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

Step 1: r₀ = b - Ax₀

r[0] = b₀

r[1] = b₁

p₀ = r₀

Step 2: alpha₀ = <r₀, r₀> / <p₀, Ap₀>

<p, Ap> = a₀*b₀² + 2*a₁*b₀*b₁ + a₂*b₁²

alpha₀ = (b₀²+b₁²) / ((b₁²)*a₂+2*(b₁*b₀*a₁)+(b₀²)*a₀)

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: x[i] = x + alpha*p

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 = <r_k+1, r_k+1> / <r_k, r_k>

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] = r[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 - 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