Changes between Version 24 and Version 25 of PolynomialExpansion


Ignore:
Timestamp:
01/26/16 14:56:32 (10 years ago)
Author:
sili
Comment:

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  • PolynomialExpansion

    v24 v25  
    1 A = {{a,,0,,, a,,1,,}, {a,,1,,, a,,2,,}}, b = {b,,0,,, b,,1,,}
    21
     2A = {{X0, X1}, {X1, X2}},  b = X3[2]
     3   
    34Step 1: r = b - Ax
     5  r[0] = X3[0]
    46
    5 r[0] = b,,0,,
    6 
    7 r[1] = b,,1,,
     7  r[1] = X3[1]
    88
    99Step 2: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
     10  p[i]*(A[i][j]*p[j]) = X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^
    1011
    11 p[i]*(A[i][j]*p[j]) = a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2
    12 
    13 alpha = (b,,0,,2+b,,1,,2) / ((b,,1,,2)*a,,2,,+2*(b,,1,,*b,,0,,*a,,1,,)+(b,,0,,2)*a,,0,,)
     12  alpha = (X3[0]^2^+X3[1]^2^) / ((X3[1]^2^)*X2+2*(X3[1]*X3[0]*X1)+(X3[0]^2^)*X0)
    1413
    1514Step 3: r[i] = r - alpha * A[i][j] * p[j]
     15  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^)
    1616
    17 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)
    18 
    19 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)
     17  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^)
    2018
    2119Step 4: x[i] = x + alpha*p[i]
     20  X[0] = (X3[0]*(X3[0]^2^ + X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
    2221
    23 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)
    24 
    25 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)
     22  X[1] = (X3[1]*(X3[0]^2^ + X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
    2623
    2724Step 5: beta = rsnew / rsold = (rk[i]*rk[i]) / (r[i]*r[i])
     25  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^
    2826
    29 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
    30 
    31 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
     27  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^
    3228
    3329Step 6: p[i] = rk[i] +beta * p
     30  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^
    3431
    35 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
    36 
    37 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
     32  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^
    3833
    3934Step 7: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
     35  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^
    4036
    41 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
    42 
    43 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))
     37  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^))
    4438
    4539Step 8: r[i] = r - alpha * A[i][j] * p[j]
     40  r[0] = 0
    4641
    47 r[0] = 0
    48 
    49 r[1] = 0
     42  r[1] = 0
    5043
    5144Step 9: x[i] = x + alpha*p[i]
     45  X[0] = (X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^)
    5246
    53 x[0] = (a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2)
     47  X[1] = (-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^)
    5448
    55 x[1] = (-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2)
     49assertion: 
     50  bncg[0] = A[0][0]*X[0] + A[0][1]*x[1] = X0*(X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^) + X1*(-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^) = X3[0]*(X0*X2 - X1^2^) / (X0*X2 - X1^2^) = X3[0]
    5651
    57 assertion:
     52  bncg[1] = A[1][0]*X[0] + A[1][1]*x[1] = X1*(X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^) + X2*(-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^) = X3[1]*(X0*X2 - X1^2^) / (X0*X2 - X1^2^) = X3[1]
    5853
    59 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,,
     54  b[0] = X3[0]
    6055
    61 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,,
    62 
    63 b[0] = b,,0,,
    64 
    65 b[1] = b,,1,,
     56  b[1] = X3[1]
    6657
    6758END