c - How to generate a very large non singular matrix A in Ax = b? -

i solving system of linear algebraic equations ax = b using jacobian method taking manual inputs. want analyze performance of solver large system. there method generate matrix i.e non singular? attaching code here.`

#include<stdio.h> #include<stdlib.h> #include<math.h>  #define tol = 0.0001  void main() {   int size,i,j,k = 0;   printf("\n enter number of equations: ");   scanf("%d",&size);   double reci = 0.0;   double *x = (double *)malloc(size*sizeof(double));   double *x_old = (double *)malloc(size*sizeof(double));    double *b = (double *)malloc(size*sizeof(double));   double *coeffmat = (double *)malloc(size*size*sizeof(double));    printf("\n enter coefficient matrix: \n");    for(i = 0; < size; i++)   {     for(j = 0; j < size; j++)     {       printf(" coeffmat[%d][%d] = ",i,j);       scanf("%lf",&coeffmat[i*size+j]);       printf("\n");       //coeffmat[i*size+j] = 1.0;     }   }    printf("\n enter b vector: \n");    for(i = 0; < size; i++)   {     x[i] = 0.0;     printf(" b[%d] = ",i);     scanf("%lf",&b[i]);       }    double sum = 0.0;    while(k < size)   {      for(i = 0; < size; i++)     {       x_old[i] = x[i];     }      for(i = 0; < size; i++)     {       sum = 0.0;       for(j = 0; j < size; j++)       {         if(i != j)         {           sum += (coeffmat[i * size + j] * x_old[j] );         }       }        x[i] = (b[i] -sum) / coeffmat[i * size + i];      }      k = k+1;   }    printf("\n solution is: ");    for(i = 0; < size; i++)   {     printf(" x[%d] = %lf \n ",i,x[i]);   }  } 

talonmies' comment mentions http://www.eecs.berkeley.edu/pubs/techrpts/1991/csd-91-658.pdf right approach (at least in principle, , in full generality).

however, not handling "very large" matrixes (e.g. because program use naive algorithms, , because don't run on large supercomputer lot of ram). naive approach of generating matrix random coefficients , testing afterwards non-singular enough.

very large matrixes have many billions of coefficients, , need powerful supercomputer e.g. terabytes of ram. don't have that, if did, program run long (you don't have parallelism), might give wrong results (read http://floating-point-gui.de/ more) don't care.

a matrix of million coefficients (e.g. 1024*1024) considered small current hardware standards (and more enough test code on current laptops or desktops, , test parallel implementations), , generating randomly of them (and computing determinant test not singular) enough, , doable. might generate them and/or check regularity external tool, e.g. scilab, r, octave, etc. once program computed solution x₀, use tool (or write program) compute ax₀ - b , check close 0 vector (there cases disappointed or surprised, since round-off errors matter).

you'll need enough pseudo random number generator perhaps simple drand48(3) considered obsolete (you should find , use better); seed random source (e.g. /dev/urandom on linux).

btw, compile code warnings & debug info (e.g. gcc -wall -wextra -g). #define tol = 0.0001 wrong (should #define tol 0.0001 or const double tol = 0.0001;). use debugger (gdb) & valgrind. add optimizations (-o2 -mcpu=native) when benchmarking. read documentation of every used function, notably <stdio.h>. check result count scanf... in c99, should not cast result of malloc, forgot test against failure, code:

double *b = malloc(size*sizeof(double)); if (!b) {perror("malloc b"); exit(exit_failure); }; 

you'll rather end, not start, printf control strings \n because stdout (not always!) line buffered. see fflush.

you should read basic linear algebra textbook...

notice writing robust , efficient programs invert matrixes or solve linear systems difficult art (which don't know @ : has programming issues, algorithmic issues, , mathematical issues; read numerical analysis book). can still phd , spend whole life working on that. please understand need ten years learn programming (or many other things).