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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 Polynomial Manipulations, Control Theory, Sets, Top |
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6 @chapter Polynomial Manipulations |
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7 |
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8 In Octave, a polynomial is represented by its coefficients (arranged |
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9 in descending order). For example, a vector |
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10 @iftex |
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11 @end iftex |
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12 @ifinfo |
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13 $c$ |
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14 @end ifinfo |
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15 of length |
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16 @iftex |
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17 @tex |
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18 $N+1$ |
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19 @end tex |
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20 @ifinfo |
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21 @var{N+1} |
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22 @end ifinfo |
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23 corresponds to the following polynomial of order |
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24 @iftex |
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25 @tex |
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26 $N$ |
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27 $$ |
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28 p (x) = c_1 x^N + ... + c_N x + c_{N+1}. |
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29 $$ |
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30 @end tex |
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31 @end iftex |
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32 @ifinfo |
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33 @var{N} |
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34 |
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35 @example |
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36 p(x) = @var{c}(1) x^@var{N} + ... + @var{c}(@var{N}) x + @var{c}(@var{N}+1). |
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37 @end example |
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38 @end ifinfo |
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39 |
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40 @deftypefn {Function File} {} compan (@var{c}) |
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41 Compute the companion matrix corresponding to polynomial coefficient |
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42 vector @var{c}. |
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43 |
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44 The companion matrix is |
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45 @iftex |
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46 @tex |
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47 $$ |
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48 A = \left[\matrix{ |
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49 -c_2/c_1 & -c_3/c_1 & \cdots & -c_N/c_1 & -c_{N+1}/c_1\cr |
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50 1 & 0 & \cdots & 0 & 0 \cr |
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51 0 & 1 & \cdots & 0 & 0 \cr |
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52 \vdots & \vdots & \ddots & \vdots & \vdots \cr |
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53 0 & 0 & \cdots & 1 & 0}\right]. |
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54 $$ |
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55 @end tex |
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56 @end iftex |
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57 @ifinfo |
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58 |
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59 @smallexample |
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60 _ _ |
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61 | -c(2)/c(1) -c(3)/c(1) ... -c(N)/c(1) -c(N+1)/c(1) | |
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62 | 1 0 ... 0 0 | |
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63 | 0 1 ... 0 0 | |
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64 A = | . . . . . | |
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65 | . . . . . | |
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66 | . . . . . | |
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67 |_ 0 0 ... 1 0 _| |
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68 @end smallexample |
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69 @end ifinfo |
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70 |
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71 The eigenvalues of the companion matrix are equal to the roots of the |
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72 polynomial. |
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73 @end deftypefn |
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74 |
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75 @deftypefn {Function File} {} conv (@var{a}, @var{b}) |
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76 Convolve two vectors. |
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77 |
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78 @code{y = conv (a, b)} returns a vector of length equal to |
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79 @code{length (a) + length (b) - 1}. |
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80 If @var{a} and @var{b} are polynomial coefficient vectors, @code{conv} |
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81 returns the coefficients of the product polynomial. |
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82 @end deftypefn |
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83 |
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84 @deftypefn {Function File} {} deconv (@var{y}, @var{a}) |
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85 Deconvolve two vectors. |
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86 |
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87 @code{[b, r] = deconv (y, a)} solves for @var{b} and @var{r} such that |
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88 @code{y = conv (a, b) + r}. |
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89 |
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90 If @var{y} and @var{a} are polynomial coefficient vectors, @var{b} will |
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91 contain the coefficients of the polynomial quotient and @var{r} will be |
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92 a remander polynomial of lowest order. |
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93 @end deftypefn |
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94 |
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95 @deftypefn {Function File} {} poly (@var{a}) |
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96 If @var{a} is a square @var{N}-by-@var{N} matrix, @code{poly (@var{a})} |
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97 is the row vector of the coefficients of @code{det (z * eye (N) - a)}, |
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98 the characteristic polynomial of @var{a}. If @var{x} is a vector, |
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99 @code{poly (@var{x})} is a vector of coefficients of the polynomial |
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100 whose roots are the elements of @var{x}. |
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101 @end deftypefn |
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102 |
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103 @deftypefn {Function File} {} polyderiv (@var{c}) |
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104 Return the coefficients of the derivative of the polynomial whose |
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105 coefficients are given by vector @var{c}. |
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106 @end deftypefn |
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107 |
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108 @deftypefn {Function File} {[@var{p}, @var{yf}] =} polyfit (@var{x}, @var{y}, @var{n}) |
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109 Return the coefficients of a polynomial @var{p}(@var{x}) of degree |
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110 @var{n} that minimizes |
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111 @iftex |
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112 @tex |
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113 $$ |
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114 \sum_{i=1}^N (p(x_i) - y_i)^2 |
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115 $$ |
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116 @end tex |
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117 @end iftex |
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118 @ifinfo |
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119 @code{sumsq (p(x(i)) - y(i))}, |
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120 @end ifinfo |
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121 to best fit the data in the least squares sense. |
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122 @end deftypefn |
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123 |
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124 If two output arguments are requested, the second contains the values of |
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125 the polynomial for each value of @var{x}. |
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126 |
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127 @deftypefn {Function File} {} polyinteg (@var{c}) |
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128 Return the coefficients of the integral of the polynomial whose |
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129 coefficients are represented by the vector @var{c}. |
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130 |
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131 The constant of integration is set to zero. |
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132 @end deftypefn |
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133 |
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134 @deftypefn {Function File} {} polyreduce (@var{c}) |
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135 Reduces a polynomial coefficient vector to a minimum number of terms by |
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136 stripping off any leading zeros. |
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137 @end deftypefn |
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138 |
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139 @deftypefn {Function File} {} polyval (@var{c}, @var{x}) |
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140 Evaluate a polynomial. |
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141 |
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142 @code{polyval (@var{c}, @var{x})} will evaluate the polynomial at the |
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143 specified value of @var{x}. |
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144 |
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145 If @var{x} is a vector or matrix, the polynomial is evaluated at each of |
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146 the elements of @var{x}. |
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147 @end deftypefn |
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148 |
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149 @deftypefn {Function File} {} polyvalm (@var{c}, @var{x}) |
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150 Evaluate a polynomial in the matrix sense. |
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151 |
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152 @code{polyvalm (@var{c}, @var{x})} will evaluate the polynomial in the |
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153 matrix sense, i.e. matrix multiplication is used instead of element by |
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154 element multiplication as is used in polyval. |
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155 |
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156 The argument @var{x} must be a square matrix. |
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157 @end deftypefn |
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158 |
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159 @deftypefn {Function File} {} residue (@var{b}, @var{a}, @var{tol}) |
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160 If @var{b} and @var{a} are vectors of polynomial coefficients, then |
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161 residue calculates the partial fraction expansion corresponding to the |
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162 ratio of the two polynomials. |
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163 @cindex partial fraction expansion |
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164 |
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165 The function @code{residue} returns @var{r}, @var{p}, @var{k}, and |
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166 @var{e}, where the vector @var{r} contains the residue terms, @var{p} |
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167 contains the pole values, @var{k} contains the coefficients of a direct |
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168 polynomial term (if it exists) and @var{e} is a vector containing the |
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169 powers of the denominators in the partial fraction terms. |
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170 |
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171 Assuming @var{b} and @var{a} represent polynomials |
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172 @iftex |
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173 @tex |
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174 $P(s)$ and $Q(s)$ |
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175 @end tex |
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176 @end iftex |
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177 @ifinfo |
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178 P (s) and Q(s) |
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179 @end ifinfo |
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180 we have: |
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181 @iftex |
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182 @tex |
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183 $$ |
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184 {P(s)\over Q(s)} = \sum_{m=1}^M {r_m\over (s-p_m)^e_m} |
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185 + \sum_{i=1}^N k_i s^{N-i}. |
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186 $$ |
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187 @end tex |
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188 @end iftex |
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189 @ifinfo |
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190 |
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191 @example |
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192 P(s) M r(m) N |
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193 ---- = SUM ------------- + SUM k(i)*s^(N-i) |
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194 Q(s) m=1 (s-p(m))^e(m) i=1 |
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195 @end example |
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196 @end ifinfo |
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197 |
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198 @noindent |
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199 where @var{M} is the number of poles (the length of the @var{r}, |
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200 @var{p}, and @var{e} vectors) and @var{N} is the length of the @var{k} |
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201 vector. |
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202 |
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203 The argument @var{tol} is optional, and if not specified, a default |
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204 value of 0.001 is assumed. The tolerance value is used to determine |
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205 whether poles with small imaginary components are declared real. It is |
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206 also used to determine if two poles are distinct. If the ratio of the |
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207 imaginary part of a pole to the real part is less than @var{tol}, the |
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208 imaginary part is discarded. If two poles are farther apart than |
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209 @var{tol} they are distinct. For example, |
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210 |
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211 @example |
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212 @group |
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213 b = [1, 1, 1]; |
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214 a = [1, -5, 8, -4]; |
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215 [r, p, k, e] = residue (b, a); |
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216 @result{} r = [-2, 7, 3] |
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217 @result{} p = [2, 2, 1] |
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218 @result{} k = [](0x0) |
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219 @result{} e = [1, 2, 1] |
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220 @end group |
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221 @end example |
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222 |
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223 @noindent |
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224 which implies the following partial fraction expansion |
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225 @iftex |
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226 @tex |
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227 $$ |
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228 {s^2+s+1\over s^3-5s^2+8s-4} = {-2\over s-2} + {7\over (s-2)^2} + {3\over s-1} |
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229 $$ |
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230 @end tex |
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231 @end iftex |
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232 @ifinfo |
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233 |
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234 @example |
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235 s^2 + s + 1 -2 7 3 |
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236 ------------------- = ----- + ------- + ----- |
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237 s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1) |
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238 @end example |
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239 @end ifinfo |
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240 @end deftypefn |
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241 |
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242 @deftypefn {Function File} {} roots (@var{v}) |
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243 |
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244 For a vector @var{v} with @var{N} components, return |
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245 the roots of the polynomial |
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246 @iftex |
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247 @tex |
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248 $$ |
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249 v_1 z^{N-1} + \cdots + v_{N-1} z + v_N. |
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250 $$ |
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251 @end tex |
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252 @end iftex |
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253 @ifinfo |
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254 |
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255 @example |
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256 v(1) * z^(N-1) + ... + v(N-1) * z + v(N). |
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257 @end example |
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258 @end ifinfo |
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259 @end deftypefn |
