A[N][N]; double v[N]; double result[N]; // Initialized result to [0,0,...,0] for (i = 0; i < N; i++) result[i] = 0.0; // Multiply matrix row with vector for (i = 0; i < N; i++) // row i { for (j = 0; j < N; j++) result[i] = result[i] + A[i][j] * v[j]; }
- Input parameters:
- Input Matrix
- Input matrix dimensions
- Input Vector
- Output parameter:
- Output Vector
Because the output vector is NOT a single value , the output value must be returned in a array variable
- Matrix-vector multiply function
// m = # rows // n = # columns void MatrixVectorMult( double *out[], double *A[], double *in[], int m, int n) { int i, j; // Set output vector to ZERO for (i = 0; i < m; i = i + 1) { out[i] = 0.0; } // Matrix vector multiply algorithm for (i = 0; i < m; i = i + 1) { for (j = 0; j < n; j = j + 1) { out[i] = out[i] + A[i*n+j]*in[j]; } } }
- How to use the function:
int main(int argc, char ** argv) { double A[8][8]; double u[8], v[8]; ... MatrixVectorMult(v, (double *)A, u, 8, 8); }
- Example Program:
(Demo above code)
- Matrix-vector multiply function: click here
- Sample main program: click here
- In many problems (e.g.,
matrices that results from discretization),
the problem matrix is very
sparse
Although sparse matrices can be stored using a two-dimensional array, it is a very bad idea to do so for several reasons:
- Very
inefficient use of computer memory:
it is not very useful to store many 0 values of the sparse matrix
- The more important reason is
deteriorated runtime speed:
the computer
program needs to perform
unnecessary computations
(e.g.,: x + 0*y = x, you can skip the multiplication and addition operations)
- Very
inefficient use of computer memory:
it is not very useful to store many 0 values of the sparse matrix
- NOTE:
-
Although the
problem matrix
is sparse,
the solution vector
is NOT sparse
- Example:
Sparse Matrix -------------------------- 0 1 2 3 4 5 6 7 0 [ 11 12 14 ] [ 3 ] [3*11 + 5*12 + 2*0 + 1*14 + 0*0 + 1*0 + 2*0 + 4*0] 1 [ 22 23 25 ] [ 5 ] [... 2 [ 31 33 34 ] [ 2 ] [... 3 [ 42 45 46 ] [ 1 ] [... 4 [ 55 ] x [ 0 ] = [... 5 [ 65 66 67 ] [ 1 ] [... 6 [ 75 77 78 ] [ 2 ] [... 7 [ 87 88 ] [ 4 ] [... A[8][8] v[8] - A two-dimensional array is wasteful in space and wasteful in computation (many multiplications involves ZERO)
- Storage methods:
- Arrays
- Linked lists
- Storing
sparse matrices
using
multiple arrays :
- Coordinate-wise
- Compressed Sparse Row (CSR)
- Compressed Sparse Column (CSC) (similar to CSR, skipped)
- These techniques can also be implemented with
linked lists
with enhanced flexibility
- Each type of representation has its strengths and its weaknesses.
Depending on what operations you want to perform with the sparse matrix, you pick the best representation...