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

Step 1: r₀ = b - Ax₀; p₀ = r₀ (No expansion)

r[0] = b₀

r[1] = b₁

Step 2: alpha₀ = <r₀, r₀> / <p₀, Ap₀> (No expansion)

<p₀, Ap₀> = b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)

alpha₀ = (b₀²+b₁²) / (b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))

Step 3: r₁ = r₀ - alpha₀*Ap₀ (No expansion)

r[0] = (b₀(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₀b₀+a₁b₁)) / (b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))

r[1] = (b₁(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₁b₀+a₂b₁)) / (b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))

Step 4: x₁ = x₀ + alpha₀*p₀ (No expansion)

x[0] = (b₀(b₀² + b₁²)) / (b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))

x[1] = (b₁(b₀² + b₁²)) / (b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))

Step 5: beta = rsnew / rsold = <r₁, r₁> / <r₀, r₀> (No expansion)

rsnew = ((b₀(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₀b₀+a₁b₁))² + (b₁(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₁b₀+a₂b₁))²) / (b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))²

beta = ((b₀(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₀b₀+a₁b₁))² + (b₁(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₁b₀+a₂b₁))²) / (b₀²+b₁²)(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))²

Step 6: p₁ = r₁ +beta*p₀ (No expansion)

p[0] = ((b₀²+b₁²)(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))(b₀(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₀b₀+a₁b₁)) + b₀((b₀(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₀b₀+a₁b₁))² + (b₁(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁) - (b₀²+b₁²)(a₁b₀+a₂b₁))²)) / (b₀²+b₁²)(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))²

p[1] = ((b₀²+b₁²)(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))(b₁(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₁b₀+a₂b₁)) + b₁((b₀(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁)) - (b₀²+b₁²)(a₀b₀+a₁b₁))² + (b₁(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁) - (b₀²+b₁²)(a₁b₀+a₂b₁))²)) / (b₀²+b₁²)(b₀(a₀b₀ + a₁b₁) + b₁(a₁b₀ + a₂b₁))²

Step 7: alpha₁ = <r₁, r₁> / <p₁, Ap₁>

<p₁, Ap₁> = p[0](a₀p[0]+a₁p[1]) + p[1](a₁p[0]+a₂p[1])

alpha₁ = (a₀b₀² + 2a₁b₀b₁ + a₂b₁²) / ((-a₁² + a₀a₂) (b₀² + b₁²))

Step 8: r₂ = r₁ - alpha₁*Ap₁

r[0] = 0

r[1] = 0

Step 9: x₂ = x₁ + alpha₁*p₁

x[0] = (a₂b₀ - a₁b₁) / (a₀a₂ - a₁²)

x[1] = (-a₁b₀ + a₀b₁) / (a₀a₂ - a₁²)

assertion:

bncg[0] = A[0][0]*x[0] + A[0][1]*x[1] = a₀(a₂b₀ - a₁b₁) / (a₀a₂ - a₁²) + a₁(-a₁b₀ + a₀b₁) / (a₀a₂ - a₁²) = b₀(a₀a₂ - a₁²) / (a₀a₂ - a₁²) = b₀

bncg[1] = A[1][0]*x[0] + A[1][1]*x[1] = a₁(a₂b₀ - a₁b₁) / (a₀a₂ - a₁²) + a₂(-a₁b₀ + a₀b₁) / (a₀a₂ - a₁²) = b₁(a₀a₂ - a₁²) / (a₀a₂ - a₁²) = b₁

b[0] = b₀

b[1] = b₁

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

END