Changes between Version 22 and Version 23 of PolynomialExpansion


Ignore:
Timestamp:
01/26/16 14:41:43 (10 years ago)
Author:
sili
Comment:

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  • PolynomialExpansion

    v22 v23  
     1A = {{X,,0,,, X,,1,,}, {X,,1,,, X,,2,,}}, b = {b,,0,,, b,,1,,}
    12
    2 A = {{X,,0,,, X,,1,,}, {X,,1,,, X,,2,,}}, b = {b,,0,,, b,,1,,}
    3    
    4  A,,0,,  A,,1,,
    5 
     3A0 A1
    64
    75Step 1: r = b - Ax
    8   r[0] = X3[0]
    96
    10   r[1] = X3[1]
     7r[0] = b,,0,,
     8
     9r[1] = b,,1,,
    1110
    1211Step 2: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
    13   p[i]*(A[i][j]*p[j]) = X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^
    1412
    15   alpha = (X3[0]^2^+X3[1]^2^) / ((X3[1]^2^)*X2+2*(X3[1]*X3[0]*X1)+(X3[0]^2^)*X0)
     13p[i]*(A[i][j]*p[j]) = X,,0,,*b,,0,,2 + 2*X,,1,,*b,,0,,*b,,1,, + X,,2,,*b,,1,,2
     14
     15alpha = (b,,0,,2+b,,1,,2) / ((b,,1,,2)*X,,2,,+2*(b,,1,,*b,,0,,*X,,1,,)+(b,,0,,2)*X,,0,,)
    1616
    1717Step 3: r[i] = r - alpha * A[i][j] * p[j]
    18   r[0] = (-X3[1]*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
    1918
    20   r[1] = (X3[0]*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
     19r[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)
     20
     21r[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)
    2122
    2223Step 4: x[i] = x + alpha*p[i]
    23   x[0] = (X3[0]*(X3[0]^2^ + X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
    2424
    25   x[1] = (X3[1]*(X3[0]^2^ + X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
     25x[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)
     26
     27x[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)
    2628
    2729Step 5: beta = rsnew / rsold = (rk[i]*rk[i]) / (r[i]*r[i])
    28   rsnew = ((X3[0]^2^ + X3[1]^2^)*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)^2^) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
    2930
    30   beta = (X1*X3[0]^2^ - X0*X3[0]*X3[1] + X2*X3[0]*X3[1] - X1*X3[1]^2^)^2^ / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
     31rsnew = ((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
     32
     33beta = (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
    3134
    3235Step 6: p[i] = rk[i] +beta * p
    33   p[0] = (-1)*((X1*X3[0] + X2*X3[1])*(X3[0]^2^ + X3[1]^2^)*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
    3436
    35   p[1] = ((X0*X3[0] + X1*X3[1])*(X3[0]^2^ + X3[1]^2^)*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
     37p[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
     38
     39p[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
    3640
    3741Step 7: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
    38   p[i]*(A[i][j]*p[j]) = ((-X1^2^ + X0*X2)*(X3[0]^2^ + X3[1]^2^)^2^*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)^2^) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^3^
    3942
    40   alpha = (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^) / ((-X1^2^ + X0*X2) (X3[0]^2^ + X3[1]^2^))
     43p[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
     44
     45alpha = (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))
    4146
    4247Step 8: r[i] = r - alpha * A[i][j] * p[j]
    43   r[0] = 0
    4448
    45   r[1] = 0
     49r[0] = 0
     50
     51r[1] = 0
    4652
    4753Step 9: x[i] = x + alpha*p[i]
    48   x[0] = (X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^)
    4954
    50   x[1] = (-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^)
     55x[0] = (X,,2,,*b,,0,, - X,,1,,*b,,1,,) / (X,,0,,*X,,2,, - X,,1,,2)
    5156
    52 assertion:  bncg[i] = A[i][j]*x[j]
    53   bncg[0] = X3[0]
     57x[1] = (-X,,1,,*b,,0,, + X,,0,,*b,,1,,) / (X,,0,,*X,,2,, - X,,1,,2)
    5458
    55   bncg[1] = X3[1]
     59assertion:
    5660
    57   b[0] = X3[0]
     61bncg[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,,
    5862
    59   b[1] = X3[1]
     63bncg[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,,
     64
     65b[0] = b,,0,,
     66
     67b[1] = b,,1,,
    6068
    6169END