source: CIVL/examples/cg/cg3_sylvester.cvl

/* 
 * CG 3x3 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 3

$input double diag1,diag2,diag3,off1,off2,off3;
$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;

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

  // temp = A*x
  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];
    }
  }

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

  // rsold = <r,r>
  rsold = 0.0;
  for(int i=0; i<n; i++) {
    rsold += r[i]*r[i];
  }

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

  for(int i=0; i<steps; i++) {
    // temp = A*p
    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];
    }
    for(int i=0; i<n; i++) {
      r[i] = tempp[i];
    }
    for(int i=0; i<n; i++) {
      tempp[i] = x[i] +alpha*p[i];
    }
    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];
      }
      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] = diag1;
  A[1][1] = diag2;
  A[2][2] = diag3;
  A[0][1] = off1;
  A[1][0] = off1;
  A[0][2] = off2;
  A[2][0] = off2;
  A[1][2] = off3;
  A[2][1] = off3;
  
  $assume(b[0]!=0 || b[1]!=0 || b[2]!=0);
  $assume(diag1 > 0);                     //assumption of Sylvester's criterion
  $assume((diag1*diag2 - off1*off1) > 0);//all principal minors are positive.
  $assume((diag1*diag2*diag3 + 2*off1*off2*off3 - diag1*off3*off3 - diag2*off2*off2 - diag3*off1*off1) > 0);
  
  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]);
  }
}