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1 ## Copyright (C) 1996, 1998 Auburn University. All rights reserved. |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by the |
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7 ## Free Software Foundation; either version 2, or (at your option) any |
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8 ## later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 ## for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301 USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {@var{x} =} zgscal (@var{f}, @var{z}, @var{n}, @var{m}, @var{p}) |
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22 ## Generalized conjugate gradient iteration to |
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23 ## solve zero-computation generalized eigenvalue problem balancing equation |
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24 ## @math{fx=z}; called by @command{zgepbal}. |
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25 ## @end deftypefn |
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26 |
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27 ## References: |
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28 ## ZGEP: Hodel, "Computation of Zeros with Balancing," 1992, submitted to LAA |
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29 ## Generalized CG: Golub and Van Loan, "Matrix Computations, 2nd ed" 1989 |
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30 |
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31 ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu> |
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32 ## Created: July 24, 1992 |
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33 ## Conversion to Octave R. Bruce Tenison July 3, 1994 |
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34 |
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35 function x = zgscal (a, b, c, d, z, n, m, p) |
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36 |
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37 ## initialize parameters: |
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38 ## Givens rotations, diagonalized 2x2 block of F, gcg vector initialization |
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39 |
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40 nmp = n+m+p; |
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41 |
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42 ## x_0 = x_{-1} = 0, r_0 = z |
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43 x = zeros(nmp,1); |
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44 xk1 = x; |
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45 xk2 = x; |
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46 rk1 = z; |
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47 k = 0; |
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48 |
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49 ## construct balancing least squares problem |
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50 F = eye(nmp); |
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51 for kk=1:nmp |
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52 F(1:nmp,kk) = zgfmul(a,b,c,d,F(:,kk)); |
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53 endfor |
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54 |
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55 [U,H,k1] = krylov(F,z,nmp,1e-12,1); |
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56 if(!issquare(H)) |
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57 if(columns(H) != k1) |
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58 error("zgscal(tzero): k1=%d, columns(H)=%d",k1,columns(H)); |
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59 elseif(rows(H) != k1+1) |
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60 error("zgscal: k1=%d, rows(H) = %d",k1,rows(H)); |
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61 elseif ( norm(H(k1+1,:)) > 1e-12*norm(H,"inf") ) |
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62 zgscal_last_row_of_H = H(k1+1,:) |
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63 error("zgscal: last row of H nonzero (norm(H)=%e)",norm(H,"inf")) |
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64 endif |
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65 H = H(1:k1,1:k1); |
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66 U = U(:,1:k1); |
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67 endif |
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68 |
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69 ## tridiagonal H can still be rank deficient, so do permuted qr |
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70 ## factorization |
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71 [qq,rr,pp] = qr(H); # H = qq*rr*pp' |
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72 nn = rank(rr); |
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73 qq = qq(:,1:nn); |
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74 rr = rr(1:nn,:); # rr may not be square, but "\" does least |
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75 xx = U*pp*(rr\qq'*(U'*z)); # squares solution, so this works |
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76 ## xx1 = pinv(F)*z; |
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77 ## zgscal_x_xx1_err = [xx,xx1,xx-xx1] |
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78 return; |
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79 |
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80 ## the rest of this is left from the original zgscal; |
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81 ## I've had some numerical problems with the GCG algorithm, |
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82 ## so for now I'm solving it with the krylov routine. |
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83 |
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84 ## initialize residual error norm |
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85 rnorm = norm(rk1,1); |
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86 |
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87 xnorm = 0; |
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88 fnorm = 1e-12 * norm([a,b;c,d],1); |
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89 |
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90 ## dummy defines for MATHTOOLS compiler |
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91 gamk2 = 0; omega1 = 0; ztmz2 = 0; |
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92 |
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93 ## do until small changes to x |
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94 len_x = length(x); |
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95 while ((k < 2*len_x) & (xnorm> 0.5) & (rnorm>fnorm))|(k == 0) |
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96 k = k+1; |
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97 |
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98 ## solve F_d z_{k-1} = r_{k-1} |
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99 zk1= zgfslv(n,m,p,rk1); |
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100 |
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101 ## Generalized CG iteration |
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102 ## gamk1 = (zk1'*F_d*zk1)/(zk1'*F*zk1); |
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103 ztMz1 = zk1'*rk1; |
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104 gamk1 = ztMz1/(zk1'*zgfmul(a,b,c,d,zk1)); |
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105 |
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106 if(rem(k,len_x) == 1) omega = 1; |
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107 else omega = 1/(1-gamk1*ztMz1/(gamk2*omega1*ztmz2)); |
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108 endif |
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109 |
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110 ## store x in xk2 to save space |
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111 xk2 = xk2 + omega*(gamk1*zk1 + xk1 - xk2); |
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112 |
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113 ## compute new residual error: rk = z - F xk, check end conditions |
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114 rk1 = z - zgfmul(a,b,c,d,xk2); |
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115 rnorm = norm(rk1); |
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116 xnorm = max(abs(xk1 - xk2)); |
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117 |
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118 ## printf("zgscal: k=%d, gamk1=%e, gamk2=%e, \nztMz1=%e ztmz2=%e\n", ... |
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119 ## k,gamk1, gamk2, ztMz1, ztmz2); |
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120 ## xk2_1_zk1 = [xk2 xk1 zk1] |
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121 ## ABCD = [a,b;c,d] |
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122 ## prompt |
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123 |
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124 ## get ready for next iteration |
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125 gamk2 = gamk1; |
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126 omega1 = omega; |
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127 ztmz2 = ztMz1; |
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128 [xk1,xk2] = swap(xk1,xk2); |
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129 endwhile |
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130 x = xk2; |
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131 |
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132 ## check convergence |
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133 if (xnorm> 0.5 & rnorm>fnorm) |
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134 warning("zgscal(tzero): GCG iteration failed; solving with pinv"); |
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135 |
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136 ## perform brute force least squares; construct F |
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137 Am = eye(nmp); |
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138 for ii=1:nmp |
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139 Am(:,ii) = zgfmul(a,b,c,d,Am(:,ii)); |
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140 endfor |
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141 |
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142 ## now solve with qr factorization |
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143 x = pinv(Am)*z; |
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144 endif |
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145 endfunction |
