1 | version="$Id: control.lib,v 1.31 2005-05-06 14:38:12 hannes Exp $"; |
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2 | category="System and Control Theory"; |
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3 | info=" |
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4 | LIBRARY: control.lib Algebraic analysis tools for System and Control Theory |
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5 | |
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6 | AUTHORS: Oleksandr Iena yena@mathematik.uni-kl.de |
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7 | @* Markus Becker mbecker@mathematik.uni-kl.de |
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8 | @* Viktor Levandovskyy levandov@mathematik.uni-kl.de |
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9 | |
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10 | SUPPORT: Forschungsschwerpunkt 'Mathematik und Praxis' (Project of Dr. E. Zerz |
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11 | and V. Levandovskyy), Uni Kaiserslautern |
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12 | |
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13 | MAIN PROCEDURES: |
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14 | control(R); analysis of controllability-related properties of R (using Ext modules) |
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15 | control2(R); analysis of controllability-related properties of R (using dimension) |
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16 | autonom(R); analysis of autonomy-related properties of R (using Ext modules) |
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17 | autonom2(R); analysis of autonomy-related properties of R (using dimension) |
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18 | |
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19 | COMPONENT PROCEDURES: |
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20 | LeftKernel(R); a left kernel of R |
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21 | RightKernel(R); a right kernel of R |
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22 | LeftInverse(R); a left inverse of R |
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23 | RightInverse(R); a right inverse of R |
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24 | smith(M); a Smith form of a module M |
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25 | colrank(M); a column rank of M as of matrix |
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26 | genericity(M); analysis of the genericity of parameters |
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27 | canonize(L); Groebnerification for modules in the output of control or autonomy procs |
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28 | iostruct(R); computes an IO-structure of behavior given by a module R |
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29 | FindTorsion(R, I); generators of the submodule of a module R, annihilated by the ideal I |
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30 | |
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31 | AUXILIARY PROCEDURES: |
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32 | ControlExample(s); set up an example from the mini database inside of the library |
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33 | declare(N,V,P,O); defines the ring easily |
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34 | view(); well-formatted output of lists, modules and matrices |
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35 | "; |
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36 | |
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37 | LIB "homolog.lib"; |
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38 | LIB "poly.lib"; |
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39 | LIB "primdec.lib"; |
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40 | LIB "matrix.lib"; |
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41 | |
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42 | //--------------------------------------------------------------- |
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43 | static proc Opt_Our() |
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44 | "USAGE: Opt_Our(); |
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45 | RETURN: intvec, where previous options are stored |
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46 | PURPOSE: save previous options and set customized options |
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47 | " |
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48 | { |
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49 | intvec v; |
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50 | v=option(get); |
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51 | option(redSB); |
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52 | option(redTail); |
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53 | return (v); |
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54 | } |
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55 | |
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56 | //------------------------------------------------------------------------- |
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57 | |
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58 | static proc space(int n) |
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59 | "USAGE:spase(n); n is an integer (number of needed spaces) |
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60 | RETURN: string consisting of n spaces |
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61 | NOTE: the procedure is used in the procedure 'view' to have a better formatted output |
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62 | "{ |
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63 | int i; |
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64 | string s=""; |
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65 | for(i=1;i<=n;i++) |
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66 | { |
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67 | s=s+" "; |
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68 | }; |
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69 | return(s); |
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70 | }; |
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71 | //----------------------------------------------------------------------------- |
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72 | proc view(M) |
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73 | "USAGE: view(M); M is of any type |
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74 | RETURN: no return value |
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75 | PURPOSE: procedure for (well-) formatted output of modules, matrices, lists of modules, matrices; shows everything even if entries are long |
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76 | NOTE: in case of other types( not 'module', 'matrix', 'list') works just as standard 'print' procedure |
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77 | EXAMPLE: example view; shows an example |
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78 | "{ |
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79 | // to be replaced with something more feasible |
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80 | if ( (typeof(M)=="module")||(typeof(M)=="matrix") ) |
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81 | { |
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82 | int @R=nrows(M); |
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83 | int @C=ncols(M); |
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84 | int i; |
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85 | int j; |
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86 | list MaxLength=list(); |
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87 | int Size=0; |
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88 | int max; |
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89 | string s; |
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90 | |
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91 | for(i=1;i<=@C;i++) |
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92 | { |
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93 | max=0; |
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94 | |
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95 | for(j=1;j<=@R;j++) |
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96 | { |
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97 | Size=size( string( M[j,i] ) ); |
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98 | if( Size>max ) |
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99 | { |
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100 | max=Size; |
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101 | }; |
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102 | }; |
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103 | MaxLength[i] = max; |
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104 | }; |
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105 | |
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106 | for(i=1;i<=@R;i++) |
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107 | { |
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108 | s=""; |
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109 | for(j=1;j<@C;j++) |
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110 | { |
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111 | s=s+string(M[i,j])+space( MaxLength[j]-size( string( M[i,j] ) ) ) +","; |
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112 | }; |
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113 | |
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114 | s=s+string(M[i,j])+space( MaxLength[j]-size( string( M[i,j] ) ) ); |
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115 | |
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116 | if (i!=@R) |
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117 | { |
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118 | s=s+","; |
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119 | }; |
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120 | print(s); |
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121 | }; |
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122 | |
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123 | return(); |
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124 | }; |
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125 | |
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126 | if(typeof(M)=="list") |
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127 | { |
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128 | int sz=size(M); |
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129 | int i; |
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130 | for(i=1;i<=sz;i++) |
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131 | { |
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132 | view(M[i]); |
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133 | print(""); |
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134 | }; |
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135 | |
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136 | return(); |
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137 | }; |
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138 | print(M); |
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139 | return(); |
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140 | } |
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141 | example |
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142 | {"EXAMPLE:";echo = 2; |
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143 | ring r; |
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144 | matrix M[1][3] = x,y,z; |
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145 | print(M); |
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146 | view(M); |
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147 | }; |
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148 | //-------------------------------------------------------------------------- |
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149 | proc RightKernel(matrix M) |
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150 | "USAGE: RightKernel(M); M a matrix |
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151 | PURPOSE: computes the right kernel of matrix M (a module of all elements v such that Mv=0) |
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152 | RETURN: module |
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153 | EXAMPLE: example RightKernel; shows an example |
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154 | "{ |
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155 | return(modulo(M,std(0))); |
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156 | } |
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157 | example |
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158 | {"EXAMPLE:";echo = 2; |
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159 | ring r = 0,(x,y,z),dp; |
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160 | matrix M[1][3] = x,y,z; |
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161 | print(M); |
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162 | matrix R = RightKernel(M); |
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163 | print(R); |
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164 | // check: |
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165 | print(M*R); |
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166 | }; |
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167 | //------------------------------------------------------------------------- |
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168 | proc LeftKernel(matrix M) |
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169 | "USAGE: LeftKernel(M); M a matrix |
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170 | PURPOSE: computes left kernel of matrix M (a module of all elements v such that vM=0) |
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171 | RETURN: module |
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172 | EXAMPLE: example LeftKernel; shows an example |
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173 | " |
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174 | { |
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175 | return( transpose( modulo( transpose(M),std(0) ) ) ); |
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176 | } |
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177 | example |
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178 | {"EXAMPLE:";echo = 2; |
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179 | ring r= 0,(x,y,z),dp; |
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180 | matrix M[3][1] = x,y,z; |
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181 | print(M); |
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182 | matrix L = LeftKernel(M); |
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183 | print(L); |
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184 | // check: |
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185 | print(L*M); |
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186 | }; |
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187 | //------------------------------------------------------------------------ |
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188 | proc LeftInverse(module M) |
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189 | "USAGE: LeftInverse(M); M a module |
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190 | PURPOSE: computes such a matrix L, that LM == Id; |
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191 | RETURN: module |
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192 | EXAMPLE: example LeftInverse; shows an example |
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193 | NOTE: exists only in the case when Id belongs to M! |
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194 | " |
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195 | { |
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196 | // it works also for the NC case; |
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197 | int NCols = ncols(M); |
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198 | module Id = freemodule(NCols); |
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199 | module N = transpose(M); |
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200 | intvec old_opt=Opt_Our(); |
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201 | Id = std(Id); |
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202 | matrix T; |
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203 | // check the correctness (Id \subseteq M) |
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204 | // via dimension: dim (M) = -1! |
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205 | int d = dim_Our(N); |
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206 | if (d != -1) |
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207 | { |
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208 | // No left inverse exists |
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209 | return(matrix(0)); |
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210 | } |
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211 | matrix T2 = lift(N, Id); |
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212 | T2 = transpose(T2); |
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213 | option(set,old_opt); // set the options back |
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214 | return(T2); |
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215 | } |
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216 | example |
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217 | { |
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218 | "EXAMPLE:";echo =2; |
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219 | // a trivial example: |
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220 | ring r = 0,(x,z),dp; |
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221 | matrix M[2][1] = 1,x2z; |
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222 | print(M); |
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223 | print( LeftInverse(M) ); |
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224 | kill r; |
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225 | // derived from the example TwoPendula: |
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226 | ring r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
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227 | matrix U[3][1]; |
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228 | U[1,1]=(-L2)*Dt^4+(g)*Dt^2; |
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229 | U[2,1]=(-L1)*Dt^4+(g)*Dt^2; |
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230 | U[3,1]=(L1*L2)*Dt^4+(-g*L1-g*L2)*Dt^2+(g^2); |
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231 | module M = module(U); |
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232 | module L = LeftInverse(M); |
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233 | print(L); |
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234 | // check |
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235 | print(L*M); |
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236 | }; |
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237 | //----------------------------------------------------------------------- |
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238 | proc RightInverse(module R) |
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239 | "USAGE: RightInverse(M); M a module |
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240 | PURPOSE: computes such a matrix L, that ML == Id; |
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241 | RETURN: module |
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242 | EXAMPLE: example RightInverse; shows an example |
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243 | NOTE: exists only in the case when Id belongs to M! |
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244 | " |
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245 | { |
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246 | return(transpose(LeftInverse(transpose(R)))); |
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247 | } |
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248 | example |
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249 | { "EXAMPLE:";echo =2; |
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250 | // a trivial example: |
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251 | ring r = 0,(x,z),dp; |
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252 | matrix M[1][2] = 1,x2+z; |
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253 | print(M); |
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254 | print( RightInverse(M) ); |
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255 | kill r; |
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256 | // derived from the TwoPendula example: |
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257 | ring r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
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258 | matrix U[1][3]; |
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259 | U[1,1]=(-L2)*Dt^4+(g)*Dt^2; |
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260 | U[1,2]=(-L1)*Dt^4+(g)*Dt^2; |
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261 | U[1,3]=(L1*L2)*Dt^4+(-g*L1-g*L2)*Dt^2+(g^2); |
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262 | module M = module(U); |
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263 | module L = RightInverse(M); |
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264 | print(L); |
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265 | // check |
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266 | print(M*L); |
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267 | }; |
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268 | //----------------------------------------------------------------------- |
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269 | static proc dim_Our(module R) |
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270 | { |
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271 | int d; |
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272 | if (attrib(R,"isSB")<>1) |
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273 | { |
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274 | R = std(R); |
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275 | } |
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276 | d = dim(R); |
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277 | return(d); |
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278 | } |
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279 | //----------------------------------------------------------------------- |
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280 | static proc Ann_Our(module R) |
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281 | { |
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282 | return(Ann(R)); |
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283 | } |
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284 | //----------------------------------------------------------------------- |
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285 | static proc Ext_Our(int i, module R, list #) |
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286 | { |
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287 | // mimicking 'Ext_R' from homolog.lib |
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288 | int ExtraArg = ( size(#)>0 ); |
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289 | if (ExtraArg) |
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290 | { |
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291 | return( Ext_R(i,R,#[1]) ); |
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292 | } |
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293 | else |
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294 | { |
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295 | return( Ext_R(i,R) ); |
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296 | } |
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297 | } |
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298 | //------------------------------------------------------------------------ |
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299 | static proc is_zero_Our |
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300 | { |
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301 | //just a copy of 'is_zero' from "poly.lib" patched with GKdim |
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302 | int d = dim_Our(std(#[1])); |
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303 | int a = ( d==-1 ); |
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304 | if( size(#) >1 ) { list L=a,d; return(L); } |
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305 | return(a); |
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306 | // return( is_zero(R) ) ; |
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307 | }; |
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308 | //------------------------------------------------------------------------ |
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309 | static proc control_output(int i, int NVars, module R, module Ext_1, list Gen) |
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310 | "USAGE: control_output(i, NVars, R, Ext_1), |
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311 | PURPOSE: where |
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312 | @* i is integer (number of first nonzero Ext or a number of variables in a basering + 1 in case that all the Exts are zero), |
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313 | @* NVars: integer, number of variables in a base ring, |
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314 | @* R: module R (cokernel representation), |
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315 | @* Ext_1: module, the first Ext(its cokernel representation) |
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316 | RETURN: list with all the contollability properties of the system which is to be returned in 'control' procedure |
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317 | NOTE: this procedure is used in 'control' procedure |
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318 | "{ |
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319 | // TODO: NVars to be replaced with the global hom. dimension of basering!!! |
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320 | // Is not clear what to do with gl.dim of qrings |
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321 | string DofS = "dimension of the system:"; |
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322 | string Fn = "number of first nonzero Ext:"; |
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323 | string Gen_mes = "Parameter constellations which might lead to a non-controllable system:"; |
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324 | |
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325 | module RK = RightKernel(R); |
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326 | int d=dim_Our(std(transpose(R))); |
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327 | |
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328 | if (i==1) |
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329 | { |
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330 | return( |
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331 | list ( Fn, |
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332 | i, |
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333 | "not controllable , image representation for controllable part:", |
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334 | RK, |
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335 | "kernel representation for controllable part:", |
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336 | LeftKernel( RK ), |
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337 | "obstruction to controllability", |
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338 | Ext_1, |
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339 | "annihilator of torsion module (of obstruction to controllability)", |
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340 | Ann_Our(Ext_1), |
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341 | DofS, |
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342 | d |
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343 | ) |
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344 | ); |
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345 | }; |
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346 | |
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347 | if(i>NVars) |
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348 | { |
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349 | return( list( Fn, |
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350 | -1, |
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351 | "strongly controllable(flat), image representation:", |
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352 | RK, |
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353 | "left inverse to image representation:", |
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354 | LeftInverse(RK), |
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355 | DofS, |
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356 | d, |
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357 | Gen_mes, |
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358 | Gen) |
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359 | ); |
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360 | }; |
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361 | |
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362 | // |
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363 | //now i<=NVars |
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364 | // |
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365 | |
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366 | if( (i==2) ) |
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367 | { |
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368 | return( list( Fn, |
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369 | i, |
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370 | "controllable, not reflexive, image representation:", |
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371 | RK, |
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372 | DofS, |
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373 | d, |
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374 | Gen_mes, |
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375 | Gen) |
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376 | ); |
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377 | }; |
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378 | |
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379 | if( (i>=3) ) |
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380 | { |
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381 | return( list ( Fn, |
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382 | i, |
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383 | "reflexive, not strongly controllable, image representation:", |
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384 | RK, |
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385 | DofS, |
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386 | d, |
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387 | Gen_mes, |
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388 | Gen) |
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389 | ); |
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390 | }; |
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391 | }; |
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392 | //------------------------------------------------------------------------- |
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393 | |
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394 | proc control(module R) |
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395 | "USAGE: control(R); R a module (R is the matrix of the system of equations to be investigated) |
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396 | PURPOSE: compute the list of all the properties concerning controllability of the system (behavior), represented by the matrix R |
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397 | RETURN: list |
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398 | EXAMPLE: example control; shows an example |
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399 | " |
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400 | { |
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401 | int i; |
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402 | int NVars=nvars(basering); |
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403 | // TODO: NVars to be replaced with the global hom. dimension of basering!!! |
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404 | int ExtIsZero; |
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405 | intvec v=Opt_Our(); |
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406 | module R_std=std(R); |
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407 | module Ext_1 = std(Ext_Our(1,R_std)); |
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408 | |
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409 | ExtIsZero=is_zero_Our(Ext_1); |
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410 | i=1; |
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411 | while( (ExtIsZero) && (i<=NVars) ) |
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412 | { |
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413 | i++; |
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414 | ExtIsZero = is_zero_Our( Ext_Our(i,R_std) ); |
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415 | }; |
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416 | matrix T=lift(R,R_std); |
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417 | list l=genericity(T); |
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418 | option(set,v); |
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419 | |
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420 | return( control_output( i, NVars, R, Ext_1, l ) ); |
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421 | } |
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422 | example |
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423 | {"EXAMPLE:";echo = 2; |
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424 | // a WindTunnel example |
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425 | ring A = (0,a, omega, zeta, k),(D1, delta),dp; |
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426 | module R; |
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427 | R = [D1+a, -k*a*delta, 0, 0], |
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428 | [0, D1, -1, 0], |
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429 | [0, omega^2, D1+2*zeta*omega, -omega^2]; |
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430 | R=transpose(R); |
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431 | view(R); |
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432 | view(control(R)); |
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433 | }; |
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434 | //-------------------------------------------------------------------------- |
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435 | proc control2(module R) |
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436 | "USAGE: control2(R); R a module (R is the matrix of the system of equations to be investigated) |
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437 | PURPOSE: computes list of all the properties concerning controllability of the system (behavior), represented by the matrix R |
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438 | RETURN: list |
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439 | EXAMPLE: example control2; shows an example |
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440 | NOTE: this procedure is analogous to 'control' but uses dimension calculations.This approach works for full row rank matrices only. |
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441 | " |
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442 | { |
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443 | if( nrows(R) != colrank(transpose(R)) ) |
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444 | { |
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445 | return ("control2 cannot be applied, since R does not have full row rank"); |
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446 | } |
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447 | intvec v=Opt_Our(); |
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448 | module R_std=std(R); |
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449 | int d=dim_Our(R_std); |
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450 | int NVars=nvars(basering); |
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451 | int i=NVars-d; |
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452 | module Ext_1=std(Ext_Our(1,R_std)); |
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453 | matrix T=lift(R,R_std); |
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454 | list l=genericity(T); |
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455 | option(set, v); |
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456 | return( control_output( i, NVars, R, Ext_1, l)); |
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457 | } |
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458 | example |
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459 | {"EXAMPLE:";echo = 2; |
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460 | //a WindTunnel example |
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461 | ring A = (0,a, omega, zeta, k),(D1, delta),dp; |
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462 | module R; |
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463 | R = [D1+a, -k*a*delta, 0, 0], |
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464 | [0, D1, -1, 0], |
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465 | [0, omega^2, D1+2*zeta*omega, -omega^2]; |
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466 | R=transpose(R); |
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467 | view(R); |
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468 | view(control2(R)); |
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469 | }; |
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470 | //------------------------------------------------------------------------ |
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471 | proc colrank(module M) |
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472 | "USAGE: proc colrank(M), M a matrix/module |
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473 | PURPOSE: compute the column rank of M as of matrix |
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474 | RETURN: int |
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475 | NOTE: this procedure uses bareiss-algorithm which might not terminate in some cases |
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476 | " |
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477 | { |
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478 | module M_red = bareiss(M)[1]; |
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479 | int NCols_red = ncols(M_red); |
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480 | return (NCols_red); |
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481 | } |
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482 | example |
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483 | {"EXAMPLE: ";echo = 2; |
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484 | // de Rham complex |
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485 | ring r=0,(D(1..3)),dp; |
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486 | module R; |
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487 | R=[0,-D(3),D(2)], |
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488 | [D(3),0,-D(1)], |
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489 | [-D(2),D(1),0]; |
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490 | R=transpose(R); |
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491 | colrank(R); |
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492 | }; |
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493 | |
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494 | //------------------------------------------------------------------------ |
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495 | static proc autonom_output( int i, int NVars, module RC, int R_rank ) |
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496 | "USAGE: proc autonom_output(i, NVars, RC, R_rank) |
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497 | i: integer, number of first nonzero Ext or |
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498 | just number of variables in a base ring + 1 in case that all the Exts are zero |
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499 | NVars: integer, number of variables in a base ring |
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500 | RC: module, kernel-representation of controllable part of the system |
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501 | R_rank: integer, column rank of the representation matrix |
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502 | PURPOSE: compute all the autonomy properties of the system which is to be returned in 'autonom' procedure |
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503 | RETURN: list |
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504 | NOTE: this procedure is used in 'autonom' procedure |
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505 | " |
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506 | { |
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507 | int d=NVars-i;//that is the dimension of the system |
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508 | string DofS="dimension of the system:"; |
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509 | string Fn = "number of first nonzero Ext:"; |
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510 | if(i==0) |
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511 | { |
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512 | return( list( Fn, |
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513 | i, |
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514 | "not autonomous", |
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515 | "kernel representation for controllable part", |
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516 | RC, |
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517 | "column rank of the matrix", |
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518 | R_rank, |
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519 | DofS, |
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520 | d ) |
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521 | ); |
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522 | }; |
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523 | |
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524 | if( i>NVars ) |
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525 | { |
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526 | return( list( Fn, |
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527 | -1, |
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528 | "trivial", |
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529 | DofS, |
---|
530 | d ) |
---|
531 | ); |
---|
532 | }; |
---|
533 | |
---|
534 | // |
---|
535 | //now i<=NVars |
---|
536 | // |
---|
537 | |
---|
538 | |
---|
539 | if( i==1 ) |
---|
540 | // in case that NVars==1 there is no sense to consider the notion |
---|
541 | // of strongly autonomous behavior, because it does not imply |
---|
542 | // that system is overdetermined in this case |
---|
543 | { |
---|
544 | return( list ( Fn, |
---|
545 | i, |
---|
546 | "autonomous, not overdetermined", |
---|
547 | DofS, |
---|
548 | d ) |
---|
549 | ); |
---|
550 | }; |
---|
551 | |
---|
552 | if( i==NVars ) |
---|
553 | { |
---|
554 | return( list( Fn, |
---|
555 | i, |
---|
556 | "strongly autonomous(fin. dimensional),in particular overdetermined", |
---|
557 | DofS, |
---|
558 | d) |
---|
559 | ); |
---|
560 | }; |
---|
561 | |
---|
562 | if( i<NVars ) |
---|
563 | { |
---|
564 | return( list ( Fn, |
---|
565 | i, |
---|
566 | "overdetermined, not strongly autonomous", |
---|
567 | DofS, |
---|
568 | d) |
---|
569 | ); |
---|
570 | }; |
---|
571 | }; |
---|
572 | //-------------------------------------------------------------------------- |
---|
573 | proc autonom2(module R) |
---|
574 | "USAGE: autonom2(R); R a module (R is a matrix of the system of equations which is to be investigated) |
---|
575 | PURPOSE: computes the list of all the properties concerning autonomy of the system (behavior), represented by the matrix R |
---|
576 | RETURN: list |
---|
577 | NOTE: this procedure is analogous to 'autonom' but uses dimension calculations |
---|
578 | EXAMPLE: example autonom2; shows an example |
---|
579 | " |
---|
580 | { |
---|
581 | int d; |
---|
582 | int NVars = nvars(basering); |
---|
583 | module RT = transpose(R); |
---|
584 | module RC; //for computation of controllable part if if exists |
---|
585 | int R_rank = ncols(R); |
---|
586 | d = dim_Our( std(RT) ); //this is the dimension of the system |
---|
587 | int i = NVars-d; //First non-zero Ext |
---|
588 | if( d==0 ) |
---|
589 | { |
---|
590 | RC=LeftKernel(RightKernel(R)); |
---|
591 | R_rank=colrank(R); |
---|
592 | } |
---|
593 | return( autonom_output(i,NVars,RC,R_rank) ); |
---|
594 | } |
---|
595 | example |
---|
596 | {"EXAMPLE:"; echo = 2; |
---|
597 | // Cauchy1 example |
---|
598 | ring r=0,(s1,s2,s3,s4),dp; |
---|
599 | module R= [s1,-s2], |
---|
600 | [s2, s1], |
---|
601 | [s3,-s4], |
---|
602 | [s4, s3]; |
---|
603 | R=transpose(R); |
---|
604 | view( R ); |
---|
605 | view( autonom2(R) ); |
---|
606 | }; |
---|
607 | //---------------------------------------------------------- |
---|
608 | proc autonom(module R) |
---|
609 | "USAGE: autonom(R); R a module (R is a matrix of the system of equations which is to be investigated) |
---|
610 | PURPOSE: find all the properties concerning autonomy of the system (behavior) represented by the matrix R |
---|
611 | RETURN: list |
---|
612 | EXAMPLE: example autonom; shows an example |
---|
613 | " |
---|
614 | { |
---|
615 | int NVars=nvars(basering); |
---|
616 | int ExtIsZero; |
---|
617 | module RT=transpose(R); |
---|
618 | module RC; |
---|
619 | int R_rank=ncols(R); |
---|
620 | ExtIsZero=is_zero_Our(Ext_Our(0,RT)); |
---|
621 | int i=0; |
---|
622 | while( (ExtIsZero)&&(i<=NVars) ) |
---|
623 | { |
---|
624 | i++; |
---|
625 | ExtIsZero = is_zero_Our(Ext_Our(i,RT)); |
---|
626 | }; |
---|
627 | if (i==0) |
---|
628 | { |
---|
629 | RC=LeftKernel(RightKernel(R)); |
---|
630 | R_rank=colrank(R); |
---|
631 | } |
---|
632 | return(autonom_output(i,NVars,RC,R_rank)); |
---|
633 | } |
---|
634 | example |
---|
635 | {"EXAMPLE:"; echo = 2; |
---|
636 | // Cauchy |
---|
637 | ring r=0,(s1,s2,s3,s4),dp; |
---|
638 | module R= [s1,-s2], |
---|
639 | [s2, s1], |
---|
640 | [s3,-s4], |
---|
641 | [s4, s3]; |
---|
642 | R=transpose(R); |
---|
643 | view( R ); |
---|
644 | view( autonom(R) ); |
---|
645 | }; |
---|
646 | |
---|
647 | |
---|
648 | //---------------------------------------------------------- |
---|
649 | proc genericity(matrix M) |
---|
650 | "USAGE: genericity(M), M is a matrix/module |
---|
651 | PURPOSE: determine parametric expressions which have been assumed to be non-zero in the process of computing the Groebner basis |
---|
652 | RETURN: list (of strings) |
---|
653 | NOTE: we strongly recommend to switch on the redSB and redTail options; |
---|
654 | @* the procedure is effective with the lift procedure for modules with parameters |
---|
655 | EXAMPLE: example genericity; shows an example |
---|
656 | " |
---|
657 | { |
---|
658 | // returns "-", if there are no parameters! |
---|
659 | if (npars(basering)==0) |
---|
660 | { |
---|
661 | return("-"); |
---|
662 | } |
---|
663 | list RT = evas_genericity(M); // list of strings |
---|
664 | if ((size(RT)==1) && (RT[1] == "")) |
---|
665 | { |
---|
666 | return("-"); |
---|
667 | } |
---|
668 | return(RT); |
---|
669 | } |
---|
670 | example |
---|
671 | { // TwoPendula |
---|
672 | "EXAMPLE:"; echo = 2; |
---|
673 | ring r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
---|
674 | module RR = |
---|
675 | [m1*L1*Dt^2, m2*L2*Dt^2, -1, (M+m1+m2)*Dt^2], |
---|
676 | [m1*L1^2*Dt^2-m1*L1*g, 0, 0, m1*L1*Dt^2], |
---|
677 | [0, m2*L2^2*Dt^2-m2*L2*g, 0, m2*L2*Dt^2]; |
---|
678 | module R = transpose(RR); |
---|
679 | module SR = std(R); |
---|
680 | matrix T = lift(R,SR); |
---|
681 | genericity(T); |
---|
682 | //-- The result might be different when computing reduced bases: |
---|
683 | matrix T2; |
---|
684 | option(redSB); |
---|
685 | option(redTail); |
---|
686 | module SR2 = std(R); |
---|
687 | T2 = lift(R,SR2); |
---|
688 | genericity(T2); |
---|
689 | } |
---|
690 | //--------------------------------------------------------------- |
---|
691 | static proc victors_genericity(matrix M) |
---|
692 | { |
---|
693 | // returns "-", if there are no parameters! |
---|
694 | if (npars(basering)==0) |
---|
695 | { |
---|
696 | return("-"); |
---|
697 | } |
---|
698 | int plevel = printlevel-voice+2; |
---|
699 | // M is a matrix over a ring with params and vars; |
---|
700 | ideal I = ideal(M); // a list of entries |
---|
701 | I = simplify(I,2); // delete 0's |
---|
702 | // decompose every coeff in every poly |
---|
703 | int i; |
---|
704 | int s = size(I); |
---|
705 | ideal NM; |
---|
706 | poly p; |
---|
707 | number num; |
---|
708 | int cl=1; |
---|
709 | intvec ZeroVec; ZeroVec[nvars(basering)] = 0; |
---|
710 | intvec W; |
---|
711 | ideal Numero, Denomiro; |
---|
712 | int cNu=0; int cDe=0; |
---|
713 | for (i=1; i<=s; i++) |
---|
714 | { |
---|
715 | // remove contents and add them as polys |
---|
716 | p = I[i]; |
---|
717 | W = leadexp(p); |
---|
718 | if (W == ZeroVec) // i.e. just a coef |
---|
719 | { |
---|
720 | num = denominator(leadcoef(p)); // from poly.lib |
---|
721 | NM[cl] = numerator(leadcoef(p)); |
---|
722 | dbprint(p,"numerator:"); |
---|
723 | dbprint(p, string(NM[cl])); |
---|
724 | cNu++; Numero[cNu]= NM[cl]; |
---|
725 | cl++; |
---|
726 | NM[cl] = num; // denominator |
---|
727 | dbprint(p,"denominator:"); |
---|
728 | dbprint(p, string(NM[cl])); |
---|
729 | cDe++; Denomiro[cDe]= NM[cl]; |
---|
730 | cl++; |
---|
731 | p = p - lead(p); // for the next cycle |
---|
732 | } |
---|
733 | if ( p!= 0) |
---|
734 | { |
---|
735 | num = content(p); |
---|
736 | p = p/num; |
---|
737 | NM[cl] = denominator(num); |
---|
738 | dbprint(p,"content denominator:"); |
---|
739 | dbprint(p, string(NM[cl])); |
---|
740 | cNu++; Numero[cNu]= NM[cl]; |
---|
741 | cl++; |
---|
742 | NM[cl] = numerator(num); |
---|
743 | dbprint(p,"content numerator:"); |
---|
744 | dbprint(p, string(NM[cl])); |
---|
745 | cDe++; Denomiro[cDe]= NM[cl]; |
---|
746 | cl++; |
---|
747 | } |
---|
748 | // it seems that the next elements will not have real influence |
---|
749 | while( p != 0) |
---|
750 | { |
---|
751 | NM[cl] = leadcoef(p); // should be all integer, i.e. non-rational |
---|
752 | dbprint(p,"coef:"); |
---|
753 | dbprint(p, string(NM[cl])); |
---|
754 | cl++; |
---|
755 | p = p - lead(p); |
---|
756 | } |
---|
757 | } |
---|
758 | NM = simplify(NM,4); // delete identical |
---|
759 | string newvars = parstr(basering); |
---|
760 | def save = basering; |
---|
761 | string NewRing = "ring @NR =" +string(char(basering))+",("+newvars+"),Dp;"; |
---|
762 | execute(NewRing); |
---|
763 | // get params as variables |
---|
764 | // create a list of non-monomials |
---|
765 | ideal @L; |
---|
766 | ideal F; |
---|
767 | ideal NM = imap(save,NM); |
---|
768 | NM = simplify(NM,8); //delete multiples |
---|
769 | poly p,q; |
---|
770 | cl = 1; |
---|
771 | int j, cf; |
---|
772 | for(i=1; i<=size(NM);i++) |
---|
773 | { |
---|
774 | p = NM[i] - lead(NM[i]); |
---|
775 | if (p!=0) |
---|
776 | { |
---|
777 | // L[cl] = p; |
---|
778 | F = factorize(NM[i],1); //non-constant factors only |
---|
779 | cf = 1; |
---|
780 | // factorize every polynomial |
---|
781 | // throw away every monomial from factorization (also constants from above ring) |
---|
782 | for (j=1; j<=size(F);j++) |
---|
783 | { |
---|
784 | q = F[j]-lead(F[j]); |
---|
785 | if (q!=0) |
---|
786 | { |
---|
787 | @L[cl] = F[j]; |
---|
788 | cl++; |
---|
789 | } |
---|
790 | } |
---|
791 | } |
---|
792 | } |
---|
793 | // return the result [in string-format] |
---|
794 | @L = simplify(@L,2+4+8); // skip zeroes, doubled and entries, diff. by a constant |
---|
795 | list SL; |
---|
796 | for (j=1; j<=size(@L);j++) |
---|
797 | { |
---|
798 | SL[j] = string(@L[j]); |
---|
799 | } |
---|
800 | setring save; |
---|
801 | return(SL); |
---|
802 | } |
---|
803 | //--------------------------------------------------------------- |
---|
804 | static proc evas_genericity(matrix M) |
---|
805 | { |
---|
806 | // called from the main genericity proc |
---|
807 | ideal I = ideal(M); |
---|
808 | I = simplify(I,2+4); |
---|
809 | int s = size(I); |
---|
810 | ideal Den; |
---|
811 | poly p; |
---|
812 | int i; |
---|
813 | for (i=1; i<=s; i++) |
---|
814 | { |
---|
815 | p = I[i]; |
---|
816 | while (p !=0) |
---|
817 | { |
---|
818 | Den = Den, denominator(leadcoef(p)); |
---|
819 | p = p-lead(p); |
---|
820 | } |
---|
821 | } |
---|
822 | Den = simplify(Den,2+4); |
---|
823 | string newvars = parstr(basering); |
---|
824 | def save = basering; |
---|
825 | string NewRing = "ring @NR =" +string(char(basering))+",("+newvars+"),Dp;"; |
---|
826 | execute(NewRing); |
---|
827 | ideal F; |
---|
828 | ideal Den = imap(save,Den); |
---|
829 | Den = simplify(Den,2); |
---|
830 | int s1 = size(Den); |
---|
831 | for (i=1; i<=s1; i++) |
---|
832 | { |
---|
833 | if (Den[i] !=1) |
---|
834 | { |
---|
835 | F= F, factorize(Den[i],1); |
---|
836 | } |
---|
837 | } |
---|
838 | F = simplify(F, 2+4+8); |
---|
839 | ideal @L = F; |
---|
840 | list SL; |
---|
841 | int c,j; |
---|
842 | string Mono; |
---|
843 | c = 1; |
---|
844 | for (j=1; j<=size(@L);j++) |
---|
845 | { |
---|
846 | if (leadcoef(@L[j]) <0) |
---|
847 | { |
---|
848 | @L[j] = -1*@L[j]; |
---|
849 | } |
---|
850 | if ( (@L[j] - lead(@L[j]))==0 ) //@L[j] is a monomial |
---|
851 | { |
---|
852 | Mono = Mono + string(@L[j])+ ","; // concatenation |
---|
853 | } |
---|
854 | else |
---|
855 | { |
---|
856 | c++; |
---|
857 | SL[c] = string(@L[j]); |
---|
858 | } |
---|
859 | } |
---|
860 | if (Mono!="") |
---|
861 | { |
---|
862 | Mono = Mono[1..size(Mono)-1]; // delete the last semicolon |
---|
863 | } |
---|
864 | SL[1] = Mono; |
---|
865 | setring save; |
---|
866 | return(SL); |
---|
867 | } |
---|
868 | |
---|
869 | //--------------------------------------------------------------- |
---|
870 | proc canonize(list L) |
---|
871 | "USAGE: canonize(L), L a list |
---|
872 | PURPOSE: modules in the list are canonized by computing their reduced minimal (= unique up to constant factor w.r.t. the given ordering) Groebner bases |
---|
873 | RETURN: list |
---|
874 | ASSUME: L is the output of control/autonomy procedures |
---|
875 | EXAMPLE: example canonize; shows an example |
---|
876 | " |
---|
877 | { |
---|
878 | list M = L; |
---|
879 | intvec v=Opt_Our(); |
---|
880 | int s = size(L); |
---|
881 | int i; |
---|
882 | for (i=2; i<=s; i=i+2) |
---|
883 | { |
---|
884 | if (typeof(M[i])=="module") |
---|
885 | { |
---|
886 | M[i] = std(M[i]); |
---|
887 | // M[i] = prune(M[i]); // mimimal embedding: no need yet |
---|
888 | // M[i] = std(M[i]); |
---|
889 | } |
---|
890 | } |
---|
891 | option(set, v); //set old values back |
---|
892 | return(M); |
---|
893 | } |
---|
894 | example |
---|
895 | { // TwoPendula with L1=L2=L |
---|
896 | "EXAMPLE:"; echo = 2; |
---|
897 | ring r=(0,m1,m2,M,g,L),Dt,dp; |
---|
898 | module RR = |
---|
899 | [m1*L*Dt^2, m2*L*Dt^2, -1, (M+m1+m2)*Dt^2], |
---|
900 | [m1*L^2*Dt^2-m1*L*g, 0, 0, m1*L*Dt^2], |
---|
901 | [0, m2*L^2*Dt^2-m2*L*g, 0, m2*L*Dt^2]; |
---|
902 | module R = transpose(RR); |
---|
903 | list C = control(R); |
---|
904 | list CC = canonize(C); |
---|
905 | view(CC); |
---|
906 | } |
---|
907 | |
---|
908 | //---------------------------------------------------------------- |
---|
909 | |
---|
910 | static proc elementof (int i, intvec v) |
---|
911 | { |
---|
912 | int b=0; |
---|
913 | for(int j=1;j<=nrows(v);j++) |
---|
914 | { |
---|
915 | if(v[j]==i) |
---|
916 | { |
---|
917 | b=1; |
---|
918 | return (b); |
---|
919 | } |
---|
920 | } |
---|
921 | return (b); |
---|
922 | } |
---|
923 | //----------------------------------------------------------------- |
---|
924 | proc iostruct(module R) |
---|
925 | "USAGE: iostruct( R ); R a module |
---|
926 | RETURN: list L with entries: string s, intvec v, module P and module Q |
---|
927 | PURPOSE: if R is the kernel-representation-matrix of some system, then we output a input-ouput representation Py=Qu of the system, the components that have been chosen as outputs(intvec v) and a comment s |
---|
928 | NOTE: the procedure uses Bareiss algorithm which might not terminate in some cases |
---|
929 | EXAMPLE: example iostruct; shows an example |
---|
930 | " |
---|
931 | { |
---|
932 | list L = bareiss(R); |
---|
933 | int R_rank = ncols(L[1]); |
---|
934 | int NCols=ncols(R); |
---|
935 | intvec v=L[2]; |
---|
936 | int temp; |
---|
937 | int NRows=nrows(v); |
---|
938 | int i,j; |
---|
939 | int b=1; |
---|
940 | module P; |
---|
941 | module Q; |
---|
942 | int n=0; |
---|
943 | |
---|
944 | while(b==1) //sort v through bubblesort |
---|
945 | { |
---|
946 | b=0; |
---|
947 | for(i=1;i<NRows;i++) |
---|
948 | { |
---|
949 | if(v[i]>v[i+1]) |
---|
950 | { |
---|
951 | temp=v[i]; |
---|
952 | v[i]=v[i+1]; |
---|
953 | v[i+1]=temp; |
---|
954 | b=1; |
---|
955 | } |
---|
956 | } |
---|
957 | } |
---|
958 | P=R[v]; //generate P |
---|
959 | for(i=1;i<=NCols;i++) //generate Q |
---|
960 | { |
---|
961 | if(elementof(i,v)==1) |
---|
962 | { |
---|
963 | i++; |
---|
964 | continue; |
---|
965 | } |
---|
966 | Q=Q,R[i]; |
---|
967 | } |
---|
968 | Q=simplify(Q,2); |
---|
969 | string s="The following components have been chosen as outputs: "; |
---|
970 | return (list(s,v,P,Q)); |
---|
971 | } |
---|
972 | example |
---|
973 | {"EXAMPLE:";echo = 2; |
---|
974 | //Example Antenna |
---|
975 | ring r = (0, K1, K2, Te, Kp, Kc),(Dt, delta), (c,dp); |
---|
976 | module RR; |
---|
977 | RR = |
---|
978 | [Dt, -K1, 0, 0, 0, 0, 0, 0, 0], |
---|
979 | [0, Dt+K2/Te, 0, 0, 0, 0, -Kp/Te*delta, -Kc/Te*delta, -Kc/Te*delta], |
---|
980 | [0, 0, Dt, -K1, 0, 0, 0, 0, 0], |
---|
981 | [0, 0, 0, Dt+K2/Te, 0, 0, -Kc/Te*delta, -Kp/Te*delta, -Kc/Te*delta], |
---|
982 | [0, 0, 0, 0, Dt, -K1, 0, 0, 0], |
---|
983 | [0, 0, 0, 0, 0, Dt+K2/Te, -Kc/Te*delta, -Kc/Te*delta, -Kp/Te*delta]; |
---|
984 | module R = transpose(RR); |
---|
985 | view(iostruct(R)); |
---|
986 | }; |
---|
987 | |
---|
988 | //--------------------------------------------------------------- |
---|
989 | static proc smdeg(matrix N) |
---|
990 | "USAGE: smdeg( N ); N a matrix |
---|
991 | RETURN: intvec |
---|
992 | PURPOSE: returns an intvec of length 2 with the index of an element of N with smallest degree |
---|
993 | " |
---|
994 | { |
---|
995 | int n = nrows(N); |
---|
996 | int m = ncols(N); |
---|
997 | int d,d_temp; |
---|
998 | intvec v; |
---|
999 | int i,j; // counter |
---|
1000 | |
---|
1001 | if (N==0) |
---|
1002 | { |
---|
1003 | v = 1,1; |
---|
1004 | return(v); |
---|
1005 | } |
---|
1006 | |
---|
1007 | for (i=1; i<=n; i++) |
---|
1008 | // hier wird ein Element ausgewaehlt(!=0) und mit dessen Grad gestartet |
---|
1009 | { |
---|
1010 | for (j=1; j<=m; j++) |
---|
1011 | { |
---|
1012 | if( deg(N[i,j])!=-1 ) |
---|
1013 | { |
---|
1014 | d=deg(N[i,j]); |
---|
1015 | break; |
---|
1016 | } |
---|
1017 | } |
---|
1018 | if (d != -1) |
---|
1019 | { |
---|
1020 | break; |
---|
1021 | } |
---|
1022 | } |
---|
1023 | for(i=1; i<=n; i++) |
---|
1024 | { |
---|
1025 | for(j=1; j<=m; j++) |
---|
1026 | { |
---|
1027 | d_temp = deg(N[i,j]); |
---|
1028 | if ( (d_temp < d) && (N[i,j]!=0) ) |
---|
1029 | { |
---|
1030 | d=d_temp; |
---|
1031 | } |
---|
1032 | } |
---|
1033 | } |
---|
1034 | for (i=1; i<=n; i++) |
---|
1035 | { |
---|
1036 | for (j=1; j<=m;j++) |
---|
1037 | { |
---|
1038 | if ( (deg(N[i,j]) == d) && (N[i,j]!=0) ) |
---|
1039 | { |
---|
1040 | v = i,j; |
---|
1041 | return(v); |
---|
1042 | } |
---|
1043 | } |
---|
1044 | } |
---|
1045 | } |
---|
1046 | //--------------------------------------------------------------- |
---|
1047 | static proc NoNon0Pol(vector v) |
---|
1048 | "USAGE: NoNon0Pol(v), v a vector |
---|
1049 | RETURN: int |
---|
1050 | PURPOSE: returns 1, if there is only one non-zero element in v and 0 else |
---|
1051 | "{ |
---|
1052 | int i,j; |
---|
1053 | int n = nrows(v); |
---|
1054 | for( j=1; j<=n; j++) |
---|
1055 | { |
---|
1056 | if (v[j] != 0) |
---|
1057 | { |
---|
1058 | i++; |
---|
1059 | } |
---|
1060 | } |
---|
1061 | if ( i!=1 ) |
---|
1062 | { |
---|
1063 | i=0; |
---|
1064 | } |
---|
1065 | return(i); |
---|
1066 | } |
---|
1067 | //--------------------------------------------------------------- |
---|
1068 | static proc extgcd_Our(poly p, poly q) |
---|
1069 | { |
---|
1070 | ideal J; //for extgcd-computations |
---|
1071 | matrix T; //----------"------------ |
---|
1072 | list L; |
---|
1073 | // the extgcd-command has a bug in versions before 2-0-7 |
---|
1074 | if ( system("version")<=2006 ) |
---|
1075 | { |
---|
1076 | J = p,q; // J = N[k-1,k-1],N[k,k]; //J is of type ideal |
---|
1077 | L[1] = liftstd(J,T); //T is of type matrix |
---|
1078 | if(J[1]==p) //this is just for the case the SINGULAR swaps the |
---|
1079 | // two elements due to ordering |
---|
1080 | { |
---|
1081 | L[2] = T[1,1]; |
---|
1082 | L[3] = T[2,1]; |
---|
1083 | } |
---|
1084 | else |
---|
1085 | { |
---|
1086 | L[2] = T[2,1]; |
---|
1087 | L[3] = T[1,1]; |
---|
1088 | } |
---|
1089 | } |
---|
1090 | else |
---|
1091 | { |
---|
1092 | L=extgcd(p,q); |
---|
1093 | // L=extgcd(N[k-1,k-1],N[k,k]); |
---|
1094 | //one can use this line if extgcd-bug is fixed |
---|
1095 | } |
---|
1096 | return(L); |
---|
1097 | } |
---|
1098 | static proc normalize_Our(matrix N, matrix Q) |
---|
1099 | "USAGE: normalize_Our(N,Q), N, Q are two matrices |
---|
1100 | PURPOSE: normalizes N and divides the columns of Q through the leading coefficients of the columns of N |
---|
1101 | RETURN: normalized matrix N and altered Q(according to the scheme mentioned in purpose). If number of columns of N and Q do not coincide, N and Q are returned unchanged |
---|
1102 | NOTE: number of columns of N and Q must coincide. |
---|
1103 | " |
---|
1104 | { |
---|
1105 | if(ncols(N) != ncols(Q)) |
---|
1106 | { |
---|
1107 | return (N,Q); |
---|
1108 | } |
---|
1109 | module M = module(N); |
---|
1110 | module S = module(Q); |
---|
1111 | int NCols = ncols(N); |
---|
1112 | number n; |
---|
1113 | for(int i=1;i<=NCols;i++) |
---|
1114 | { |
---|
1115 | n = leadcoef(M[i]); |
---|
1116 | if( n != 0 ) |
---|
1117 | { |
---|
1118 | M[i]=M[i]/n; |
---|
1119 | S[i]=S[i]/n; |
---|
1120 | } |
---|
1121 | } |
---|
1122 | N = matrix(M); |
---|
1123 | Q = matrix(S); |
---|
1124 | return (N,Q); |
---|
1125 | } |
---|
1126 | |
---|
1127 | //--------------------------------------------------------------- |
---|
1128 | proc smith( module M ) |
---|
1129 | "USAGE: smith(M), M a module or a matrix, |
---|
1130 | PURPOSE: computes the Smith form of a matrix |
---|
1131 | RETURN: a list of length 4 with the following entries: |
---|
1132 | @* [1]: The Smith-Form S of M, |
---|
1133 | @* [2]: the rank of M, |
---|
1134 | @* [3]: a unimodular matrix U, |
---|
1135 | @* [4]: a unimodular matrix V, |
---|
1136 | such that U*M*V=S. An warning is returned when no Smith Form exists. |
---|
1137 | NOTE: The Smith form only exists over PIDs (principal ideal domains). Use global ordering for computations! |
---|
1138 | " |
---|
1139 | { |
---|
1140 | if (nvars(basering)>1) //if more than one variable, return empty list |
---|
1141 | { |
---|
1142 | string s="The Smith-Form only exists for principal ideal domains"; |
---|
1143 | return (s); |
---|
1144 | } |
---|
1145 | matrix N = matrix(M); //Typecasting |
---|
1146 | int n = nrows(N); |
---|
1147 | int m = ncols(N); |
---|
1148 | matrix P = unitmat(n); //left transformation matrix |
---|
1149 | matrix Q = unitmat(m); //right transformation matrix |
---|
1150 | int k, i, j, deg_temp; |
---|
1151 | poly tmp; |
---|
1152 | vector v; |
---|
1153 | list L; //for extgcd-computation |
---|
1154 | intmat f[1][n]; //to save degrees |
---|
1155 | matrix lambda[1][n]; //to save leadcoefficients |
---|
1156 | intmat g[1][m]; //to save degrees |
---|
1157 | matrix mu[1][m]; //to save leadcoefficients |
---|
1158 | int ii; //counter |
---|
1159 | |
---|
1160 | while ((k!=n) && (k!=m) ) |
---|
1161 | { |
---|
1162 | k++; |
---|
1163 | while ((k<=n) && (k<=m)) //outer while-loop for column-operations |
---|
1164 | { |
---|
1165 | while(k<=m ) //inner while-loop for row-operations |
---|
1166 | { |
---|
1167 | if( (n>m) && (k < n) && (k<m)) |
---|
1168 | { |
---|
1169 | if( simplify((ideal(submat(N,k+1..n,k+1..m))),2)== 0) |
---|
1170 | { |
---|
1171 | return(N,k-1,P,Q); |
---|
1172 | } |
---|
1173 | } |
---|
1174 | i,j = smdeg(submat(N,k..n,k..m)); //choose smallest degree in the remaining submatrix |
---|
1175 | i=i+(k-1); //indices adjusted to the whole matrix |
---|
1176 | j=j+(k-1); |
---|
1177 | if(i!=k) //take the element with smallest degree in the first position |
---|
1178 | { |
---|
1179 | N=permrow(N,i,k); |
---|
1180 | P=permrow(P,i,k); |
---|
1181 | } |
---|
1182 | if(j!=k) |
---|
1183 | { |
---|
1184 | N=permcol(N,j,k); |
---|
1185 | Q=permcol(Q,j,k); |
---|
1186 | } |
---|
1187 | if(NoNon0Pol(N[k])==1) |
---|
1188 | { |
---|
1189 | break; |
---|
1190 | } |
---|
1191 | tmp=leadcoef(N[k,k]); |
---|
1192 | deg_temp=ord(N[k,k]); //ord outputs the leading degree of N[k,k] |
---|
1193 | for(ii=k+1;ii<=n;ii++) |
---|
1194 | { |
---|
1195 | lambda[1,ii]=leadcoef(N[ii,k])/tmp; |
---|
1196 | f[1,ii]=deg(N[ii,k])-deg_temp; |
---|
1197 | } |
---|
1198 | for(ii=k+1;ii<=n;ii++) |
---|
1199 | { |
---|
1200 | N = addrow(N,k,-lambda[1,ii]*var(1)^f[1,ii],ii); |
---|
1201 | P = addrow(P,k,-lambda[1,ii]*var(1)^f[1,ii],ii); |
---|
1202 | N,Q=normalize_Our(N,Q); |
---|
1203 | } |
---|
1204 | } |
---|
1205 | if (k>n) |
---|
1206 | { |
---|
1207 | break; |
---|
1208 | } |
---|
1209 | if(NoNon0Pol(transpose(N)[k])==1) |
---|
1210 | { |
---|
1211 | break; |
---|
1212 | } |
---|
1213 | tmp=leadcoef(N[k,k]); |
---|
1214 | deg_temp=ord(N[k,k]); //ord outputs the leading degree of N[k][k] |
---|
1215 | |
---|
1216 | for(ii=k+1;ii<=m;ii++) |
---|
1217 | { |
---|
1218 | mu[1,ii]=leadcoef(N[k,ii])/tmp; |
---|
1219 | g[1,ii]=deg(N[k,ii])-deg_temp; |
---|
1220 | } |
---|
1221 | for(ii=k+1;ii<=m;ii++) |
---|
1222 | { |
---|
1223 | N=addcol(N,k,-mu[1,ii]*var(1)^g[1,ii],ii); |
---|
1224 | Q=addcol(Q,k,-mu[1,ii]*var(1)^g[1,ii],ii); |
---|
1225 | N,Q=normalize_Our(N,Q); |
---|
1226 | } |
---|
1227 | } |
---|
1228 | if( (k!=1) && (k<n) && (k<m) ) |
---|
1229 | { |
---|
1230 | L = extgcd_Our(N[k-1,k-1],N[k,k]); |
---|
1231 | if ( N[k-1,k-1]!=L[1] ) //means that N[k-1,k-1] is not a divisor of N[k,k] |
---|
1232 | { |
---|
1233 | N=addrow(N,k-1,L[2],k); |
---|
1234 | P=addrow(P,k-1,L[2],k); |
---|
1235 | N,Q=normalize_Our(N,Q); |
---|
1236 | |
---|
1237 | N=addcol(N,k,-L[3],k-1); |
---|
1238 | Q=addcol(Q,k,-L[3],k-1); |
---|
1239 | N,Q=normalize_Our(N,Q); |
---|
1240 | k=k-2; |
---|
1241 | } |
---|
1242 | } |
---|
1243 | } |
---|
1244 | if( (k<=n) && (k<=m) ) |
---|
1245 | { |
---|
1246 | if( N[k,k]==0) |
---|
1247 | { |
---|
1248 | return(N,k-1,P,Q); |
---|
1249 | } |
---|
1250 | } |
---|
1251 | return(N,k,P,Q); |
---|
1252 | } |
---|
1253 | example |
---|
1254 | { |
---|
1255 | "EXAMPLE:";echo = 2; |
---|
1256 | option(redSB); |
---|
1257 | option(redTail); |
---|
1258 | ring r=0,x,dp; |
---|
1259 | // see what happens when the matrix is already in Smith-Form |
---|
1260 | module M = [x,0,0],[0,x2,0],[0,0,x3]; |
---|
1261 | list L = smith(M); |
---|
1262 | print(L[1]); |
---|
1263 | matrix N=matrix(M); |
---|
1264 | matrix B=L[3]*N*L[4]; |
---|
1265 | print(B); |
---|
1266 | //------- and yet another example -------------- |
---|
1267 | module M2=[x2,x,3x3-4],[2x2-1,4x,5x2],[2x5,3x,4x]; |
---|
1268 | print(M2); |
---|
1269 | list P=smith(M2); |
---|
1270 | print(P[1]); |
---|
1271 | matrix N2=matrix(M2); |
---|
1272 | matrix B2=P[3]*N2*P[4]; |
---|
1273 | print(B2); |
---|
1274 | } |
---|
1275 | //--------------------------------------------------------------- |
---|
1276 | static proc list_tex(L, string name,link l,int nr_loop) |
---|
1277 | "USAGE: list_tex(L,name,l), where L is a list, name a string, l a link |
---|
1278 | writes the content of list L in a tex-file 'name' |
---|
1279 | RETURN: nothing |
---|
1280 | " |
---|
1281 | { |
---|
1282 | if(typeof(L)!="list") //in case L is not a list |
---|
1283 | { |
---|
1284 | texobj(name,L); |
---|
1285 | } |
---|
1286 | if(size(L)==0) |
---|
1287 | { |
---|
1288 | } |
---|
1289 | else |
---|
1290 | { |
---|
1291 | string t; |
---|
1292 | for (int i=1;i<=size(L);i++) |
---|
1293 | { |
---|
1294 | while(1) |
---|
1295 | { |
---|
1296 | if(typeof(L[i])=="string") //Fehler hier fuer normalen output->nur wenn string in liste dann verbatim |
---|
1297 | { |
---|
1298 | t=L[i]; |
---|
1299 | if(nr_loop==1) |
---|
1300 | { |
---|
1301 | write(l,"\\begin\{center\}"); |
---|
1302 | write(l,"\\begin\{verbatim\}"); |
---|
1303 | } |
---|
1304 | write(l,t); |
---|
1305 | if(nr_loop==0) |
---|
1306 | { |
---|
1307 | write(l,"\\par"); |
---|
1308 | } |
---|
1309 | if(nr_loop==1) |
---|
1310 | { |
---|
1311 | write(l,"\\end\{verbatim\}"); |
---|
1312 | write(l,"\\end\{center\}"); |
---|
1313 | } |
---|
1314 | break; |
---|
1315 | } |
---|
1316 | if(typeof(L[i])=="module") |
---|
1317 | { |
---|
1318 | texobj(name,matrix(L[i])); |
---|
1319 | break; |
---|
1320 | } |
---|
1321 | if(typeof(L[i])=="list") |
---|
1322 | { |
---|
1323 | list_tex(L[i],name,l,1); |
---|
1324 | break; |
---|
1325 | } |
---|
1326 | write(l,"\\begin\{center\}"); |
---|
1327 | texobj(name,L[i]); |
---|
1328 | write(l,"\\end\{center\}"); |
---|
1329 | write(l,"\\par"); |
---|
1330 | break; |
---|
1331 | } |
---|
1332 | } |
---|
1333 | } |
---|
1334 | } |
---|
1335 | example |
---|
1336 | { |
---|
1337 | "EXAMPLE:";echo = 2; |
---|
1338 | } |
---|
1339 | //--------------------------------------------------------------- |
---|
1340 | proc verbatim_tex(string s, link l) |
---|
1341 | "USAGE: verbatim_tex(s,l), where s is a string and l a link |
---|
1342 | PURPOSE: writes the content of s in verbatim-environment in the file |
---|
1343 | specified by link |
---|
1344 | RETURN: nothing |
---|
1345 | " |
---|
1346 | { |
---|
1347 | write(l,"\\begin{verbatim}"); |
---|
1348 | write(l,s); |
---|
1349 | write(l,"\\end{verbatim}"); |
---|
1350 | write(l,"\\par"); |
---|
1351 | } |
---|
1352 | example |
---|
1353 | { |
---|
1354 | "EXAMPLE:";echo = 2; |
---|
1355 | } |
---|
1356 | //--------------------------------------------------------------- |
---|
1357 | proc FindTorsion(module R, ideal TAnn) |
---|
1358 | "USAGE: FindTorsion(R, I); R an ideal/matrix/module, I an ideal |
---|
1359 | PURPOSE: computes the Groebner basis of the submodule of R, annihilated by I |
---|
1360 | ETURN: module |
---|
1361 | NOTE: especially helpful, when I is the annihilator of the t(R) - the torsion submodule of R. In this case, the result is the explicit presentation of t(R) as |
---|
1362 | the submodule of R |
---|
1363 | EXAMPLE: example FindTorsion; shows an example |
---|
1364 | " |
---|
1365 | { |
---|
1366 | // motivation: let R be a module, |
---|
1367 | // TAnn is the annihilator of t(R)\subset R |
---|
1368 | // compute the generators of t(R) explicitly |
---|
1369 | ideal AS = TAnn; |
---|
1370 | module S = R; |
---|
1371 | if (attrib(S,"isSB")<>1) |
---|
1372 | { |
---|
1373 | S = std(S); |
---|
1374 | } |
---|
1375 | if (attrib(AS,"isSB")<>1) |
---|
1376 | { |
---|
1377 | AS = std(AS); |
---|
1378 | } |
---|
1379 | int nc = ncols(S); |
---|
1380 | module To = quotient(S,AS); |
---|
1381 | To = std(NF(To,S)); |
---|
1382 | return(To); |
---|
1383 | } |
---|
1384 | example |
---|
1385 | { |
---|
1386 | "EXAMPLE:";echo = 2; |
---|
1387 | // Flexible Rod |
---|
1388 | ring A = 0,(D1, D2), (c,dp); |
---|
1389 | module R= [D1, -D1*D2, -1], [2*D1*D2, -D1-D1*D2^2, 0]; |
---|
1390 | module RR = transpose(R); |
---|
1391 | list L = control(RR); |
---|
1392 | // here, we have the annihilator: |
---|
1393 | ideal LAnn = D1; // = L[10] |
---|
1394 | module Tr = FindTorsion(RR,LAnn); |
---|
1395 | print(RR); // the module itself |
---|
1396 | print(Tr); // generators of the torsion submodule |
---|
1397 | } |
---|
1398 | |
---|
1399 | |
---|
1400 | proc ControlExample(string s) |
---|
1401 | "USAGE: ControlExample(s); s a string |
---|
1402 | PURPOSE: set up an example from the mini database by initalizing a ring and a module in a ring |
---|
1403 | RETURN: ring |
---|
1404 | NOTE: in order to see the list of available examples, execute @code{ControlExample(\"show\");} |
---|
1405 | @* To use ab example, one has to do the following. Suppose one calls the ring, where the example will be activated, A. Then, by executing |
---|
1406 | @* @code{def A = ControlExample(\"Antenna\");} and @code{setring A;}, |
---|
1407 | @* A will become a basering from the example \"Antenna\" with |
---|
1408 | the predefined system module R (transposed). |
---|
1409 | After that one can just execute @code{control(R);} respectively |
---|
1410 | @code{autonom(R);} to perform the control resp. autonomy analysis of R. |
---|
1411 | EXAMPLE: example ControlExample; shows an example |
---|
1412 | "{ |
---|
1413 | list E, S, D; // E=official name, S=synonym, D=description |
---|
1414 | E[1] = "Cauchy1"; S[1] = "cauchy1"; D[1] = "1-dimensional Cauchy equation"; |
---|
1415 | E[2] = "Cauchy2"; S[2] = "cauchy2"; D[2] = "2-dimensional Cauchy equation"; |
---|
1416 | E[3] = "Control1"; S[3] = "control1"; D[3] = "example of a simple noncontrollable system"; |
---|
1417 | E[4] = "Control2"; S[4] = "control2"; D[4] = "example of a simple controllable system"; |
---|
1418 | E[5] = "Antenna"; S[5] = "antenna"; D[5] = "antenna"; |
---|
1419 | E[6] = "Einstein"; S[6] = "einstein"; D[6] = "Einstein equations in vacuum"; |
---|
1420 | E[7] = "FlexibleRod"; S[7] = "flexible rod"; D[7] = "flexible rod"; |
---|
1421 | E[8] = "TwoPendula"; S[8] = "two pendula"; D[8] = "two pendula mounted on a cart"; |
---|
1422 | E[9] = "WindTunnel"; S[9] = "wind tunnel";D[9] = "wind tunnel"; |
---|
1423 | E[10] = "Zerz1"; S[10] = "zerz1"; D[10] = "example from the lecture of Eva Zerz"; |
---|
1424 | // all the examples so far |
---|
1425 | int i; |
---|
1426 | if ( (s=="show") || (s=="Show") ) |
---|
1427 | { |
---|
1428 | print("The list of examples:"); |
---|
1429 | for (i=1; i<=size(E); i++) |
---|
1430 | { |
---|
1431 | printf("name: %s, desc: %s", E[i],D[i]); |
---|
1432 | } |
---|
1433 | return(); |
---|
1434 | } |
---|
1435 | string t; |
---|
1436 | for (i=1; i<=size(E); i++) |
---|
1437 | { |
---|
1438 | if ( (s==E[i]) || (s==S[i]) ) |
---|
1439 | { |
---|
1440 | t = "def @A = ex"+E[i]+"();"; |
---|
1441 | execute(t); |
---|
1442 | return(@A); |
---|
1443 | } |
---|
1444 | } |
---|
1445 | "No example found"; |
---|
1446 | return(); |
---|
1447 | } |
---|
1448 | example |
---|
1449 | { |
---|
1450 | "EXAMPLE:";echo = 2; |
---|
1451 | ControlExample("show"); // let us see all available examples: |
---|
1452 | def B = ControlExample("TwoPendula"); // let us set up a particular example |
---|
1453 | setring B; |
---|
1454 | print(R); |
---|
1455 | } |
---|
1456 | |
---|
1457 | //---------------------------------------------------------- |
---|
1458 | // |
---|
1459 | //Some example rings with defined systems |
---|
1460 | //---------------------------------------------------------- |
---|
1461 | //autonomy: |
---|
1462 | // |
---|
1463 | //---------------------------------------------------------- |
---|
1464 | static proc exCauchy1() |
---|
1465 | { |
---|
1466 | ring @r=0,(s1,s2),dp; |
---|
1467 | module R= [s1,-s2], |
---|
1468 | [s2, s1]; |
---|
1469 | R=transpose(R); |
---|
1470 | export R; |
---|
1471 | return(@r); |
---|
1472 | }; |
---|
1473 | //---------------------------------------------------------- |
---|
1474 | static proc exCauchy2() |
---|
1475 | { |
---|
1476 | ring @r=0,(s1,s2,s3,s4),dp; |
---|
1477 | module R= [s1,-s2], |
---|
1478 | [s2, s1], |
---|
1479 | [s3,-s4], |
---|
1480 | [s4, s3]; |
---|
1481 | R=transpose(R); |
---|
1482 | export R; |
---|
1483 | return(@r); |
---|
1484 | }; |
---|
1485 | //---------------------------------------------------------- |
---|
1486 | static proc exZerz1() |
---|
1487 | { |
---|
1488 | ring @r=0,(d1,d2),dp; |
---|
1489 | module R=[d1^2-d2], |
---|
1490 | [d2^2-1]; |
---|
1491 | R=transpose(R); |
---|
1492 | export R; |
---|
1493 | return(@r); |
---|
1494 | }; |
---|
1495 | //---------------------------------------------------------- |
---|
1496 | //control |
---|
1497 | //---------------------------------------------------------- |
---|
1498 | static proc exControl1() |
---|
1499 | { |
---|
1500 | ring @r=0,(s1,s2,s3),dp; |
---|
1501 | module R=[0,-s3,s2], |
---|
1502 | [s3,0,-s1]; |
---|
1503 | R=transpose(R); |
---|
1504 | export R; |
---|
1505 | return(@r); |
---|
1506 | }; |
---|
1507 | //---------------------------------------------------------- |
---|
1508 | static proc exControl2() |
---|
1509 | { |
---|
1510 | ring @r=0,(s1,s2,s3),dp; |
---|
1511 | module R=[0,-s3,s2], |
---|
1512 | [s3,0,-s1], |
---|
1513 | [-s2,s1,0]; |
---|
1514 | R=transpose(R); |
---|
1515 | export R; |
---|
1516 | return(@r); |
---|
1517 | }; |
---|
1518 | //---------------------------------------------------------- |
---|
1519 | static proc exAntenna() |
---|
1520 | { |
---|
1521 | ring @r = (0, K1, K2, Te, Kp, Kc),(Dt, delta), dp; |
---|
1522 | module R; |
---|
1523 | R = [Dt, -K1, 0, 0, 0, 0, 0, 0, 0], |
---|
1524 | [0, Dt+K2/Te, 0, 0, 0, 0, -Kp/Te*delta, -Kc/Te*delta, -Kc/Te*delta], |
---|
1525 | [0, 0, Dt, -K1, 0, 0, 0, 0, 0], |
---|
1526 | [0, 0, 0, Dt+K2/Te, 0, 0, -Kc/Te*delta, -Kp/Te*delta, -Kc/Te*delta], |
---|
1527 | [0, 0, 0, 0, Dt, -K1, 0, 0, 0], |
---|
1528 | [0, 0, 0, 0, 0, Dt+K2/Te, -Kc/Te*delta, -Kc/Te*delta, -Kp/Te*delta]; |
---|
1529 | |
---|
1530 | R=transpose(R); |
---|
1531 | export R; |
---|
1532 | return(@r); |
---|
1533 | }; |
---|
1534 | |
---|
1535 | //---------------------------------------------------------- |
---|
1536 | |
---|
1537 | static proc exEinstein() |
---|
1538 | { |
---|
1539 | ring @r = 0,(D(1..4)),dp; |
---|
1540 | module R = |
---|
1541 | [D(2)^2+D(3)^2-D(4)^2, D(1)^2, D(1)^2, -D(1)^2, -2*D(1)*D(2), 0, 0, -2*D(1)*D(3), 0, 2*D(1)*D(4)], |
---|
1542 | [D(2)^2, D(1)^2+D(3)^2-D(4)^2, D(2)^2, -D(2)^2, -2*D(1)*D(2), -2*D(2)*D(3), 0, 0, 2*D(2)*D(4), 0], |
---|
1543 | [D(3)^2, D(3)^2, D(1)^2+D(2)^2-D(4)^2, -D(3)^2, 0, -2*D(2)*D(3), 2*D(3)*D(4), -2*D(1)*D(3), 0, 0], |
---|
1544 | [D(4)^2, D(4)^2, D(4)^2, D(1)^2+D(2)^2+D(3)^2, 0, 0, -2*D(3)*D(4), 0, -2*D(2)*D(4), -2*D(1)*D(4)], |
---|
1545 | [0, 0, D(1)*D(2), -D(1)*D(2), D(3)^2-D(4)^2, -D(1)*D(3), 0, -D(2)*D(3), D(1)*D(4), D(2)*D(4)], |
---|
1546 | [D(2)*D(3), 0, 0, -D(2)*D(3),-D(1)*D(3), D(1)^2-D(4)^2, D(2)*D(4), -D(1)*D(2), D(3)*D(4), 0], |
---|
1547 | [D(3)*D(4), D(3)*D(4), 0, 0, 0, -D(2)*D(4), D(1)^2+D(2)^2, -D(1)*D(4), -D(2)*D(3), -D(1)*D(3)], |
---|
1548 | [0, D(1)*D(3), 0, -D(1)*D(3), -D(2)*D(3), -D(1)*D(2), D(1)*D(4), D(2)^2-D(4)^2, 0, D(3)*D(4)], |
---|
1549 | [D(2)*D(4), 0, D(2)*D(4), 0, -D(1)*D(4), -D(3)*D(4), -D(2)*D(3), 0, D(1)^2+D(3)^2, -D(1)*D(2)], |
---|
1550 | [0, D(1)*D(4), D(1)*D(4), 0, -D(2)*D(4), 0, -D(1)*D(3), -D(3)*D(4), -D(1)*D(2), D(2)^2+D(3)^2]; |
---|
1551 | |
---|
1552 | R=transpose(R); |
---|
1553 | export R; |
---|
1554 | return(@r); |
---|
1555 | }; |
---|
1556 | |
---|
1557 | //---------------------------------------------------------- |
---|
1558 | static proc exFlexibleRod() |
---|
1559 | { |
---|
1560 | ring @r = 0,(D1, delta), dp; |
---|
1561 | module R; |
---|
1562 | R = [D1, -D1*delta, -1], [2*D1*delta, -D1-D1*delta^2, 0]; |
---|
1563 | |
---|
1564 | R=transpose(R); |
---|
1565 | export R; |
---|
1566 | return(@r); |
---|
1567 | }; |
---|
1568 | |
---|
1569 | //---------------------------------------------------------- |
---|
1570 | static proc exTwoPendula() |
---|
1571 | { |
---|
1572 | ring @r=(0,m1,m2,M,g,L1,L2),Dt,dp; |
---|
1573 | module R = [m1*L1*Dt^2, m2*L2*Dt^2, -1, (M+m1+m2)*Dt^2], |
---|
1574 | [m1*L1^2*Dt^2-m1*L1*g, 0, 0, m1*L1*Dt^2], |
---|
1575 | [0, m2*L2^2*Dt^2-m2*L2*g, 0, m2*L2*Dt^2]; |
---|
1576 | |
---|
1577 | R=transpose(R); |
---|
1578 | export R; |
---|
1579 | return(@r); |
---|
1580 | }; |
---|
1581 | //---------------------------------------------------------- |
---|
1582 | static proc exWindTunnel() |
---|
1583 | { |
---|
1584 | ring @r = (0,a, omega, zeta, k),(D1, delta),dp; |
---|
1585 | module R = [D1+a, -k*a*delta, 0, 0], |
---|
1586 | [0, D1, -1, 0], |
---|
1587 | [0, omega^2, D1+2*zeta*omega, -omega^2]; |
---|
1588 | |
---|
1589 | R=transpose(R); |
---|
1590 | export R; |
---|
1591 | return(@r); |
---|
1592 | }; |
---|
1593 | |
---|
1594 | |
---|
1595 | //--------------------------------------------------------------- |
---|
1596 | proc declare(string NameOfRing, string Variables, list #) |
---|
1597 | "USAGE: declare(NameOfRing, Variables,[Parameters, Ordering]); where |
---|
1598 | @* NameOfRing is string with name of ring, |
---|
1599 | @* Variables is a string with names of variables separated by commas (e.g. \"x,y,z\"), |
---|
1600 | @* Parameters is string of parameters in the ring separated by commas (e.g. \"a,b,c\"), |
---|
1601 | @* Ordering is string with name of ordering (by default, the ordering (dp,C) is used). |
---|
1602 | PURPOSE: define the ring easily |
---|
1603 | RETURN: no return value |
---|
1604 | EXAMPLE: example declare; shows an example |
---|
1605 | " |
---|
1606 | { |
---|
1607 | if(size(#)==0) |
---|
1608 | { |
---|
1609 | execute("ring "+NameOfRing+"=0,("+Variables+"),dp;"); |
---|
1610 | } |
---|
1611 | else |
---|
1612 | { |
---|
1613 | if(size(#)==1) |
---|
1614 | { |
---|
1615 | execute( "ring " + NameOfRing + "=(0," + #[1] + "),(" + Variables + "),dp;" ); |
---|
1616 | } |
---|
1617 | else |
---|
1618 | { |
---|
1619 | if( (size(#[1])!=0)&&(#[1]!=" ") ) |
---|
1620 | { |
---|
1621 | execute( "ring " + NameOfRing + "=(0," + #[1] + "),(" + Variables + "),("+#[2]+");" ); |
---|
1622 | } |
---|
1623 | else |
---|
1624 | { |
---|
1625 | execute( "ring " + NameOfRing + "=0,("+Variables+"),("+#[2]+");" ); |
---|
1626 | }; |
---|
1627 | }; |
---|
1628 | }; |
---|
1629 | keepring(basering); |
---|
1630 | } |
---|
1631 | example |
---|
1632 | {"EXAMPLE:";echo = 2; |
---|
1633 | string v="x,y,z"; |
---|
1634 | string p="q,p"; |
---|
1635 | string Ord ="c,lp"; |
---|
1636 | //---------------------------------- |
---|
1637 | declare("Ring_1",v); |
---|
1638 | print(nameof(basering)); |
---|
1639 | print(basering); |
---|
1640 | //---------------------------------- |
---|
1641 | declare("Ring_2",v,p); |
---|
1642 | print(basering); |
---|
1643 | print(nameof(basering)); |
---|
1644 | //---------------------------------- |
---|
1645 | declare("Ring_3",v,p,Ord); |
---|
1646 | print(basering); |
---|
1647 | print(nameof(basering)); |
---|
1648 | //---------------------------------- |
---|
1649 | declare("Ring_4",v,"",Ord); |
---|
1650 | print(basering); |
---|
1651 | print(nameof(basering)); |
---|
1652 | //---------------------------------- |
---|
1653 | declare("Ring_5",v," ",Ord); |
---|
1654 | print(basering); |
---|
1655 | print(nameof(basering)); |
---|
1656 | } |
---|
1657 | // |
---|
1658 | // maybe reasonable to add this in declare |
---|
1659 | // |
---|
1660 | // print("Please enter your representation matrix in the following form: |
---|
1661 | // module R=[1st row],[2nd row],..."); |
---|
1662 | // print("Type the command: R=transpose(R)"); |
---|
1663 | // print(" To compute controllability please enter: control(R)"); |
---|
1664 | // print(" To compute autonomy please enter: autonom(R)"); |
---|