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1 ## Copyright (C) 1993, 1998, 1999 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, 59 Temple Place, Suite 330, Boston, MA 02111 USA. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{u}, @var{h}, @var{nu}] =} krylov (@var{a}, @var{v}, @var{k}, @var{eps1}, @var{pflg}); |
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21 ## construct orthogonal basis U of block Krylov subspace; |
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22 ## [v a*v a^2*v ... a^(k+1)*v]; |
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23 ## method used: householder reflections to guard against loss of |
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24 ## orthogonality |
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25 ## eps1: threshhold for 0 (default: 1e-12) |
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26 ## pflg: flag to use row pivoting (improves numerical behavior) |
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27 ## 0 [default]: no pivoting; prints a warning message if trivial |
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28 ## null space is corrupted |
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29 ## 1 : pivoting performed |
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30 ## |
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31 ## outputs: |
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32 ## u: orthogonal basis of block krylov subspace |
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33 ## h: Hessenberg matrix; if v is a vector then a u = u h |
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34 ## otherwise h is meaningless |
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35 ## nu: dimension of span of krylov subspace (based on eps1) |
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36 ## if b is a vector and k > m-1, krylov returns h = the Hessenberg |
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37 ## decompostion of a. |
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38 ## |
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39 ## Reference: Hodel and Misra, "Partial Pivoting in the Computation of |
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40 ## Krylov Subspaces", to be submitted to Linear Algebra and its |
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41 ## Applications |
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42 ## @end deftypefn |
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43 |
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44 ## Author: A. Scottedward Hodel <a.s.hodel@eng.auburn.edu> |
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45 |
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46 function [Uret,H,nu] = krylov(A,V,k,eps1,pflg); |
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47 |
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48 defeps = 1e-12; |
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49 |
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50 if (nargin < 3 || nargin > 5) |
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51 usage ("[U, nu] = krylov (A, V, k, eps1, pflg)") |
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52 elseif (nargin < 5) |
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53 pflg = 0; # default permutation flag |
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54 endif |
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55 |
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56 if(nargin < 4) |
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57 eps1 = defeps; # default tolerance parameter |
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58 endif |
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59 |
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60 if (isempty (eps1)) |
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61 eps1 = defeps; |
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62 endif |
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63 |
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64 na = issquare (A); |
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65 if (! na) |
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66 error ("A(%d x %d) must be square", rows (A), columns (A)); |
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67 endif |
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68 |
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69 [m, kb] = size(V); |
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70 if (m != na) |
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71 error("A(%d x %d), V(%d x %d): argument dimensions do not match", |
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72 na, na, m, kb) |
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73 endif |
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74 |
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75 if (! isscalar (k)) |
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76 error ("krylov: third argument must be a scalar integer"); |
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77 endif |
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78 |
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79 Vnrm = norm (V, Inf); |
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80 |
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81 ## check for trivial solution |
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82 if (Vnrm == 0) |
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83 Uret = []; |
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84 nu = 0; |
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85 return; |
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86 endif |
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87 |
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88 # identify trivial null space |
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89 abm = max (abs ([A, V]')); |
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90 nzidx = find (abm != 0); |
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91 zidx = find (abm == 0); |
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92 |
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93 # set up vector of pivot points |
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94 pivot_vec = 1:na; |
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95 |
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96 iter = 0; |
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97 alpha = []; |
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98 nh = 0; |
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99 while (length(alpha) < na) && (columns(V) > 0) && (iter < k) |
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100 iter++; |
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101 |
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102 ## get orthogonal basis of V |
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103 jj = 1; |
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104 while (jj <= columns (V) && length (alpha) < na) |
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105 ## index of next Householder reflection |
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106 nu = length(alpha)+1; |
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107 |
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108 short_pv = pivot_vec(nu:na); |
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109 q = V(:,jj); |
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110 short_q = q(short_pv); |
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111 |
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112 if (norm (short_q) < eps1) |
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113 ## insignificant column; delete |
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114 nv = columns (V); |
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115 if (jj != nv) |
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116 [V(:,jj), V(:,nv)] = swap (V(:,jj), V(:,nv)); |
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117 ## XXX FIXME XXX -- H columns should be swapped too. Not done |
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118 ## since Block Hessenberg structure is lost anyway. |
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119 endif |
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120 V = V(:,1:(nv-1)); |
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121 ## one less reflection |
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122 nu--; |
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123 else |
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124 ## new householder reflection |
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125 if (pflg) |
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126 ## locate max magnitude element in short_q |
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127 asq = abs (short_q); |
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128 maxv = max (asq); |
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129 maxidx = find (asq == maxv); |
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130 pivot_idx = short_pv(maxidx(1)); |
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131 |
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132 ## see if need to change the pivot list |
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133 if (pivot_idx != pivot_vec(nu)) |
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134 swapidx = maxidx(1) + (nu-1); |
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135 [pivot_vec(nu), pivot_vec(swapidx)] = ... |
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136 swap (pivot_vec(nu), pivot_vec(swapidx)); |
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137 endif |
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138 endif |
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139 |
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140 ## isolate portion of vector for reflection |
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141 idx = pivot_vec(nu:na); |
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142 jdx = pivot_vec(1:nu); |
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143 |
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144 [hv, av, z] = housh (q(idx), 1, 0); |
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145 alpha(nu) = av; |
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146 U(idx,nu) = hv; |
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147 |
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148 # reduce V per the reflection |
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149 V(idx,:) = V(idx,:) - av*hv*(hv' * V(idx,:)); |
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150 if(iter > 1) |
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151 ## XXX FIXME XXX -- not done correctly for block case |
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152 H(nu,nu-1) = V(pivot_vec(nu),jj); |
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153 endif |
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154 |
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155 ## advance to next column of V |
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156 jj++; |
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157 endif |
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158 endwhile |
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159 |
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160 ## check for oversize V (due to full rank) |
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161 if ((columns (V) > na) && (length (alpha) == na)) |
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162 ## trim to size |
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163 V = V(:,1:na); |
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164 elseif (columns(V) > na) |
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165 krylov_V = V |
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166 krylov_na = na |
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167 krylov_length_alpha = length (alpha) |
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168 error ("This case should never happen; submit a bug report"); |
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169 endif |
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170 |
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171 if (columns (V) > 0) |
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172 ## construct next Q and multiply |
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173 Q = zeros (size (V)); |
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174 for kk = 1:columns (Q) |
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175 Q(pivot_vec(nu-columns(Q)+kk),kk) = 1; |
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176 endfor |
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177 |
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178 ## apply Householder reflections |
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179 for ii = nu:-1:1 |
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180 idx = pivot_vec(ii:na); |
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181 hv = U(idx,ii); |
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182 av = alpha(ii); |
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183 Q(idx,:) = Q(idx,:) - av*hv*(hv'*Q(idx,:)); |
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184 endfor |
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185 endif |
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186 |
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187 ## multiply to get new vector; |
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188 V = A*Q; |
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189 ## project off of previous vectors |
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190 nu = length (alpha); |
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191 for i = 1:nu |
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192 hv = U(:,i); |
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193 av = alpha(i); |
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194 V = V - av*hv*(hv'*V); |
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195 H(i,nu-columns(V)+(1:columns(V))) = V(pivot_vec(i),:); |
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196 end |
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197 |
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198 endwhile |
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199 |
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200 ## Back out complete U matrix |
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201 ## back out U matrix ; |
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202 j1 = columns (U); |
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203 for i = j1:-1:1; |
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204 idx = pivot_vec(i:na); |
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205 hv = U(idx,i); |
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206 av = alpha(i); |
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207 U(:,i) = zeros (na, 1); |
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208 U(idx(1),i) = 1; |
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209 U(idx,i:j1) = U(idx,i:j1)-av*hv*(hv'*U(idx,i:j1)); |
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210 endfor |
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211 |
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212 nu = length (alpha); |
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213 Uret = U; |
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214 if (max (max (abs (Uret(zidx,:)))) > 0) |
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215 warning ("krylov: trivial null space corrupted; set pflg = 1 or eps1 > %e", |
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216 eps1); |
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217 endif |
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218 |
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219 endfunction |
