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1 function [r, p, k, e] = residue(b,a,toler) |
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2 #[r p k e] = residue(b,a) |
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3 #If b and a are vectors of polynomial coefficients, then residue |
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4 #calculates the partial fraction expansion corresponding to the |
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5 #ratio of the two polynomials. The vector r contains the residue |
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6 #terms, p contains the pole values, k contains the coefficients of |
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7 #a direct polynomial term (if it exists) and e is a vector containing |
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8 #the powers of the denominators in the partial fraction terms. |
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9 #Assuming b and a represent polynomials P(s) and Q(s) we have: |
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10 # |
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11 # P(s) M r(m) N |
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12 # ---- = # ------------- + # k(n)*s^(N-n) |
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13 # Q(s) m=1 (s-p(m))^e(m) n=1 |
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14 # |
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15 #(# represents summation) where M is the number of poles (the length of |
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16 #the r, p, and e vectors) and N is the length of the k vector. |
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17 # |
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18 #[r p k e] = residue(b,a,tol) |
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19 #This form of the function call may be used to set a tolerance value. |
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20 #The default value is 0.001. The tolerance value is used to determine |
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21 #whether poles with small imaginary components are declared real. It is |
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22 #also used to determine if two poles are distinct. If the ratio of the |
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23 #imaginary part of a pole to the real part is less than tol, the imaginary |
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24 #part is discarded. If two poles are farther apart than tol they are |
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25 #distinct. |
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26 # |
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27 #Example: |
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28 # b = [1 1 1]; |
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29 # a = [1 -5 8 -4]; |
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30 # |
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31 # [r p k e] = residue(b,a) |
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32 # |
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33 # returns |
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34 # |
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35 # r = [-2 7 3]; p = [2 2 1]; k = []; e = [1 2 1]; |
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36 # |
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37 # which implies the following partial fraction expansion |
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38 # |
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39 # s^2 + s + 1 -2 7 3 |
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40 # ------------------- = ----- + ------- + ----- |
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41 # s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1) |
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42 # |
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43 #SEE ALSO: poly, roots, conv, deconv, polyval, polyderiv, polyinteg |
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44 |
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45 # Author: |
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46 # Tony Richardson |
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47 # amr@mpl.ucsd.edu |
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48 # June 1994 |
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49 |
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50 # Here's the method used to find the residues. |
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51 # The partial fraction expansion can be written as: |
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52 # |
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53 # |
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54 # P(s) D M(k) A(k,m) |
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55 # ---- = # # ------------- |
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56 # Q(s) k=1 m=1 (s - pr(k))^m |
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57 # |
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58 # (# is used to represent a summation) where D is the number of |
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59 # distinct roots, pr(k) is the kth distinct root, M(k) is the |
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60 # multiplicity of the root, and A(k,m) is the residue cooresponding |
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61 # to the kth distinct root with multiplicity m. For example, |
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62 # |
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63 # s^2 A(1,1) A(2,1) A(2,2) |
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64 # ------------------- = ------ + ------ + ------- |
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65 # s^3 + 4s^2 + 5s + 2 (s+2) (s+1) (s+1)^2 |
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66 # |
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67 # In this case there are two distinct roots (D=2 and pr = [-2 -1]), |
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68 # the first root has multiplicity one and the second multiplicity |
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69 # two (M = [1 2]) The residues are actually stored in vector format as |
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70 # r = [ A(1,1) A(2,1) A(2,2) ]. |
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71 # |
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72 # We then multiply both sides by Q(s). Continuing the example: |
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73 # |
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74 # s^2 = r(1)*(s+1)^2 + r(2)*(s+1)*(s+2) + r(3)*(s+2) |
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75 # |
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76 # or |
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77 # |
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78 # s^2 = r(1)*(s^2+2s+1) + r(2)*(s^2+3s+2) +r(3)*(s+2) |
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79 # |
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80 # The coefficients of the polynomials on the right are stored |
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81 # in a row vector called rhs, while the coefficients of the |
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82 # polynomial on the left is stored in a row vector called lhs. |
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83 # If the multiplicity of any root is greater than one we'll |
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84 # also need derivatives of this equation of order up to the |
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85 # maximum multiplicity minus one. The derivative coefficients |
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86 # are stored in successive rows of lhs and rhs. |
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87 # |
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88 # For our example lhs and rhs would be: |
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89 # |
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90 # | 1 0 0 | |
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91 # lhs = | | |
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92 # | 0 2 0 | |
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93 # |
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94 # | 1 2 1 1 3 2 0 1 2 | |
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95 # rhs = | | |
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96 # | 0 2 2 0 2 3 0 0 1 | |
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97 # |
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98 # We then form a vector B and a matrix A obtained by evaluating the |
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99 # polynomials in lhs and rhs at the pole values. If a pole has a |
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100 # multiplicity greater than one we also evaluate the derivative |
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101 # polynomials (successive rows) at the pole value. |
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102 # |
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103 # For our example we would have |
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104 # |
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105 # | 4| | 1 0 0 | | r(1) | |
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106 # | 1| = | 0 0 1 | * | r(2) | |
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107 # |-2| | 0 1 1 | | r(3) | |
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108 # |
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109 # We then solve for the residues using matrix division. |
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110 |
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111 if(nargin < 2 || nargin > 3) |
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112 error("usage: residue(b,a[,toler])"); |
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113 endif |
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114 |
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115 if (nargin == 2) |
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116 # Set the default tolerance level |
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117 toler = .001; |
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118 endif |
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119 |
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120 # Make sure both polynomials are in reduced form. |
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121 a = polyreduce(a); |
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122 b = polyreduce(b); |
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123 |
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124 b = b/a(1); |
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125 a = a/a(1); |
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126 |
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127 la = length(a); |
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128 lb = length(b); |
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129 |
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130 # Handle special cases here. |
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131 if(la == 0 || lb == 0) |
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132 k = r = p = e = []; |
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133 return; |
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134 elseif (la == 1) |
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135 k = b/a; |
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136 r = p = e = []; |
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137 return; |
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138 endif |
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139 |
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140 # Find the poles. |
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141 p = roots(a); |
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142 lp = length(p); |
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143 |
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144 # Determine if the poles are (effectively) real. |
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145 index = find(abs(imag(p) ./ real(p)) < toler); |
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146 if (length(index) != 0) |
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147 p(index) = real(p(index)); |
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148 endif |
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149 |
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150 # Find the direct term if there is one. |
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151 if(lb>=la) |
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152 # Also returns the reduced numerator. |
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153 [k, b] = deconv(b,a); |
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154 lb = length(b); |
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155 else |
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156 k = []; |
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157 endif |
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158 |
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159 if(lp == 1) |
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160 r = polyval(b,p); |
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161 e = 1; |
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162 return; |
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163 endif |
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164 |
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165 |
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166 # We need to determine the number and multiplicity of the roots. |
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167 # D is the number of distinct roots. |
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168 # M is a vector of length D containing the multiplicity of each root. |
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169 # pr is a vector of length D containing only the distinct roots. |
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170 # e is a vector of length lp which indicates the power in the partial |
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171 # fraction expansion of each term in p. |
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172 |
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173 # Set initial values. We'll remove elements from pr as we find |
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174 # multiplicities. We'll shorten M afterwards. |
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175 e = ones(lp,1); |
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176 M = zeros(lp,1); |
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177 pr = p; |
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178 D = 1; M(1) = 1; |
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179 |
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180 old_p_index = 1; new_p_index = 2; |
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181 M_index = 1; pr_index = 2; |
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182 while(new_p_index<=lp) |
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183 if(abs(p(new_p_index) - p(old_p_index)) < toler) |
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184 # We've found a multiple pole. |
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185 M(M_index) = M(M_index) + 1; |
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186 e(new_p_index) = e(new_p_index-1) + 1; |
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187 # Remove the pole from pr. |
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188 pr(pr_index) = []; |
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189 else |
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190 # It's a different pole. |
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191 D++; M_index++; M(M_index) = 1; |
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192 old_p_index = new_p_index; pr_index++; |
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193 endif |
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194 new_p_index++; |
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195 endwhile |
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196 |
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197 # Shorten M to it's proper length |
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198 M = M(1:D); |
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199 |
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200 # Now set up the polynomial matrices. |
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201 |
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202 MM = max(M); |
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203 # Left hand side polynomial |
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204 lhs = zeros(MM,lb); |
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205 rhs = zeros(MM,lp*lp); |
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206 lhs(1,:) = b; |
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207 rhi = 1; |
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208 dpi = 1; |
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209 mpi = 1; |
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210 while(dpi<=D) |
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211 for ind = 1:M(dpi) |
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212 if(mpi>1 && (mpi+ind)<=lp) |
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213 cp = [p(1:mpi-1); p(mpi+ind:lp)]; |
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214 elseif (mpi==1) |
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215 cp = p(mpi+ind:lp); |
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216 else |
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217 cp = p(1:mpi-1); |
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218 endif |
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219 rhs(1,rhi:rhi+lp-1) = prepad(poly(cp),lp); |
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220 rhi = rhi + lp; |
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221 endfor |
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222 mpi = mpi + M(dpi); |
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223 dpi++; |
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224 endwhile |
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225 if(MM > 1) |
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226 for index = 2:MM |
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227 lhs(index,:) = prepad(polyderiv(lhs(index-1,:)),lb); |
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228 ind = 1; |
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229 for rhi = 1:lp |
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230 cp = rhs(index-1,ind:ind+lp-1); |
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231 rhs(index,ind:ind+lp-1) = prepad(polyderiv(cp),lp); |
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232 ind = ind + lp; |
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233 endfor |
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234 endfor |
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235 endif |
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236 |
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237 # Now lhs contains the numerator polynomial and as many derivatives as are |
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238 # required. rhs is a matrix of polynomials, the first row contains the |
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239 # corresponding polynomial for each residue and successive rows are |
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240 # derivatives. |
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241 |
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242 # Now we need to evaluate the first row of lhs and rhs at each distinct |
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243 # pole value. If there are multiple poles we will also need to evaluate |
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244 # the derivatives at the pole value also. |
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245 |
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246 B = zeros(lp,1); |
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247 A = zeros(lp,lp); |
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248 |
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249 dpi = 1; |
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250 row = 1; |
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251 while(dpi<=D) |
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252 for mi = 1:M(dpi) |
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253 B(row) = polyval(lhs(mi,:),pr(dpi)); |
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254 ci = 1; |
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255 for col = 1:lp |
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256 cp = rhs(mi,ci:ci+lp-1); |
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257 A(row,col) = polyval(cp,pr(dpi)); |
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258 ci = ci + lp; |
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259 endfor |
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260 row++; |
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261 endfor |
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262 dpi++; |
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263 endwhile |
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264 |
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265 # Solve for the residues. |
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266 r = A\B; |
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267 |
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268 endfunction |
