Changes between Version 57 and Version 58 of PolynomialExpansion

Timestamp:

01/27/16 09:03:23 (10 years ago)

Author:

sili

Comment:

--

PolynomialExpansion

@@ -4 +4 @@
 
 Step 1: r,,0,, = b - Ax,,0,,; p,,0,, = r,,0,,  ('''No expansion''')
-  r[0] = b,,0,,
+  r,,0,,[0] = b,,0,,
 
-  r[1] = b,,1,,
+  r,,0,,[1] = b,,1,,
 
 Step 2: alpha,,0,, = <r,,0,,, r,,0,,> / <p,,0,,, Ap,,0,,>  ('''No expansion''')
@@ -14 +14 @@
 
 Step 3: r,,1,, = r,,0,, - alpha,,0,,*Ap,,0,,  ('''No expansion''')
-  r[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,,[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] = (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,,))
+  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[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,,[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] = (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,,))
+  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''')
@@ -29 +29 @@
 
 Step 6: p,,1,, = r,,1,, +beta*p,,0,,  ('''No expansion''')
-  p[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,,))(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,,((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,,))^2^ + (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,,))^2^)) / (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,,))^2^
+  p,,1,,[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,,))(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,,((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,,))^2^ + (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,,))^2^)) / (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,,))^2^
 
-  p[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,,))(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,,1,,((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,,))^2^ + (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,,))^2^)) / (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,,))^2^
+  p,,1,,[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,,))(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,,1,,((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,,))^2^ + (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,,))^2^)) / (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,,))^2^
 
 Step 7: alpha,,1,, = <r,,1,,, r,,1,,> / <p,,1,,, Ap,,1,,>  ('''No expansion''')
-  <p,,1,,, Ap,,1,,> = p[0](a,,0,,p[0]+a,,1,,p[1]) + p[1](a,,1,,p[0]+a,,2,,p[1])
+  <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,,(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,,))^2^ + (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,,))^2^) / (p[0](a,,0,,p[0]+a,,1,,p[1]) + p[1](a,,1,,p[0]+a,,2,,p[1]))(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))^2^
+  alpha,,1,, = ((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,,))^2^ + (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,,))^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]))(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))^2^
 
 Step 8: r,,2,, = r,,1,, - alpha,,1,,*Ap,,1,,  ('''Need to expand for cancellation''')
 
-  r[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,,)) - (a,,0,,p[0]+a,,1,,p[1])((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,,))^2^ + (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,,))^2^) / (p[0](a,,0,,p[0]+a,,1,,p[1]) + p[1](a,,1,,p[0]+a,,2,,p[1]))(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))^2^ = 0
+  r,,2,,[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,,)) - (a,,0,,p[0]+a,,1,,p[1])((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,,))^2^ + (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,,))^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]))(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))^2^ = 0
 
-  r[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,,)) - (a,,1,,p[0]+a,,2,,p[1])((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,,))^2^ + (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,,))^2^) / (p[0](a,,0,,p[0]+a,,1,,p[1]) + p[1](a,,1,,p[0]+a,,2,,p[1]))(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))^2^ = 0
+  r,,2,,[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,,)) - (a,,1,,p[0]+a,,2,,p[1])((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,,))^2^ + (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,,))^2^) / (p,,1,,[0](a,,0,,p[0]+a,,1,,p,,1,,[1]) + p,,1,,[1](a,,1,,p,,1,,[0]+a,,2,,p,,1,,[1]))(b,,0,,(a,,0,,b,,0,, + a,,1,,b,,1,,) + b,,1,,(a,,1,,b,,0,, + a,,2,,b,,1,,))^2^ = 0
 
 Step 9: x,,2,, = x,,1,, + alpha,,1,,*p,,1,,
-  x[0] = (a,,2,,b,,0,, - a,,1,,b,,1,,) / (a,,0,,a,,2,, - a,,1,,^2^)
+  x,,2,,[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^)
+  x,,2,,[1] = (-a,,1,,b,,0,, + a,,0,,b,,1,,) / (a,,0,,a,,2,, - a,,1,,^2^)
 
 assertion:   ('''Need to expand for cancellation''')