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1 @c Copyright (C) 1996 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 Special Matrices, Matrix Manipulation, Input and Output, Top |
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6 @chapter Special Matrices |
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7 |
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8 Octave provides a number of functions for creating special matrix forms. |
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9 In nearly all cases, it is best to use the built-in functions for this |
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10 purpose than to try to use other tricks to achieve the same effect. |
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11 |
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12 @menu |
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13 * Special Utility Matrices:: |
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14 * Famous Matrices:: |
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15 @end menu |
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16 |
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17 @node Special Utility Matrices, Famous Matrices, Special Matrices, Special Matrices |
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18 @section Special Utility Matrices |
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19 |
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20 @deftypefn {Built-in Function} {} eye (@var{x}) |
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21 @deftypefnx {Built-in Function} {} eye (@var{n}, @var{m}) |
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22 Returns an identity matrix. If invoked with a single scalar argument, |
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23 @code{eye} returns a square matrix with the dimension specified. If you |
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24 supply two scalar arguments, @code{eye} takes them to be the number of |
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25 rows and columns. If given a vector with two elements, @code{eye} uses |
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26 the values of the elements as the number of rows and columns, |
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27 respecively. For example, |
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28 |
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29 @example |
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30 @group |
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31 eye (3) |
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32 |
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33 @result{} 1 0 0 |
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34 0 1 0 |
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35 0 0 1 |
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36 @end group |
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37 @end example |
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38 |
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39 The following expressions all produce the same result: |
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40 |
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41 @example |
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42 @group |
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43 eye (2) |
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44 eye (2, 2) |
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45 eye (size ([1, 2; 3, 4]) |
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46 @end group |
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47 @end example |
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48 |
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49 For compatibility with @sc{Matlab}, calling @code{eye} with no arguments |
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50 is equivalent to calling it with an argument of 1. |
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51 @end deftypefn |
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52 |
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53 @deftypefn {Built-in Function} {} ones (@var{x}) |
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54 @deftypefnx {Built-in Function} {} ones (@var{n}, @var{m}) |
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55 Returns a matrix whose elements are all 1. The arguments are handled |
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56 the same as the arguments for @code{eye}. |
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57 |
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58 If you need to create a matrix whose values are all the same, you should |
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59 use an expression like |
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60 |
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61 @example |
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62 val_matrix = val * ones (n, m) |
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63 @end example |
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64 @end deftypefn |
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65 |
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66 @deftypefn {Built-in Function} {} zeros (@var{x}) |
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67 @deftypefnx {Built-in Function} {} zeros (@var{n}, @var{m}) |
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68 Returns a matrix whose elements are all 0. The arguments are handled |
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69 the same as the arguments for @code{eye}. |
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70 @end deftypefn |
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71 |
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72 @deftypefn {Built-in Function} {} rand (@var{x}) |
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73 @deftypefnx {Built-in Function} {} rand (@var{n}, @var{m}) |
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74 @deftypefnx {Built-in Function} {} rand (@code{"seed"}, @var{x}) |
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75 Returns a matrix with random elements uniformly distributed on the |
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76 interval (0, 1). The arguments are handled the same as the arguments |
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77 for @code{eye}. In |
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78 addition, you can set the seed for the random number generator using the |
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79 form |
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80 |
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81 @example |
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82 randn ("seed", @var{x}) |
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83 @end example |
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84 |
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85 @noindent |
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86 where @var{x} is a scalar value. If called as |
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87 |
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88 @example |
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89 rand ("seed") |
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90 @end example |
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91 |
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92 @noindent |
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93 @code{rand} returns the current value of the seed. |
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94 @end deftypefn |
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95 |
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96 @deftypefn {Built-in Function} {} randn (@var{x}) |
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97 @deftypefnx {Built-in Function} {} randn (@var{n}, @var{m}) |
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98 @deftypefnx {Built-in Function} {} randn (@code{"seed"}, @var{x}) |
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99 Returns a matrix with normally distributed random elements. The |
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100 arguments are handled the same as the arguments for @code{eye}. In |
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101 addition, you can set the seed for the random number generator using the |
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102 form |
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103 |
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104 @example |
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105 randn ("seed", @var{x}) |
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106 @end example |
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107 |
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108 @noindent |
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109 where @var{x} is a scalar value. If called as |
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110 |
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111 @example |
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112 randn ("seed") |
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113 @end example |
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114 |
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115 @noindent |
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116 @code{randn} returns the current value of the seed. |
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117 @end deftypefn |
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118 |
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119 The @code{rand} and @code{randn} functions use separate generators. |
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120 This ensures that |
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121 |
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122 @example |
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123 @group |
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124 rand ("seed", 13); |
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125 randn ("seed", 13); |
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126 u = rand (100, 1); |
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127 n = randn (100, 1); |
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128 @end group |
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129 @end example |
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130 |
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131 @noindent |
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132 and |
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133 |
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134 @example |
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135 @group |
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136 rand ("seed", 13); |
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137 randn ("seed", 13); |
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138 u = zeros (100, 1); |
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139 n = zeros (100, 1); |
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140 for i = 1:100 |
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141 u(i) = rand (); |
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142 n(i) = randn (); |
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143 end |
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144 @end group |
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145 @end example |
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146 |
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147 @noindent |
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148 produce equivalent results. |
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149 |
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150 Normally, @code{rand} and @code{randn} obtain their initial |
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151 seeds from the system clock, so that the sequence of random numbers is |
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152 not the same each time you run Octave. If you really do need for to |
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153 reproduce a sequence of numbers exactly, you can set the seed to a |
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154 specific value. |
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155 |
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156 If it is invoked without arguments, @code{rand} and @code{randn} return a |
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157 single element of a random sequence. |
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158 |
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159 The @code{rand} and @code{randn} functions use Fortran code from RANLIB, |
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160 a library of fortran routines for random number generation, compiled by |
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161 Barry W. Brown and James Lovato of the Department of Biomathematics at |
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162 The University of Texas, M.D. Anderson Cancer Center, Houston, TX 77030. |
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163 |
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164 @deftypefn {Built-in Function} {} diag (@var{v}, @var{k}) |
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165 Returns a diagonal matrix with vector @var{v} on diagonal @var{k}. The |
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166 second argument is optional. If it is positive, the vector is placed on |
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167 the @var{k}-th super-diagonal. If it is negative, it is placed on the |
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168 @var{-k}-th sub-diagonal. The default value of @var{k} is 0, and the |
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169 vector is placed on the main diagonal. For example, |
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170 |
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171 @example |
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172 @group |
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173 diag ([1, 2, 3], 1) |
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174 |
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175 @result{} 0 1 0 0 |
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176 0 0 2 0 |
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177 0 0 0 3 |
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178 0 0 0 0 |
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179 @end group |
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180 @end example |
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181 @end deftypefn |
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182 |
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183 The functions @code{linspace} and @code{logspace} make it very easy to |
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184 create vectors with evenly or logarithmically spaced elements. |
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185 @xref{Ranges}. |
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186 |
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187 @deftypefn {Function File} {} linspace (@var{base}, @var{limit}, @var{n}) |
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188 creates a row vector with @var{n} (@var{n} greater than 1) linearly |
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189 spaced elements between @var{base} and @var{limit}. The @var{base} and |
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190 @var{limit} are always included in the range. If @var{base} is greater |
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191 than @var{limit}, the elements are stored in decreasing order. If the |
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192 number of points is not specified, a value of 100 is used. |
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193 |
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194 The @code{linspace} function always returns a row vector, regardless of |
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195 the value of @code{prefer_column_vectors}. |
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196 @end deftypefn |
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197 |
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198 @deftypefn {Function File} {} logspace (@var{base}, @var{limit}, @var{n}) |
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199 Similar to @code{linspace} except that the values are logarithmically |
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200 spaced. |
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201 |
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202 If @var{limit} is equal to |
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203 @iftex |
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204 @tex |
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205 $\pi$, |
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206 @end tex |
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207 @end iftex |
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208 @ifinfo |
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209 pi, |
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210 @end ifinfo |
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211 the points are between |
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212 @iftex |
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213 @tex |
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214 $10^{base}$ and $\pi$, |
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215 @end tex |
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216 @end iftex |
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217 @ifinfo |
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218 10^base and pi, |
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219 @end ifinfo |
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220 @emph{not} |
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221 @iftex |
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222 @tex |
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223 $10^{base}$ and $10^{\pi}$, |
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224 @end tex |
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225 @end iftex |
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226 @ifinfo |
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227 10^base and 10^pi, |
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228 @end ifinfo |
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229 in order to be compatible with the corresponding @sc{Matlab} function. |
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230 @end deftypefn |
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231 |
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232 @node Famous Matrices, , Special Utility Matrices, Special Matrices |
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233 @section Famous Matrices |
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234 |
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235 The following functions return famous matrix forms. |
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236 |
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237 @deftypefn {Function File} {} hadamard (@var{k}) |
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238 Return the Hadamard matrix of order n = 2^k. |
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239 @end deftypefn |
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240 |
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241 @deftypefn {Function File} {} hankel (@var{c}, @var{r}) |
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242 Return the Hankel matrix constructed given the first column @var{c}, and |
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243 (optionally) the last row @var{r}. If the last element of @var{c} is |
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244 not the same as the first element of @var{r}, the last element of |
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245 @var{c} is used. If the second argument is omitted, the last row is |
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246 taken to be the same as the first column. |
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247 |
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248 A Hankel matrix formed from an m-vector @var{c}, and an n-vector |
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249 @var{r}, has the elements |
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250 @iftex |
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251 @tex |
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252 $$ |
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253 H (i, j) = \cases{c_{i+j-1},&$i+j-1\le m$;\cr r_{i+j-m},&otherwise.\cr} |
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254 $$ |
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255 @end tex |
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256 @end iftex |
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257 @ifinfo |
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258 |
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259 @example |
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260 @group |
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261 H (i, j) = c (i+j-1), i+j-1 <= m; |
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262 H (i, j) = r (i+j-m), otherwise |
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263 @end group |
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264 @end example |
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265 @end ifinfo |
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266 @end deftypefn |
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267 |
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268 @deftypefn {Function File} {} hilb (@var{n}) |
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269 Return the Hilbert matrix of order @var{n}. The |
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270 @iftex |
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271 @tex |
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272 $i,\,j$ |
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273 @end tex |
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274 @end iftex |
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275 @ifinfo |
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276 i, j |
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277 @end ifinfo |
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278 element of a Hilbert matrix is defined as |
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279 @iftex |
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280 @tex |
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281 $$ |
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282 H (i, j) = {1 \over (i + j - 1)} |
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283 $$ |
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284 @end tex |
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285 @end iftex |
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286 @ifinfo |
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287 |
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288 @example |
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289 H (i, j) = 1 / (i + j - 1) |
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290 @end example |
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291 @end ifinfo |
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292 @end deftypefn |
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293 |
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294 @deftypefn {Function File} {} invhilb (@var{n}) |
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295 Return the inverse of a Hilbert matrix of order @var{n}. This is exact. |
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296 Compare with the numerical calculation of @code{inverse (hilb (n))}, |
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297 which suffers from the ill-conditioning of the Hilbert matrix, and the |
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298 finite precision of your computer's floating point arithmetic. |
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299 @end deftypefn |
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300 |
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301 @deftypefn {Function File} {} toeplitz (@var{c}, @var{r}) |
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302 Return the Toeplitz matrix constructed given the first column @var{c}, |
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303 and (optionally) the first row @var{r}. If the first element of @var{c} |
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304 is not the same as the first element of @var{r}, the first element of |
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305 @var{c} is used. If the second argument is omitted, the first row is |
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306 taken to be the same as the first column. |
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307 |
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308 A square Toeplitz matrix has the form |
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309 @iftex |
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310 @tex |
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311 $$ |
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312 \left[\matrix{c_0 & r_1 & r_2 & \ldots & r_n\cr |
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313 c_1 & c_0 & r_1 & & c_{n-1}\cr |
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314 c_2 & c_1 & c_0 & & c_{n-2}\cr |
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315 \vdots & & & & \vdots\cr |
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316 c_n & c_{n-1} & c_{n-2} & \ldots & c_0}\right]. |
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317 $$ |
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318 @end tex |
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319 @end iftex |
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320 @ifinfo |
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321 |
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322 @example |
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323 @group |
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324 c(0) r(1) r(2) ... r(n) |
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325 c(1) c(0) r(1) r(n-1) |
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326 c(2) c(1) c(0) r(n-2) |
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327 . . |
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328 . . |
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329 . . |
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330 |
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331 c(n) c(n-1) c(n-2) ... c(0) |
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332 @end group |
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333 @end example |
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334 @end ifinfo |
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335 @end deftypefn |
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336 |
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337 @deftypefn {Function File} {} vander (@var{c}) |
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338 Return the Vandermonde matrix whose next to last column is @var{c}. |
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339 |
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340 A Vandermonde matrix has the form |
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341 @iftex |
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342 @tex |
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343 $$ |
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344 \left[\matrix{c_0^n & \ldots & c_0^2 & c_0 & 1\cr |
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345 c_1^n & \ldots & c_1^2 & c_1 & 1\cr |
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346 \vdots & & \vdots & \vdots & \vdots\cr |
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347 c_n^n & \ldots & c_n^2 & c_n & 1}\right]. |
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348 $$ |
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349 @end tex |
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350 @end iftex |
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351 @ifinfo |
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352 |
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353 @example |
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354 @group |
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355 c(0)^n ... c(0)^2 c(0) 1 |
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356 c(1)^n ... c(1)^2 c(1) 1 |
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357 . . . . |
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358 . . . . |
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359 . . . . |
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360 |
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361 c(n)^n ... c(n)^2 c(n) 1 |
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362 @end group |
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363 @end example |
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364 @end ifinfo |
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365 @end deftypefn |
