Mercurial > octave
annotate libinterp/corefcn/sqrtm.cc @ 27919:1891570abac8
update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2020.
author | John W. Eaton <jwe@octave.org> |
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date | Mon, 06 Jan 2020 22:29:51 -0500 |
parents | b442ec6dda5c |
children | bd51beb6205e |
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4331 | 1 /* |
2 | |
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update Octave Project Developers copyright for the new year
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3 Copyright (C) 2001-2020 The Octave Project Developers |
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4 |
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5 See the file COPYRIGHT.md in the top-level directory of this distribution |
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6 or <https://octave.org/COPYRIGHT.html/>. |
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7 |
4331 | 8 |
7016 | 9 This file is part of Octave. |
4331 | 10 |
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11 Octave is free software: you can redistribute it and/or modify it |
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12 under the terms of the GNU General Public License as published by |
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13 the Free Software Foundation, either version 3 of the License, or |
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14 (at your option) any later version. |
7016 | 15 |
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16 Octave is distributed in the hope that it will be useful, but |
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17 WITHOUT ANY WARRANTY; without even the implied warranty of |
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18 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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19 GNU General Public License for more details. |
4331 | 20 |
21 You should have received a copy of the GNU General Public License | |
7016 | 22 along with Octave; see the file COPYING. If not, see |
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23 <https://www.gnu.org/licenses/>. |
4331 | 24 |
25 */ | |
26 | |
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27 #if defined (HAVE_CONFIG_H) |
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28 # include "config.h" |
4331 | 29 #endif |
30 | |
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31 #include "schur.h" |
4331 | 32 #include "lo-ieee.h" |
33 #include "lo-mappers.h" | |
10608 | 34 #include "oct-norm.h" |
4331 | 35 |
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36 #include "defun.h" |
4331 | 37 #include "error.h" |
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38 #include "errwarn.h" |
4331 | 39 #include "utils.h" |
10608 | 40 #include "xnorm.h" |
4331 | 41 |
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42 template <typename Matrix> |
10608 | 43 static void |
44 sqrtm_utri_inplace (Matrix& T) | |
4331 | 45 { |
10608 | 46 typedef typename Matrix::element_type element_type; |
4331 | 47 |
10608 | 48 const element_type zero = element_type (); |
4331 | 49 |
10608 | 50 bool singular = false; |
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51 |
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52 // The following code is equivalent to this triple loop: |
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53 // |
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54 // n = rows (T); |
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55 // for j = 1:n |
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56 // T(j,j) = sqrt (T(j,j)); |
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57 // for i = j-1:-1:1 |
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58 // T(i,j) /= (T(i,i) + T(j,j)); |
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59 // k = 1:i-1; |
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60 // T(k,j) -= T(k,i) * T(i,j); |
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61 // endfor |
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62 // endfor |
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63 // |
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64 // this is an in-place, cache-aligned variant of the code |
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65 // given in Higham's paper. |
4331 | 66 |
10608 | 67 const octave_idx_type n = T.rows (); |
68 element_type *Tp = T.fortran_vec (); | |
5275 | 69 for (octave_idx_type j = 0; j < n; j++) |
4331 | 70 { |
10608 | 71 element_type *colj = Tp + n*j; |
72 if (colj[j] != zero) | |
73 colj[j] = sqrt (colj[j]); | |
74 else | |
75 singular = true; | |
4331 | 76 |
10608 | 77 for (octave_idx_type i = j-1; i >= 0; i--) |
78 { | |
79 const element_type *coli = Tp + n*i; | |
80 const element_type colji = colj[i] /= (coli[i] + colj[j]); | |
81 for (octave_idx_type k = 0; k < i; k++) | |
82 colj[k] -= coli[k] * colji; | |
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83 } |
4331 | 84 } |
85 | |
10608 | 86 if (singular) |
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87 warning_with_id ("Octave:sqrtm:SingularMatrix", |
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88 "sqrtm: matrix is singular, may not have a square root"); |
4331 | 89 } |
90 | |
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91 template <typename Matrix, typename ComplexMatrix, typename ComplexSCHUR> |
10608 | 92 static octave_value |
93 do_sqrtm (const octave_value& arg) | |
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94 { |
10608 | 95 |
96 octave_value retval; | |
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97 |
10608 | 98 MatrixType mt = arg.matrix_type (); |
99 | |
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100 bool iscomplex = arg.iscomplex (); |
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101 |
10608 | 102 typedef typename Matrix::element_type real_type; |
103 | |
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104 real_type cutoff = 0; |
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105 real_type one = 1; |
10608 | 106 real_type eps = std::numeric_limits<real_type>::epsilon (); |
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107 |
10608 | 108 if (! iscomplex) |
109 { | |
110 Matrix x = octave_value_extract<Matrix> (arg); | |
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111 |
10608 | 112 if (mt.is_unknown ()) // if type is not known, compute it now. |
113 arg.matrix_type (mt = MatrixType (x)); | |
114 | |
115 switch (mt.type ()) | |
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116 { |
10608 | 117 case MatrixType::Upper: |
118 case MatrixType::Diagonal: | |
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119 if (! x.diag ().any_element_is_negative ()) |
10608 | 120 { |
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121 // Do it in real arithmetic. |
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122 sqrtm_utri_inplace (x); |
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123 retval = x; |
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124 retval.matrix_type (mt); |
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125 } |
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126 else |
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127 iscomplex = true; |
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128 break; |
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129 |
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130 case MatrixType::Lower: |
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131 if (! x.diag ().any_element_is_negative ()) |
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132 { |
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133 x = x.transpose (); |
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134 sqrtm_utri_inplace (x); |
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135 retval = x.transpose (); |
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136 retval.matrix_type (mt); |
10608 | 137 } |
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138 else |
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139 iscomplex = true; |
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140 break; |
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141 |
10608 | 142 default: |
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143 iscomplex = true; |
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144 break; |
10608 | 145 } |
146 | |
147 if (iscomplex) | |
148 cutoff = 10 * x.rows () * eps * xnorm (x, one); | |
149 } | |
150 | |
151 if (iscomplex) | |
152 { | |
153 ComplexMatrix x = octave_value_extract<ComplexMatrix> (arg); | |
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154 |
10608 | 155 if (mt.is_unknown ()) // if type is not known, compute it now. |
156 arg.matrix_type (mt = MatrixType (x)); | |
157 | |
158 switch (mt.type ()) | |
159 { | |
160 case MatrixType::Upper: | |
161 case MatrixType::Diagonal: | |
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162 sqrtm_utri_inplace (x); |
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163 retval = x; |
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164 retval.matrix_type (mt); |
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165 break; |
10608 | 166 |
167 case MatrixType::Lower: | |
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168 x = x.transpose (); |
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169 sqrtm_utri_inplace (x); |
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170 retval = x.transpose (); |
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171 retval.matrix_type (mt); |
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172 break; |
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173 |
10608 | 174 default: |
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175 { |
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176 ComplexMatrix u; |
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177 |
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178 do |
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179 { |
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180 ComplexSCHUR schur_fact (x, "", true); |
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181 x = schur_fact.schur_matrix (); |
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182 u = schur_fact.unitary_matrix (); |
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183 } |
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184 while (0); // schur no longer needed. |
10608 | 185 |
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186 sqrtm_utri_inplace (x); |
10608 | 187 |
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188 x = u * x; // original x no longer needed. |
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189 ComplexMatrix res = xgemm (x, u, blas_no_trans, blas_conj_trans); |
10608 | 190 |
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191 if (cutoff > 0 && xnorm (imag (res), one) <= cutoff) |
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192 retval = real (res); |
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193 else |
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194 retval = res; |
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195 } |
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196 break; |
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197 } |
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198 } |
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199 |
10608 | 200 return retval; |
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201 } |
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202 |
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203 DEFUN (sqrtm, args, nargout, |
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204 doc: /* -*- texinfo -*- |
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205 @deftypefn {} {@var{s} =} sqrtm (@var{A}) |
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206 @deftypefnx {} {[@var{s}, @var{error_estimate}] =} sqrtm (@var{A}) |
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207 Compute the matrix square root of the square matrix @var{A}. |
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208 |
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209 Ref: @nospell{N.J. Higham}. @cite{A New sqrtm for @sc{matlab}}. Numerical |
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210 Analysis Report No.@: 336, Manchester @nospell{Centre} for Computational |
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211 Mathematics, Manchester, England, January 1999. |
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212 @seealso{expm, logm} |
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213 @end deftypefn */) |
4331 | 214 { |
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215 if (args.length () != 1) |
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216 print_usage (); |
4331 | 217 |
218 octave_value arg = args(0); | |
219 | |
5275 | 220 octave_idx_type n = arg.rows (); |
221 octave_idx_type nc = arg.columns (); | |
4331 | 222 |
10608 | 223 if (n != nc || arg.ndims () > 2) |
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224 err_square_matrix_required ("sqrtm", "A"); |
4331 | 225 |
20892 | 226 octave_value_list retval (nargout > 1 ? 3 : 1); |
227 | |
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228 if (nargout > 1) |
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229 { |
20892 | 230 // FIXME: Octave does not calculate a condition number with respect to |
231 // sqrtm. Should this return NaN instead of -1? | |
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232 retval(2) = -1.0; |
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233 } |
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234 |
10608 | 235 if (arg.is_diag_matrix ()) |
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236 // sqrtm of a diagonal matrix is just sqrt. |
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237 retval(0) = arg.sqrt (); |
10608 | 238 else if (arg.is_single_type ()) |
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239 retval(0) = do_sqrtm<FloatMatrix, FloatComplexMatrix, |
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240 octave::math::schur<FloatComplexMatrix>> (arg); |
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241 else if (arg.isnumeric ()) |
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242 retval(0) = do_sqrtm<Matrix, ComplexMatrix, |
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243 octave::math::schur<ComplexMatrix>> (arg); |
4331 | 244 |
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245 if (nargout > 1) |
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246 { |
10608 | 247 // This corresponds to generic code |
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248 // |
10608 | 249 // norm (s*s - x, "fro") / norm (x, "fro"); |
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250 |
10608 | 251 octave_value s = retval(0); |
252 retval(1) = xfrobnorm (s*s - arg) / xfrobnorm (arg); | |
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253 } |
4331 | 254 |
255 return retval; | |
256 } | |
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257 |
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258 /* |
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259 %!assert (sqrtm (2*ones (2)), ones (2), 3*eps) |
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260 |
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261 ## The following two tests are from the reference in the docstring above. |
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262 %!test |
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263 %! warning ("off", "Octave:sqrtm:SingularMatrix", "local"); |
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264 %! x = [0 1; 0 0]; |
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265 %! assert (any (isnan (sqrtm (x))(:))); |
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266 |
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267 %!test |
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268 %! x = eye (4); x(2,2) = x(3,3) = 2^-26; x(1,4) = 1; |
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269 %! z = eye (4); z(2,2) = z(3,3) = 2^-13; z(1,4) = 0.5; |
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270 %! [y, err] = sqrtm (x); |
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271 %! assert (y, z); |
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272 %! assert (err, 0); # Yes, this one has to hold exactly |
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273 */ |