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


Step 1: r,,0,, = b - Ax,,0,,; p,,0,, = r,,0,,  ('''No expansion''')
  r,,0,,[0] = b,,0,,

  r,,0,,[1] = b,,1,,

Step 2: alpha,,0,, = <r,,0,,, r,,0,,> / <p,,0,,, Ap,,0,,>  ('''No expansion''')
  <p,,0,,, Ap,,0,,> = b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,)

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

Step 3: r,,1,, = r,,0,, - alpha,,0,,*Ap,,0,,  ('''No expansion''')
  r,,1,,[0] = (b,,0,,(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,)) - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)) / (b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))

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

Step 4: x,,1,, = x,,0,, + alpha,,0,,*p,,0,,  ('''No expansion''')
  x,,1,,[0] = (b,,0,,(b,,0,,^2^ + b,,1,,^2^)) / (b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))

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

Step 5: beta = rsnew / rsold = <r,,1,,, r,,1,,> / <r,,0,,, r,,0,,>  ('''No expansion''')
  To make it simple to read, let's devote m = b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,)
  
  rsnew =
   [b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^\\
   ------------------------------------------------------------------------\\
   m^2^

  beta = 
   [b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^\\
   -------------------------------------------------------------------------\\ 
   (b,,0,,^2^+b,,1,,^2^)m^2^

Step 6: p,,1,, = r,,1,, +beta*p,,0,,  ('''No expansion''')
  p,,1,,[0] = 
   m(b,,0,,^2^+b,,1,,^2^)[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)] + b,,0,,{[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^}\\
   ------------------------------------------------------------------------------------------------------------------------------------\\ 
   (b,,0,,^2^+b,,1,,^2^)m^2^

  p,,1,,[1] = 
   m(b,,0,,^2^+b,,1,,^2^)[b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)] + b,,1,,{[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^}\\
   ------------------------------------------------------------------------------------------------------------------------------------\\ 
   (b,,0,,^2^+b,,1,,^2^)m^2^

Step 7: alpha,,1,, = <r,,1,,, r,,1,,> / <p,,1,,, Ap,,1,,>  ('''No expansion''')
  <p,,1,,, Ap,,1,,> = p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])

  alpha,,1,, = 
   [b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^\\
   ----------------------------------------------------------------------------\\
   m^2^[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])]

  If expand and simplified:

  alpha,,1,, = (a,,0,,b,,0,,^2^+2a,,1,,b,,0,,b,,1,,+a,,2,,b,,1,,^2^) / [(a,,0,,a,,2,,-a,,1,,^2^)(b,,0,,^2^+b,,1,,^2^)]

Step 8: r,,2,, = r,,1,, - alpha,,1,,*Ap,,1,,  ('''Need to expand for cancellation''')
  r,,2,,[0] = r,,1,,[0] - alpha,,1,,*(A[0][0]*p1[0]+A[0][1]*p1[1])
 
  r,,2,,[0] = 
   m[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)][p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])] - (a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]){[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^}\\
   --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\\
   m^2^[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])]

  r,,2,,[1] = r,,1,,[1] - alpha,,1,,*(A[1][0]*p1[0]+A[1][1]*p1[1])

  r,,2,,[1] = 
   m[b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)][p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])] - (a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1]){[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^}\\
   --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\\
   m^2^[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])]

Step 9: x,,2,, = x,,1,, + alpha,,1,,*p,,1,,
  If not expand:

  x,,2,,[0] = x,,1,,[0] + alpha,,1,,*p,,1,,[0]

  x,,2,,[0] =
   mb,,0,,(b,,0,,^2^+b,,1,,^2^)[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])] + p,,1,,[0]{[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^}\\
   ------------------------------------------------------------------------------------------------------------------------------------------------------------------\\
   m^2^[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])]

  x,,2,,[1] = x,,1,,[1] + alpha,,1,,*p,,1,,[1]

  x,,2,,[1] =
   mb,,1,,(b,,0,,^2^+b,,1,,^2^)[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])] + p,,1,,[1]{[b,,0,,m - (b,,0,,^2^+b,,1,,^2^)(a,,0,,b,,0,,+a,,1,,b,,1,,)]^2^ + [b,,1,,m - (b,,0,,^2^+b,,1,,^2^)(a,,1,,b,,0,,+a,,2,,b,,1,,)]^2^}\\
   ------------------------------------------------------------------------------------------------------------------------------------------------------------------\\
   m^2^[p,,1,,[0](a,,0,,p,,1,,[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1])]

  If expand and simplified:

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

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

assertion:   ('''Need to expand for cancellation''')
  bncg[0] = A[0][0]*x,,2,,[0] + A[0][1]*x,,2,,[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,,2,,[0] + A[1][1]*x,,2,,[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,,

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

END