A = {{a₀, a₁}, {a₁, a₂}}, b = {b₀, b₁}, x₀ = {0, 0}

Step 1: r₀ = b - Ax₀; p₀ = r₀

r[0] = b₀

r[1] = b₁

Step 2: alpha₀ = <r₀, r₀> / <p₀, Ap₀>

<p₀, Ap₀> = b₀*(a₀*b₀ + a₁*b₁) + b₁*(a₁*b₀ + a₂*b₁)

alpha₀ = (b₀²+b₁²) / b₀*(a₀*b₀ + a₁*b₁) + b₁*(a₁*b₀ + a₂*b₁)

Step 3: r₁ = r₀ - alpha₀ * Ap₀

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: x₁ = x₀ + alpha*p₀

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 = <r₁, r₁> / <r₀, r₀>

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: p₁ = r₁ +beta * p₀

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: alpha₁ = <r₁, r₁> / <p₁, Ap₁>

<p₁, Ap₁> = ((-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

alpha₁ = (a₀*b₀² + 2*a₁*b₀*b₁ + a₂*b₁²) / ((-a₁² + a₀*a₂) (b₀² + b₁²))

Step 8: r₂ = r₁ - alpha₁ * Ap₁

r[0] = 0

r[1] = 0

Step 9: x₂ = x₁ + alpha₁*p₁

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 - a₁²) / (a₀*a₂ - a₁²) = 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 - a₁²) / (a₀*a₂ - a₁²) = b₁

b[0] = b₀

b[1] = b₁

assert(bncg[i] == b[i])

END