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

A0 A1

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]) = X,,0,,*b,,0,,2 + 2*X,,1,,*b,,0,,*b,,1,, + X,,2,,*b,,1,,2

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

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

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

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

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

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

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

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

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

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

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

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

p[1] = ((X,,0,,*b,,0,, + X,,1,,*b,,1,,)*(b,,0,,2 + b,,1,,2)*(-X,,1,,*b,,0,,2 + X,,0,,*b,,0,,*b,,1,, - X,,2,,*b,,0,,*b,,1,, + X,,1,,*b,,1,,2)) / (X,,0,,*b,,0,,2 + 2*X,,1,,*b,,0,,*b,,1,, + X,,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]) = ((-X,,1,,2 + X,,0,,*X,,2,,)*(b,,0,,2 + b,,1,,2)2*(-X,,1,,*b,,0,,2 + X,,0,,*b,,0,,*b,,1,, - X,,2,,*b,,0,,*b,,1,, + X,,1,,*b,,1,,2)2) / (X,,0,,*b,,0,,2 + 2*X,,1,,*b,,0,,*b,,1,, + X,,2,,*b,,1,,2)3

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

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

assertion: 

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

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

b[0] = b,,0,,

b[1] = b,,1,,

END