source: CIVL/examples/cg/cg2_sylvester.cvl@ 897c8d7

/* 
 * CG 2x2 case with positive definite assumption based on Sylvester's criteria.
 * From wiki:
 * Sylvester's criterion states that a Hermitian matrix M is positive-definite 
 * if and only if all the following matrices have a positive determinant:
 * the upper left 1-by-1 corner of M,
 * the upper left 2-by-2 corner of M,
 * the upper left 3-by-3 corner of M,
 * M itself.
 * In other words, all of the leading principal minors must be positive.
 */
#include <civlc.cvh>
#include <stdio.h>
#define n 2

$input double a1,a2,a3;
$input double b[n];
double x[n];
double xcg[n];

void cg(double A[n][n], double b[n], double x[n], int steps) {
  double r[n];
  double p[n];
  double temp[n];
  double tempp[n];
  double rsold;
  double rsnew;
  double rsfrac;
  double alpha;

  for(int i=0; i<n; i++) x[i] = 0;
  for(int i=0; i<n; i++) {
    temp[i] = 0.0;
    for(int j=0; j<n; j++) {
      temp[i] += A[i][j]*x[j];
    }
  }
  for(int i=0; i<n; i++) {
    r[i] = b[i] -temp[i];
  }
  rsold = 0.0;
  for(int i=0; i<n; i++) {
    rsold += r[i]*r[i];
  }
  for(int i=0; i<n; i++) {
    p[i] = r[i];
  }
  for(int i=0; i<steps; i++) {
    for(int i=0; i<n; i++) {
      temp[i] = 0.0;
      for(int j=0; j<n; j++) {
        temp[i] += A[i][j]*p[j];
      }
    }
    alpha = 0.0;
    for(int i=0; i<n; i++) {
      alpha += p[i]*temp[i];
    }
    
    $assume(alpha !=0);

    alpha = rsold/alpha;

    // tempp = r-alpha*temp
    for(int i=0; i<n; i++) {
      tempp[i] = r[i] -alpha*temp[i];
    }

    // r = tempp
    for(int i=0; i<n; i++) {
      r[i] = tempp[i];
    }

    // tempp = x+alpha*p
    for(int i=0; i<n; i++) {
      tempp[i] = x[i] +alpha*p[i];
    }

    // x = tempp
    for(int i=0; i<n; i++) {
      x[i] = tempp[i];
    }

    if(i<steps-1) {
      // rsnew = <r,r>
      rsnew = 0.0;
      for(int i=0; i<n; i++) {
        rsnew += r[i]*r[i];
      }

      $assume(rsold !=0);

      rsfrac = rsnew/rsold;
      for(int i=0; i<n; i++) {
        tempp[i] = r[i] +rsfrac*p[i];
      }

      // p = tempp
      for(int i=0; i<n; i++) {
        p[i] = tempp[i];
      }
      rsold = rsnew;
    }
  }
}

void main() {
  double bncg[n];
  double A[n][n];
  A[0][0] = a1;
  A[0][1] = a2;
  A[1][0] = a2;
  A[1][1] = a3;
  $assume(b[0]!=0 || b[1]!=0);
  $assume(a1 > 0);              //assumption of Sylvester's criterion
  $assume( (a1*a3 - a2*a2) > 0);//all principal minors are positive.
  cg(A,b,xcg,n);
  printf("\n================Solution x:================\n");
  for(int i=0; i<n; i++) {
    printf("x[%d] = %f\n\n",i, xcg[i]);
  }
  for(int i=0; i<n; i++) {
    bncg[i] = 0;
    for(int j=0; j<n; j++) {
      bncg[i] += A[i][j]*xcg[j];
    }
    $assert(bncg[i] == b[i]);
  }
}