Changes between Version 23 and Version 24 of PolynomialExpansion


Ignore:
Timestamp:
01/26/16 14:44:04 (10 years ago)
Author:
sili
Comment:

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Unmodified
Added
Removed
Modified

  • PolynomialExpansion

    v23 v24  
    1 A = {{X,,0,,, X,,1,,}, {X,,1,,, X,,2,,}}, b = {b,,0,,, b,,1,,}
    2 
    3 A0 A1
     1A = {{a,,0,,, a,,1,,}, {a,,1,,, a,,2,,}}, b = {b,,0,,, b,,1,,}
    42
    53Step 1: r = b - Ax
    … …  
    119Step 2: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
    1210
    13 p[i]*(A[i][j]*p[j]) = X,,0,,*b,,0,,2 + 2*X,,1,,*b,,0,,*b,,1,, + X,,2,,*b,,1,,2
     11p[i]*(A[i][j]*p[j]) = a,,0,,*b,,0,,2 + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,2
    1412
    15 alpha = (b,,0,,2+b,,1,,2) / ((b,,1,,2)*X,,2,,+2*(b,,1,,*b,,0,,*X,,1,,)+(b,,0,,2)*X,,0,,)
     13alpha = (b,,0,,2+b,,1,,2) / ((b,,1,,2)*a,,2,,+2*(b,,1,,*b,,0,,*a,,1,,)+(b,,0,,2)*a,,0,,)
    1614
    1715Step 3: r[i] = r - alpha * A[i][j] * p[j]
    1816
    19 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)
     17r[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)
    2018
    21 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)
     19r[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)
    2220
    2321Step 4: x[i] = x + alpha*p[i]
    2422
    25 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)
     23x[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)
    2624
    27 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)
     25x[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)
    2826
    2927Step 5: beta = rsnew / rsold = (rk[i]*rk[i]) / (r[i]*r[i])
    3028
    31 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
     29rsnew = ((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
    3230
    33 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
     31beta = (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
    3432
    3533Step 6: p[i] = rk[i] +beta * p
    3634
    37 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
     35p[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
    3836
    39 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
     37p[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
    4038
    4139Step 7: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
    4240
    43 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
     41p[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
    4442
    45 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))
     43alpha = (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))
    4644
    4745Step 8: r[i] = r - alpha * A[i][j] * p[j]
    … …  
    5351Step 9: x[i] = x + alpha*p[i]
    5452
    55 x[0] = (X,,2,,*b,,0,, - X,,1,,*b,,1,,) / (X,,0,,*X,,2,, - X,,1,,2)
     53x[0] = (a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2)
    5654
    57 x[1] = (-X,,1,,*b,,0,, + X,,0,,*b,,1,,) / (X,,0,,*X,,2,, - X,,1,,2)
     55x[1] = (-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,2)
    5856
    5957assertion:
    6058
    61 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,,
     59bncg[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,,
    6260
    63 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,,
     61bncg[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,,
    6462
    6563b[0] = b,,0,,