A = {{X₀, X₁}, {X₁, X₂}}, b = {b₀, b₁}

A0 A1

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]) = X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2

alpha = (b₀2+b₁2) / ((b₁2)*X₂+2*(b₁*b₀*X₁)+(b₀2)*X₀)

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

r[0] = (-b₁*(-X₁*b₀2 + X₀*b₀*b₁ - X₂*b₀*b₁ + X₁*b₁2)) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)

r[1] = (b₀*(-X₁*b₀2 + X₀*b₀*b₁ - X₂*b₀*b₁ + X₁*b₁2)) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)

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

x[0] = (b₀*(b₀2 + b₁2)) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)

x[1] = (b₁*(b₀2 + b₁2)) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)

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

rsnew = ((b₀2 + b₁2)*(-X₁*b₀2 + X₀*b₀*b₁ - X₂*b₀*b₁ + X₁*b₁2)2) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)2

beta = (X₁*b₀2 - X₀*b₀*b₁ + X₂*b₀*b₁ - X₁*b₁2)2 / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)2

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

p[0] = (-1)*((X₁*b₀ + X₂*b₁)*(b₀2 + b₁2)*(-X₁*b₀2 + X₀*b₀*b₁ - X₂*b₀*b₁ + X₁*b₁2)) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)2

p[1] = ((X₀*b₀ + X₁*b₁)*(b₀2 + b₁2)*(-X₁*b₀2 + X₀*b₀*b₁ - X₂*b₀*b₁ + X₁*b₁2)) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)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)*(b₀2 + b₁2)2*(-X₁*b₀2 + X₀*b₀*b₁ - X₂*b₀*b₁ + X₁*b₁2)2) / (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2)3

alpha = (X₀*b₀2 + 2*X₁*b₀*b₁ + X₂*b₁2) / ((-X₁2 + X₀*X₂) (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] = (X₂*b₀ - X₁*b₁) / (X₀*X₂ - X₁2)

x[1] = (-X₁*b₀ + X₀*b₁) / (X₀*X₂ - X₁2)

assertion:

bncg[0] = A[0][0]*X[0] + A[0][1]*X[1] = X₀*(X₂*b₀ - X₁*b₁) / (X₀*X₂ - X₁2) + X₁*(-X₁*b₀ + X₀*b₁) / (X₀*X₂ - X₁2) = b₀*(X₀*X₂-X₁^{2) / (X₀*X₂-X₁}2) = b₀

bncg[1] = A[1][0]*X[0] + A[1][1]*X[1] = X₁*(X₂*b₀ - X₁*b₁) / (X₀*X₂ - X₁2) + X₂*(-X₁*b₀ + X₀*b₁) / (X₀*X₂ - X₁2) = b₁*(X₀*X₂-X₁^{2) / (X₀*X₂-X₁}2) = b₁

b[0] = b₀

b[1] = b₁

END