A = {{a₀, a₁}, {a₁, a₂}}, b = {b₀, b₁}

Step 1: r = b - Ax

r[0] = b₀

r[1] = b₁

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

p[i]*(A[i][j]*p[j]) = a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2

alpha = (b₀2+b₁2) / ((b₁2)*a₂+2*(b₁*b₀*a₁)+(b₀2)*a₀)

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

r[0] = (-b₁*(-a₁*b₀2 + a₀*b₀*b₁ - a₂*b₀*b₁ + a₁*b₁2)) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)

r[1] = (b₀*(-a₁*b₀2 + a₀*b₀*b₁ - a₂*b₀*b₁ + a₁*b₁2)) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)

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

x[0] = (b₀*(b₀2 + b₁2)) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)

x[1] = (b₁*(b₀2 + b₁2)) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)

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

rsnew = ((b₀2 + b₁2)*(-a₁*b₀2 + a₀*b₀*b₁ - a₂*b₀*b₁ + a₁*b₁2)2) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)2

beta = (a₁*b₀2 - a₀*b₀*b₁ + a₂*b₀*b₁ - a₁*b₁2)2 / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)2

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

p[0] = (-1)*((a₁*b₀ + a₂*b₁)*(b₀2 + b₁2)*(-a₁*b₀2 + a₀*b₀*b₁ - a₂*b₀*b₁ + a₁*b₁2)) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)2

p[1] = ((a₀*b₀ + a₁*b₁)*(b₀2 + b₁2)*(-a₁*b₀2 + a₀*b₀*b₁ - a₂*b₀*b₁ + a₁*b₁2)) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)2

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

p[i]*(A[i][j]*p[j]) = ((-a₁2 + a₀*a₂)*(b₀2 + b₁2)2*(-a₁*b₀2 + a₀*b₀*b₁ - a₂*b₀*b₁ + a₁*b₁2)2) / (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2)3

alpha = (a₀*b₀2 + 2*a₁*b₀*b₁ + a₂*b₁2) / ((-a₁2 + a₀*a₂) (b₀2 + b₁2))

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] = (a₂*b₀ - a₁*b₁) / (a₀*a₂ - a₁2)

x[1] = (-a₁*b₀ + a₀*b₁) / (a₀*a₂ - a₁2)

assertion:

bncg[0] = A[0][0]*X[0] + A[0][1]*X[1] = a₀*(a₂*b₀ - a₁*b₁) / (a₀*a₂ - a₁2) + a₁*(-a₁*b₀ + a₀*b₁) / (a₀*a₂ - a₁2) = b₀*(a₀*a₂-a₁^{2) / (a₀*a₂-a₁}2) = b₀

bncg[1] = A[1][0]*X[0] + A[1][1]*X[1] = a₁*(a₂*b₀ - a₁*b₁) / (a₀*a₂ - a₁2) + a₂*(-a₁*b₀ + a₀*b₁) / (a₀*a₂ - a₁2) = b₁*(a₀*a₂-a₁^{2) / (a₀*a₂-a₁}2) = b₁

b[0] = b₀

b[1] = b₁

END