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1 @c Copyright (C) 1996, 1997 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Matrix Manipulation, Arithmetic, Plotting, Top |
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6 @chapter Matrix Manipulation |
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7 |
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8 There are a number of functions available for checking to see if the |
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9 elements of a matrix meet some condition, and for rearranging the |
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10 elements of a matrix. For example, Octave can easily tell you if all |
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11 the elements of a matrix are finite, or are less than some specified |
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12 value. Octave can also rotate the elements, extract the upper- or |
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13 lower-triangular parts, or sort the columns of a matrix. |
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14 |
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15 @menu |
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16 * Finding Elements and Checking Conditions:: |
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17 * Rearranging Matrices:: |
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18 * Special Utility Matrices:: |
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19 * Famous Matrices:: |
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20 @end menu |
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21 |
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22 @node Finding Elements and Checking Conditions, Rearranging Matrices, Matrix Manipulation, Matrix Manipulation |
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23 @section Finding Elements and Checking Conditions |
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24 |
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25 The functions @code{any} and @code{all} are useful for determining |
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26 whether any or all of the elements of a matrix satisfy some condition. |
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27 The @code{find} function is also useful in determining which elements of |
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28 a matrix meet a specified condition. |
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29 |
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30 @deftypefn {Built-in Function} {} any (@var{x}) |
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31 For a vector argument, return 1 if any element of the vector is |
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32 nonzero. |
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33 |
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34 For a matrix argument, return a row vector of ones and |
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35 zeros with each element indicating whether any of the elements of the |
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36 corresponding column of the matrix are nonzero. For example, |
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37 |
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38 @example |
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39 @group |
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40 any (eye (2, 4)) |
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41 @result{} [ 1, 1, 0, 0 ] |
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42 @end group |
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43 @end example |
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44 |
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45 To see if any of the elements of a matrix are nonzero, you can use a |
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46 statement like |
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47 |
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48 @example |
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49 any (any (a)) |
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50 @end example |
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51 @end deftypefn |
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52 |
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53 @deftypefn {Built-in Function} {} all (@var{x}) |
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54 The function @code{all} behaves like the function @code{any}, except |
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55 that it returns true only if all the elements of a vector, or all the |
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56 elements in a column of a matrix, are nonzero. |
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57 @end deftypefn |
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58 |
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59 Since the comparison operators (@pxref{Comparison Ops}) return matrices |
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60 of ones and zeros, it is easy to test a matrix for many things, not just |
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61 whether the elements are nonzero. For example, |
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62 |
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63 @example |
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64 @group |
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65 all (all (rand (5) < 0.9)) |
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66 @result{} 0 |
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67 @end group |
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68 @end example |
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69 |
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70 @noindent |
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71 tests a random 5 by 5 matrix to see if all of its elements are less |
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72 than 0.9. |
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73 |
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74 Note that in conditional contexts (like the test clause of @code{if} and |
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75 @code{while} statements) Octave treats the test as if you had typed |
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76 @code{all (all (condition))}. |
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77 |
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78 @deftypefn {Function File} {[@var{err}, @var{y1}, ...] =} common_size (@var{x1}, ...) |
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79 Determine if all input arguments are either scalar or of common |
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80 size. If so, @var{err} is zero, and @var{yi} is a matrix of the |
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81 common size with all entries equal to @var{xi} if this is a scalar or |
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82 @var{xi} otherwise. If the inputs cannot be brought to a common size, |
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83 errorcode is 1, and @var{yi} is @var{xi}. For example, |
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84 |
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85 @example |
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86 @group |
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87 [errorcode, a, b] = common_size ([1 2; 3 4], 5) |
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88 @result{} errorcode = 0 |
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89 @result{} a = [ 1, 2; 3, 4 ] |
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90 @result{} b = [ 5, 5; 5, 5 ] |
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91 @end group |
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92 @end example |
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93 |
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94 @noindent |
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95 This is useful for implementing functions where arguments can either |
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96 be scalars or of common size. |
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97 @end deftypefn |
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98 |
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99 @deftypefn {Function File} {} diff (@var{x}, @var{k}) |
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100 If @var{x} is a vector of length @var{n}, @code{diff (@var{x})} is the |
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101 vector of first differences |
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102 @iftex |
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103 @tex |
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104 $x_2 - x_1, \ldots{}, x_n - x_{n-1}$. |
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105 @end tex |
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106 @end iftex |
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107 @ifinfo |
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108 @var{x}(2) - @var{x}(1), @dots{}, @var{x}(n) - @var{x}(n-1). |
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109 @end ifinfo |
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110 |
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111 If @var{x} is a matrix, @code{diff (@var{x})} is the matrix of column |
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112 differences. |
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113 |
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114 The second argument is optional. If supplied, @code{diff (@var{x}, |
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115 @var{k})}, where @var{k} is a nonnegative integer, returns the |
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116 @var{k}-th differences. |
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117 @end deftypefn |
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118 |
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119 @deftypefn {Mapping Function} {} isinf (@var{x}) |
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120 Return 1 for elements of @var{x} that are infinite and zero |
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121 otherwise. For example, |
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122 |
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123 @example |
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124 @group |
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125 isinf ([13, Inf, NaN]) |
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126 @result{} [ 0, 1, 0 ] |
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127 @end group |
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128 @end example |
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129 @end deftypefn |
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130 |
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131 @deftypefn {Mapping Function} {} isnan (@var{x}) |
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132 Return 1 for elements of @var{x} that are NaN values and zero |
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133 otherwise. For example, |
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134 |
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135 @example |
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136 @group |
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137 isnan ([13, Inf, NaN]) |
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138 @result{} [ 0, 0, 1 ] |
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139 @end group |
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140 @end example |
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141 @end deftypefn |
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142 |
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143 @deftypefn {Mapping Function} {} finite (@var{x}) |
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144 Return 1 for elements of @var{x} that are NaN values and zero |
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145 otherwise. For example, |
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146 |
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147 @example |
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148 @group |
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149 finite ([13, Inf, NaN]) |
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150 @result{} [ 1, 0, 0 ] |
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151 @end group |
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152 @end example |
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153 @end deftypefn |
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154 |
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155 @deftypefn {Loadable Function} {} find (@var{x}) |
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156 Return a vector of indices of nonzero elements of a matrix. To obtain a |
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157 single index for each matrix element, Octave pretends that the columns |
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158 of a matrix form one long vector (like Fortran arrays are stored). For |
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159 example, |
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160 |
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161 @example |
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162 @group |
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163 find (eye (2)) |
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164 @result{} [ 1; 4 ] |
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165 @end group |
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166 @end example |
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167 |
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168 If two outputs are requested, @code{find} returns the row and column |
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169 indices of nonzero elements of a matrix. For example, |
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170 |
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171 @example |
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172 @group |
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173 [i, j] = find (2 * eye (2)) |
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174 @result{} i = [ 1; 2 ] |
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175 @result{} j = [ 1; 2 ] |
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176 @end group |
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177 @end example |
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178 |
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179 If three outputs are requested, @code{find} also returns a vector |
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180 containing the nonzero values. For example, |
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181 |
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182 @example |
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183 @group |
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184 [i, j, v] = find (3 * eye (2)) |
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185 @result{} i = [ 1; 2 ] |
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186 @result{} j = [ 1; 2 ] |
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187 @result{} v = [ 3; 3 ] |
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188 @end group |
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189 @end example |
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190 @end deftypefn |
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191 |
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192 @node Rearranging Matrices, Special Utility Matrices, Finding Elements and Checking Conditions, Matrix Manipulation |
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193 @section Rearranging Matrices |
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194 |
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195 @deftypefn {Function File} {} fliplr (@var{x}) |
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196 Return a copy of @var{x} with the order of the columns reversed. For |
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197 example, |
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198 |
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199 @example |
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200 @group |
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201 fliplr ([1, 2; 3, 4]) |
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202 @result{} 2 1 |
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203 4 3 |
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204 @end group |
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205 @end example |
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206 @end deftypefn |
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207 |
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208 @deftypefn {Function File} {} flipud (@var{x}) |
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209 Return a copy of @var{x} with the order of the rows reversed. For |
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210 example, |
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211 |
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212 @example |
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213 @group |
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214 flipud ([1, 2; 3, 4]) |
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215 @result{} 3 4 |
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216 1 2 |
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217 @end group |
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218 @end example |
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219 @end deftypefn |
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220 |
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221 @deftypefn {Function File} {} rot90 (@var{x}, @var{n}) |
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222 Return a copy of @var{x} with the elements rotated counterclockwise in |
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223 90-degree increments. The second argument is optional, and specifies |
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224 how many 90-degree rotations are to be applied (the default value is 1). |
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225 Negative values of @var{n} rotate the matrix in a clockwise direction. |
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226 For example, |
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227 |
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228 @example |
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229 @group |
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230 rot90 ([1, 2; 3, 4], -1) |
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231 @result{} 3 1 |
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232 4 2 |
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233 @end group |
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234 @end example |
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235 |
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236 @noindent |
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237 rotates the given matrix clockwise by 90 degrees. The following are all |
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238 equivalent statements: |
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239 |
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240 @example |
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241 @group |
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242 rot90 ([1, 2; 3, 4], -1) |
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243 @equiv{} |
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244 rot90 ([1, 2; 3, 4], 3) |
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245 @equiv{} |
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246 rot90 ([1, 2; 3, 4], 7) |
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247 @end group |
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248 @end example |
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249 @end deftypefn |
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250 |
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251 @deftypefn {Function File} {} reshape (@var{a}, @var{m}, @var{n}) |
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252 Return a matrix with @var{m} rows and @var{n} columns whose elements are |
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253 taken from the matrix @var{a}. To decide how to order the elements, |
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254 Octave pretends that the elements of a matrix are stored in column-major |
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255 order (like Fortran arrays are stored). |
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256 |
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257 For example, |
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258 |
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259 @example |
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260 @group |
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261 reshape ([1, 2, 3, 4], 2, 2) |
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262 @result{} 1 3 |
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263 2 4 |
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264 @end group |
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265 @end example |
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266 |
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267 If the variable @code{do_fortran_indexing} is nonzero, the |
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268 @code{reshape} function is equivalent to |
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269 |
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270 @example |
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271 @group |
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272 retval = zeros (m, n); |
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273 retval (:) = a; |
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274 @end group |
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275 @end example |
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276 |
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277 @noindent |
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278 but it is somewhat less cryptic to use @code{reshape} instead of the |
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279 colon operator. Note that the total number of elements in the original |
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280 matrix must match the total number of elements in the new matrix. |
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281 @end deftypefn |
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282 |
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283 @deftypefn {Function File} {} shift (@var{x}, @var{b}) |
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284 If @var{x} is a vector, perform a circular shift of length @var{b} of |
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285 the elements of @var{x}. |
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286 |
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287 If @var{x} is a matrix, do the same for each column of @var{x}. |
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288 @end deftypefn |
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289 |
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290 @deftypefn {Loadable Function} {[@var{s}, @var{i}] =} sort (@var{x}) |
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291 Return a copy of @var{x} with the elements elements arranged in |
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292 increasing order. For matrices, @code{sort} orders the elements in each |
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293 column. |
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294 |
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295 For example, |
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296 |
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297 @example |
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298 @group |
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299 sort ([1, 2; 2, 3; 3, 1]) |
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300 @result{} 1 1 |
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301 2 2 |
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302 3 3 |
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303 @end group |
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304 @end example |
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305 |
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306 The @code{sort} function may also be used to produce a matrix |
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307 containing the original row indices of the elements in the sorted |
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308 matrix. For example, |
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309 |
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310 @example |
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311 @group |
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312 [s, i] = sort ([1, 2; 2, 3; 3, 1]) |
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313 @result{} s = 1 1 |
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314 2 2 |
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315 3 3 |
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316 @result{} i = 1 3 |
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317 2 1 |
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318 3 2 |
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319 @end group |
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320 @end example |
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321 @end deftypefn |
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322 |
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323 Since the @code{sort} function does not allow sort keys to be specified, |
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324 it can't be used to order the rows of a matrix according to the values |
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325 of the elements in various columns@footnote{For example, to first sort |
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326 based on the values in column 1, and then, for any values that are |
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327 repeated in column 1, sort based on the values found in column 2, etc.} |
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328 in a single call. Using the second output, however, it is possible to |
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329 sort all rows based on the values in a given column. Here's an example |
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330 that sorts the rows of a matrix based on the values in the second |
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331 column. |
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332 |
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333 @example |
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334 @group |
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335 a = [1, 2; 2, 3; 3, 1]; |
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336 [s, i] = sort (a (:, 2)); |
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337 a (i, :) |
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338 @result{} 3 1 |
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339 1 2 |
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340 2 3 |
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341 @end group |
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342 @end example |
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343 |
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344 @deftypefn {Function File} {} tril (@var{a}, @var{k}) |
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345 @deftypefnx {Function File} {} triu (@var{a}, @var{k}) |
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346 Return a new matrix formed by extracting extract the lower (@code{tril}) |
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347 or upper (@code{triu}) triangular part of the matrix @var{a}, and |
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348 setting all other elements to zero. The second argument is optional, |
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349 and specifies how many diagonals above or below the main diagonal should |
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350 also be set to zero. |
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351 |
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352 The default value of @var{k} is zero, so that @code{triu} and |
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353 @code{tril} normally include the main diagonal as part of the result |
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354 matrix. |
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355 |
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356 If the value of @var{k} is negative, additional elements above (for |
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357 @code{tril}) or below (for @code{triu}) the main diagonal are also |
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358 selected. |
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359 |
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360 The absolute value of @var{k} must not be greater than the number of |
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361 sub- or super-diagonals. |
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362 |
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363 For example, |
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364 |
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365 @example |
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366 @group |
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367 tril (ones (3), -1) |
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368 @result{} 0 0 0 |
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369 1 0 0 |
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370 1 1 0 |
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371 @end group |
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372 @end example |
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373 |
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374 @noindent |
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375 and |
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376 |
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377 @example |
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378 @group |
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379 tril (ones (3), 1) |
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380 @result{} 1 1 0 |
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381 1 1 1 |
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382 1 1 1 |
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383 @end group |
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384 @end example |
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385 @end deftypefn |
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386 |
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387 @deftypefn {Function File} {} vec (@var{x}) |
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388 Return the vector obtained by stacking the columns of the matrix @var{x} |
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389 one above the other. |
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390 @end deftypefn |
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391 |
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392 @deftypefn {Function File} {} vech (@var{x}) |
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393 Return the vector obtained by eliminating all supradiagonal elements of |
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394 the square matrix @var{x} and stacking the result one column above the |
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395 other. |
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396 @end deftypefn |
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397 |
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398 @node Special Utility Matrices, Famous Matrices, Rearranging Matrices, Matrix Manipulation |
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399 @section Special Utility Matrices |
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400 |
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401 @deftypefn {Built-in Function} {} eye (@var{x}) |
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402 @deftypefnx {Built-in Function} {} eye (@var{n}, @var{m}) |
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403 Return an identity matrix. If invoked with a single scalar argument, |
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404 @code{eye} returns a square matrix with the dimension specified. If you |
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405 supply two scalar arguments, @code{eye} takes them to be the number of |
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406 rows and columns. If given a vector with two elements, @code{eye} uses |
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407 the values of the elements as the number of rows and columns, |
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408 respectively. For example, |
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409 |
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410 @example |
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411 @group |
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412 eye (3) |
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413 @result{} 1 0 0 |
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414 0 1 0 |
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415 0 0 1 |
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416 @end group |
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417 @end example |
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418 |
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419 The following expressions all produce the same result: |
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420 |
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421 @example |
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422 @group |
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423 eye (2) |
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424 @equiv{} |
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425 eye (2, 2) |
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426 @equiv{} |
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427 eye (size ([1, 2; 3, 4]) |
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428 @end group |
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429 @end example |
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430 |
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431 For compatibility with @sc{Matlab}, calling @code{eye} with no arguments |
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432 is equivalent to calling it with an argument of 1. |
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433 @end deftypefn |
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434 |
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435 @deftypefn {Built-in Function} {} ones (@var{x}) |
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436 @deftypefnx {Built-in Function} {} ones (@var{n}, @var{m}) |
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437 Return a matrix whose elements are all 1. The arguments are handled |
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438 the same as the arguments for @code{eye}. |
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439 |
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440 If you need to create a matrix whose values are all the same, you should |
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441 use an expression like |
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442 |
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443 @example |
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444 val_matrix = val * ones (n, m) |
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445 @end example |
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446 @end deftypefn |
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447 |
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448 @deftypefn {Built-in Function} {} zeros (@var{x}) |
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449 @deftypefnx {Built-in Function} {} zeros (@var{n}, @var{m}) |
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450 Return a matrix whose elements are all 0. The arguments are handled |
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451 the same as the arguments for @code{eye}. |
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452 @end deftypefn |
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453 |
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454 @deftypefn {Loadable Function} {} rand (@var{x}) |
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455 @deftypefnx {Loadable Function} {} rand (@var{n}, @var{m}) |
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456 @deftypefnx {Loadable Function} {} rand (@code{"seed"}, @var{x}) |
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457 Return a matrix with random elements uniformly distributed on the |
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458 interval (0, 1). The arguments are handled the same as the arguments |
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459 for @code{eye}. In |
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460 addition, you can set the seed for the random number generator using the |
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461 form |
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462 |
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463 @example |
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464 rand ("seed", @var{x}) |
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465 @end example |
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466 |
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467 @noindent |
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468 where @var{x} is a scalar value. If called as |
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469 |
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470 @example |
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471 rand ("seed") |
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|
472 @end example |
|
|
473 |
|
|
474 @noindent |
|
|
475 @code{rand} returns the current value of the seed. |
|
|
476 @end deftypefn |
|
|
477 |
|
|
478 @deftypefn {Loadable Function} {} randn (@var{x}) |
|
|
479 @deftypefnx {Loadable Function} {} randn (@var{n}, @var{m}) |
|
|
480 @deftypefnx {Loadable Function} {} randn (@code{"seed"}, @var{x}) |
|
|
481 Return a matrix with normally distributed random elements. The |
|
|
482 arguments are handled the same as the arguments for @code{eye}. In |
|
|
483 addition, you can set the seed for the random number generator using the |
|
|
484 form |
|
|
485 |
|
|
486 @example |
|
|
487 randn ("seed", @var{x}) |
|
|
488 @end example |
|
|
489 |
|
|
490 @noindent |
|
|
491 where @var{x} is a scalar value. If called as |
|
|
492 |
|
|
493 @example |
|
|
494 randn ("seed") |
|
|
495 @end example |
|
|
496 |
|
|
497 @noindent |
|
|
498 @code{randn} returns the current value of the seed. |
|
|
499 @end deftypefn |
|
|
500 |
|
|
501 The @code{rand} and @code{randn} functions use separate generators. |
|
|
502 This ensures that |
|
|
503 |
|
|
504 @example |
|
|
505 @group |
|
|
506 rand ("seed", 13); |
|
|
507 randn ("seed", 13); |
|
|
508 u = rand (100, 1); |
|
|
509 n = randn (100, 1); |
|
|
510 @end group |
|
|
511 @end example |
|
|
512 |
|
|
513 @noindent |
|
|
514 and |
|
|
515 |
|
|
516 @example |
|
|
517 @group |
|
|
518 rand ("seed", 13); |
|
|
519 randn ("seed", 13); |
|
|
520 u = zeros (100, 1); |
|
|
521 n = zeros (100, 1); |
|
|
522 for i = 1:100 |
|
|
523 u(i) = rand (); |
|
|
524 n(i) = randn (); |
|
|
525 end |
|
|
526 @end group |
|
|
527 @end example |
|
|
528 |
|
|
529 @noindent |
|
|
530 produce equivalent results. |
|
|
531 |
|
|
532 Normally, @code{rand} and @code{randn} obtain their initial |
|
|
533 seeds from the system clock, so that the sequence of random numbers is |
|
|
534 not the same each time you run Octave. If you really do need for to |
|
|
535 reproduce a sequence of numbers exactly, you can set the seed to a |
|
|
536 specific value. |
|
|
537 |
|
|
538 If it is invoked without arguments, @code{rand} and @code{randn} return a |
|
|
539 single element of a random sequence. |
|
|
540 |
|
|
541 The @code{rand} and @code{randn} functions use Fortran code from |
|
|
542 @sc{Ranlib}, a library of fortran routines for random number generation, |
|
|
543 compiled by Barry W. Brown and James Lovato of the Department of |
|
|
544 Biomathematics at The University of Texas, M.D. Anderson Cancer Center, |
|
|
545 Houston, TX 77030. |
|
|
546 |
|
|
547 @deftypefn {Built-in Function} {} diag (@var{v}, @var{k}) |
|
|
548 Return a diagonal matrix with vector @var{v} on diagonal @var{k}. The |
|
|
549 second argument is optional. If it is positive, the vector is placed on |
|
|
550 the @var{k}-th super-diagonal. If it is negative, it is placed on the |
|
|
551 @var{-k}-th sub-diagonal. The default value of @var{k} is 0, and the |
|
|
552 vector is placed on the main diagonal. For example, |
|
|
553 |
|
|
554 @example |
|
|
555 @group |
|
|
556 diag ([1, 2, 3], 1) |
|
|
557 @result{} 0 1 0 0 |
|
|
558 0 0 2 0 |
|
|
559 0 0 0 3 |
|
|
560 0 0 0 0 |
|
|
561 @end group |
|
|
562 @end example |
|
|
563 @end deftypefn |
|
|
564 |
|
|
565 @c XXX FIXME XXX -- is this really worth documenting? |
|
|
566 @c |
|
|
567 @c @defvr {Built-in Variable} ok_to_lose_imaginary_part |
|
|
568 @c If the value of @code{ok_to_lose_imaginary_part} is nonzero, implicit |
|
|
569 @c conversions of complex numbers to real numbers are allowed (for example, |
|
|
570 @c by fsolve). If the value is @code{"warn"}, the conversion is allowed, |
|
|
571 @c but a warning is printed. Otherwise, an error message is printed and |
|
|
572 @c control is returned to the top level. The default value is |
|
|
573 @c @code{"warn"}. |
|
|
574 @c |
|
|
575 @c XXX FIXME XXX -- this is here because it is used by @code{ones}, |
|
|
576 @c @code{zeros}, @code{rand}, etc. |
|
|
577 @c @end defvr |
|
|
578 |
|
|
579 The functions @code{linspace} and @code{logspace} make it very easy to |
|
|
580 create vectors with evenly or logarithmically spaced elements. |
|
|
581 @xref{Ranges}. |
|
|
582 |
|
|
583 @deftypefn {Function File} {} linspace (@var{base}, @var{limit}, @var{n}) |
|
|
584 Return a row vector with @var{n} linearly spaced elements between |
|
|
585 @var{base} and @var{limit}. The number of elements, @var{n}, must be |
|
|
586 greater than 1. The @var{base} and @var{limit} are always included in |
|
|
587 the range. If @var{base} is greater than @var{limit}, the elements are |
|
|
588 stored in decreasing order. If the number of points is not specified, a |
|
|
589 value of 100 is used. |
|
|
590 |
|
|
591 The @code{linspace} function always returns a row vector, regardless of |
|
|
592 the value of @code{prefer_column_vectors}. |
|
|
593 @end deftypefn |
|
|
594 |
|
|
595 @deftypefn {Function File} {} logspace (@var{base}, @var{limit}, @var{n}) |
|
|
596 Similar to @code{linspace} except that the values are logarithmically |
|
|
597 spaced from |
|
|
598 @iftex |
|
|
599 @tex |
|
|
600 $10^{base}$ to $10^{limit}$. |
|
|
601 @end tex |
|
|
602 @end iftex |
|
|
603 @ifinfo |
|
|
604 10^base to 10^limit. |
|
|
605 @end ifinfo |
|
|
606 |
|
|
607 If @var{limit} is equal to |
|
|
608 @iftex |
|
|
609 @tex |
|
|
610 $\pi$, |
|
|
611 @end tex |
|
|
612 @end iftex |
|
|
613 @ifinfo |
|
|
614 pi, |
|
|
615 @end ifinfo |
|
|
616 the points are between |
|
|
617 @iftex |
|
|
618 @tex |
|
|
619 $10^{base}$ and $\pi$, |
|
|
620 @end tex |
|
|
621 @end iftex |
|
|
622 @ifinfo |
|
|
623 10^base and pi, |
|
|
624 @end ifinfo |
|
|
625 @emph{not} |
|
|
626 @iftex |
|
|
627 @tex |
|
|
628 $10^{base}$ and $10^{\pi}$, |
|
|
629 @end tex |
|
|
630 @end iftex |
|
|
631 @ifinfo |
|
|
632 10^base and 10^pi, |
|
|
633 @end ifinfo |
|
|
634 in order to be compatible with the corresponding @sc{Matlab} function. |
|
|
635 @end deftypefn |
|
|
636 |
|
|
637 @defvr {Built-in Variable} treat_neg_dim_as_zero |
|
|
638 If the value of @code{treat_neg_dim_as_zero} is nonzero, expressions |
|
|
639 like |
|
|
640 |
|
|
641 @example |
|
|
642 eye (-1) |
|
|
643 @end example |
|
|
644 |
|
|
645 @noindent |
|
|
646 produce an empty matrix (i.e., row and column dimensions are zero). |
|
|
647 Otherwise, an error message is printed and control is returned to the |
|
|
648 top level. The default value is 0. |
|
|
649 @end defvr |
|
|
650 |
|
|
651 @node Famous Matrices, , Special Utility Matrices, Matrix Manipulation |
|
|
652 @section Famous Matrices |
|
|
653 |
|
|
654 The following functions return famous matrix forms. |
|
|
655 |
|
|
656 @deftypefn {Function File} {} hankel (@var{c}, @var{r}) |
|
|
657 Return the Hankel matrix constructed given the first column @var{c}, and |
|
|
658 (optionally) the last row @var{r}. If the last element of @var{c} is |
|
|
659 not the same as the first element of @var{r}, the last element of |
|
|
660 @var{c} is used. If the second argument is omitted, the last row is |
|
|
661 taken to be the same as the first column. |
|
|
662 |
|
|
663 A Hankel matrix formed from an m-vector @var{c}, and an n-vector |
|
|
664 @var{r}, has the elements |
|
|
665 @iftex |
|
|
666 @tex |
|
|
667 $$ |
|
|
668 H (i, j) = \cases{c_{i+j-1},&$i+j-1\le m$;\cr r_{i+j-m},&otherwise.\cr} |
|
|
669 $$ |
|
|
670 @end tex |
|
|
671 @end iftex |
|
|
672 @ifinfo |
|
|
673 |
|
|
674 @example |
|
|
675 @group |
|
|
676 H (i, j) = c (i+j-1), i+j-1 <= m; |
|
|
677 H (i, j) = r (i+j-m), otherwise |
|
|
678 @end group |
|
|
679 @end example |
|
|
680 @end ifinfo |
|
|
681 @end deftypefn |
|
|
682 |
|
|
683 @deftypefn {Function File} {} hilb (@var{n}) |
|
|
684 Return the Hilbert matrix of order @var{n}. The |
|
|
685 @iftex |
|
|
686 @tex |
|
|
687 $i,\,j$ |
|
|
688 @end tex |
|
|
689 @end iftex |
|
|
690 @ifinfo |
|
|
691 i, j |
|
|
692 @end ifinfo |
|
|
693 element of a Hilbert matrix is defined as |
|
|
694 @iftex |
|
|
695 @tex |
|
|
696 $$ |
|
|
697 H (i, j) = {1 \over (i + j - 1)} |
|
|
698 $$ |
|
|
699 @end tex |
|
|
700 @end iftex |
|
|
701 @ifinfo |
|
|
702 |
|
|
703 @example |
|
|
704 H (i, j) = 1 / (i + j - 1) |
|
|
705 @end example |
|
|
706 @end ifinfo |
|
|
707 @end deftypefn |
|
|
708 |
|
|
709 @deftypefn {Function File} {} invhilb (@var{n}) |
|
|
710 Return the inverse of a Hilbert matrix of order @var{n}. This is exact. |
|
|
711 Compare with the numerical calculation of @code{inverse (hilb (n))}, |
|
|
712 which suffers from the ill-conditioning of the Hilbert matrix, and the |
|
|
713 finite precision of your computer's floating point arithmetic. |
|
|
714 @end deftypefn |
|
|
715 |
|
|
716 @deftypefn {Function File} {} sylvester_matrix (@var{k}) |
|
|
717 Return the Sylvester matrix of order |
|
|
718 @iftex |
|
|
719 @tex |
|
|
720 $n = 2^k$. |
|
|
721 @end tex |
|
|
722 @end iftex |
|
|
723 @ifinfo |
|
|
724 n = 2^k. |
|
|
725 @end ifinfo |
|
|
726 @end deftypefn |
|
|
727 |
|
|
728 @deftypefn {Function File} {} toeplitz (@var{c}, @var{r}) |
|
|
729 Return the Toeplitz matrix constructed given the first column @var{c}, |
|
|
730 and (optionally) the first row @var{r}. If the first element of @var{c} |
|
|
731 is not the same as the first element of @var{r}, the first element of |
|
|
732 @var{c} is used. If the second argument is omitted, the first row is |
|
|
733 taken to be the same as the first column. |
|
|
734 |
|
|
735 A square Toeplitz matrix has the form |
|
|
736 @iftex |
|
|
737 @tex |
|
|
738 $$ |
|
|
739 \left[\matrix{c_0 & r_1 & r_2 & \ldots & r_n\cr |
|
|
740 c_1 & c_0 & r_1 & & c_{n-1}\cr |
|
|
741 c_2 & c_1 & c_0 & & c_{n-2}\cr |
|
|
742 \vdots & & & & \vdots\cr |
|
|
743 c_n & c_{n-1} & c_{n-2} & \ldots & c_0}\right]. |
|
|
744 $$ |
|
|
745 @end tex |
|
|
746 @end iftex |
|
|
747 @ifinfo |
|
|
748 |
|
|
749 @example |
|
|
750 @group |
|
|
751 c(0) r(1) r(2) ... r(n) |
|
|
752 c(1) c(0) r(1) r(n-1) |
|
|
753 c(2) c(1) c(0) r(n-2) |
|
|
754 . . |
|
|
755 . . |
|
|
756 . . |
|
|
757 |
|
|
758 c(n) c(n-1) c(n-2) ... c(0) |
|
|
759 @end group |
|
|
760 @end example |
|
|
761 @end ifinfo |
|
|
762 @end deftypefn |
|
|
763 |
|
|
764 @deftypefn {Function File} {} vander (@var{c}) |
|
|
765 Return the Vandermonde matrix whose next to last column is @var{c}. |
|
|
766 |
|
|
767 A Vandermonde matrix has the form |
|
|
768 @iftex |
|
|
769 @tex |
|
|
770 $$ |
|
|
771 \left[\matrix{c_0^n & \ldots & c_0^2 & c_0 & 1\cr |
|
|
772 c_1^n & \ldots & c_1^2 & c_1 & 1\cr |
|
|
773 \vdots & & \vdots & \vdots & \vdots\cr |
|
|
774 c_n^n & \ldots & c_n^2 & c_n & 1}\right]. |
|
|
775 $$ |
|
|
776 @end tex |
|
|
777 @end iftex |
|
|
778 @ifinfo |
|
|
779 |
|
|
780 @example |
|
|
781 @group |
|
|
782 c(0)^n ... c(0)^2 c(0) 1 |
|
|
783 c(1)^n ... c(1)^2 c(1) 1 |
|
|
784 . . . . |
|
|
785 . . . . |
|
|
786 . . . . |
|
|
787 |
|
|
788 c(n)^n ... c(n)^2 c(n) 1 |
|
|
789 @end group |
|
|
790 @end example |
|
|
791 @end ifinfo |
|
|
792 @end deftypefn |
