A = {{a,,0,,, a,,1,,}, {a,,1,,, a,,2,,}}, b = {b,,0,,, b,,1,,}

Step 1: r = b - Ax

r[0] = b,,0,,

r[1] = b,,1,,

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

p[i]*(A[i][j]*p[j]) = a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2

alpha = (b,,0,,2+b,,1,,2) / ((b,,1,,2)*a,,2,,+2*(b,,1,,*b,,0,,*a,,1,,)+(b,,0,,2)*a,,0,,)

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

r[0] = (-b,,1,,*(-a,,1,,*b,,0,,2 + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,2)) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)

r[1] = (b,,0,,*(-a,,1,,*b,,0,,2 + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,2)) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)

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

x[0] = (b,,0,,*(b,,0,,2 + b,,1,,2)) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)

x[1] = (b,,1,,*(b,,0,,2 + b,,1,,2)) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)

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

rsnew = ((b,,0,,2 + b,,1,,2)*(-a,,1,,*b,,0,,2 + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,2)2) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)2

beta = (a,,1,,*b,,0,,2 - a,,0,,*b,,0,,*b,,1,, + a,,2,,*b,,0,,*b,,1,, - a,,1,,*b,,1,,2)2 / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)2

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

p[0] = (-1)*((a,,1,,*b,,0,, + a,,2,,*b,,1,,)*(b,,0,,2 + b,,1,,2)*(-a,,1,,*b,,0,,2 + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,2)) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)2

p[1] = ((a,,0,,*b,,0,, + a,,1,,*b,,1,,)*(b,,0,,2 + b,,1,,2)*(-a,,1,,*b,,0,,2 + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,2)) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)2

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

p[i]*(A[i][j]*p[j]) = ((-a,,1,,2 + a,,0,,*a,,2,,)*(b,,0,,2 + b,,1,,2)2*(-a,,1,,*b,,0,,2 + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,2)2) / (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2)3

alpha = (a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2) / ((-a,,1,,2 + a,,0,,*a,,2,,) (b,,0,,2 + b,,1,,2))

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] = (a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2)

x[1] = (-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2)

assertion: 

bncg[0] = A[0][0]*X[0] + A[0][1]*X[1] = a,,0,,*(a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2) + a,,1,,*(-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2) = b,,0,,*(a,,0,,*a,,2,,-a,,1,,^2) / (a,,0,,*a,,2,,-a,,1,,^2) = b,,0,,

bncg[1] = A[1][0]*X[0] + A[1][1]*X[1] = a,,1,,*(a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2) + a,,2,,*(-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2) = b,,1,,*(a,,0,,*a,,2,,-a,,1,,^2) / (a,,0,,*a,,2,,-a,,1,,^2) = b,,1,,

b[0] = b,,0,,

b[1] = b,,1,,

END