A = {{X0, X1}, {X1, X2}}, b = X3[2]

Step 1: r = b - Ax

r[0] = X3[0]

r[1] = X3[1]

Step 2: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])

p[i]*(A[i][j]*p[j]) = X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²

alpha = (X3[0]²+X3[1]²) / ((X3[1]²)*X2+2*(X3[1]*X3[0]*X1)+(X3[0]²)*X0)

Step 3: r[i] = r - alpha * A[i][j] * p[j]

r[0] = (-X3[1]*(-X1*X3[0]² + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]²)) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)

r[1] = (X3[0]*(-X1*X3[0]² + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]²)) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)

Step 4: x[i] = x + alpha*p[i]

x[0] = (X3[0]*(X3[0]² + X3[1]²)) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)

x[1] = (X3[1]*(X3[0]² + X3[1]²)) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)

Step 5: beta = rsnew / rsold = (rk[i]*rk[i]) / (r[i]*r[i])

rsnew = ((X3[0]² + X3[1]²)*(-X1*X3[0]² + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]²)²) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)²

beta = (X1*X3[0]² - X0*X3[0]*X3[1] + X2*X3[0]*X3[1] - X1*X3[1]²)² / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)²

Step 6: p[i] = rk[i] +beta * p

p[0] = (-1)*((X1*X3[0] + X2*X3[1])*(X3[0]² + X3[1]²)*(-X1*X3[0]² + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]²)) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)²

p[1] = ((X0*X3[0] + X1*X3[1])*(X3[0]² + X3[1]²)*(-X1*X3[0]² + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]²)) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)²

Step 7: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])

p[i]*(A[i][j]*p[j]) = ((-X1² + X0*X2)*(X3[0]² + X3[1]²)²*(-X1*X3[0]² + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]²)²) / (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²)³

alpha = (X0*X3[0]² + 2*X1*X3[0]*X3[1] + X2*X3[1]²) / ((-X1² + X0*X2) (X3[0]² + X3[1]²))

Step 8: r[i] = r - alpha * A[i][j] * p[j]

r[0] = 0

r[1] = 0

Step 9: x[i] = x + alpha*p[i]

x[0] = (X2*X3[0] - X1*X3[1]) / (X0*X2 - X1²)

x[1] = (-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1²)

assertion: bncg[i] = A[i][j]*x[j]

bncg[0] = X3[0]

bncg[1] = X3[1]

b[0] = X3[0]

b[1] = X3[1]

END