1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: findifs.lib,v 1.6 2009-04-06 17:43:33 motsak Exp $"; |
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3 | category="Applications"; |
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4 | info=" |
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5 | LIBRARY: findifs.lib Tools for the finite difference schemes |
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6 | AUTHORS: Viktor Levandovskyy, levandov@risc.uni-linz.ac.at |
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7 | |
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8 | THEORY: We provide the presentation of difference operators in a polynomial, |
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9 | semi-factorized and a nodal form. Running @code{findifs_example();} |
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10 | will show how we generate finite difference schemes of linear PDE's |
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11 | from given approximations. |
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12 | |
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13 | PROCEDURES: |
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14 | decoef(P,n); decompose poly P into summands with respect to the number n |
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15 | difpoly2tex(S,P[,Q]); present the difference scheme in the nodal form |
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16 | |
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17 | AUXILIARY PROCEDURES: |
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18 | findifs_example(); containes a guided explanation of our approach |
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19 | exp2pt(P[,L]); convert a polynomial M into the TeX format, in nodal form |
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20 | mon2pt(P[,L]); convert a monomial M into the TeX format, in nodal form |
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21 | texcoef(n); converts the number n into TeX |
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22 | npar(n); search for 'n' among the parameters and returns its number |
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23 | magnitude(P); compute the square of the magnitude of a complex expression |
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24 | replace(s,what,with); replace in s all the substrings with a given string |
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25 | xchange(w,a,b); exchange two substrings of a string |
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26 | par2tex(s); convert special characters to TeX in s |
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27 | "; |
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28 | |
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29 | |
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30 | LIB "latex.lib"; |
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31 | LIB "poly.lib"; |
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32 | |
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33 | // 1. GLOBAL ASSUME: in the ring we have first Tx, then Tt: [FIXED, not needed anymore]! |
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34 | // 2. map vars other than Tx,Tt to parameters instead or just ignore them [?] |
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35 | // 3. clear the things with brackets |
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36 | // 4. todo: content resp lcmZ, gcdZ |
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37 | |
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38 | proc xchange(string where, string a, string b) |
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39 | "USAGE: xchange(w,a,b); w,a,b strings |
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40 | RETURN: string |
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41 | PURPOSE: exchanges substring 'a' with a substring 'b' in the string w |
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42 | NOTE: |
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43 | EXAMPLE: example xchange; shows examples |
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44 | "{ |
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45 | // replaces a<->b in where |
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46 | // assume they are of the same size [? seems to work] |
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47 | string s = "H"; |
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48 | string t; |
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49 | t = replace(where,a,s); |
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50 | t = replace(t,b,a); |
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51 | t = replace(t,s,b); |
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52 | return(t); |
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53 | } |
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54 | example |
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55 | { |
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56 | " EXAMPLE:"; echo=2; |
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57 | ring r = (0,dt,dh,A),Tt,dp; |
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58 | poly p = (Tt*dt+dh+1)^2+2*A; |
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59 | string s = texpoly("",p); |
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60 | s; |
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61 | string t = xchange(s,"dh","dt"); |
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62 | t; |
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63 | } |
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64 | |
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65 | proc par2tex(string s) |
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66 | "USAGE: par2tex(s); s a string |
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67 | RETURN: string |
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68 | PURPOSE: converts special characters to TeX in s |
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69 | NOTE: the convention is the following: |
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70 | 'Tx' goes to 'T_x', |
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71 | 'dx' to '\\tri x' (the same for dt, dy, dz), |
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72 | 'theta', 'ro', 'A', 'V' are converted to greek letters. |
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73 | EXAMPLE: example par2tex; shows examples |
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74 | "{ |
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75 | // s is a tex string with a poly |
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76 | // replace theta with \theta |
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77 | // A with \lambda |
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78 | // dt with \tri t |
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79 | // dh with \tri h |
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80 | // Tx with T_x, Ty with T_y |
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81 | // Tt with T_t |
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82 | // V with \nu |
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83 | // ro with \rho |
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84 | // dx with \tri x |
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85 | // dy with \tri y |
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86 | string t = s; |
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87 | t = replace(t,"Tt","T_t"); |
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88 | t = replace(t,"Tx","T_x"); |
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89 | t = replace(t,"Ty","T_y"); |
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90 | t = replace(t,"dt","\\tri t"); |
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91 | t = replace(t,"dh","\\tri h"); |
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92 | t = replace(t,"dx","\\tri x"); |
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93 | t = replace(t,"dy","\\tri y"); |
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94 | t = replace(t,"theta","\\theta"); |
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95 | t = replace(t,"A","\\lambda"); |
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96 | t = replace(t,"V","\\nu"); |
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97 | t = replace(t,"ro","\\rho"); |
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98 | return(t); |
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99 | } |
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100 | example |
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101 | { |
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102 | " EXAMPLE:"; echo=2; |
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103 | ring r = (0,dt,theta,A),Tt,dp; |
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104 | poly p = (Tt*dt+theta+1)^2+2*A; |
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105 | string s = texfactorize("",p); |
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106 | s; |
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107 | par2tex(s); |
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108 | string T = texfactorize("",p*(-theta*A)); |
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109 | par2tex(T); |
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110 | } |
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111 | |
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112 | proc replace(string s, string what, string with) |
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113 | "USAGE: replace(s,what,with); s,what,with strings |
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114 | RETURN: string |
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115 | PURPOSE: replaces in 's' all the substrings 'what' with substring 'with' |
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116 | NOTE: |
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117 | EXAMPLE: example replace; shows examples |
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118 | "{ |
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119 | // clear: replace in s, "what" with "with" |
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120 | int ss = size(s); |
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121 | int cn = find(s,what); |
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122 | if ( (cn==0) || (cn>ss)) |
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123 | { |
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124 | return(s); |
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125 | } |
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126 | int gn = 0; // global counter |
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127 | int sw = size(what); |
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128 | int swith = size(with); |
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129 | string out=""; |
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130 | string tmp; |
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131 | gn = 0; |
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132 | while(cn!=0) |
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133 | { |
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134 | // "cn:"; cn; |
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135 | // "gn"; gn; |
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136 | tmp = ""; |
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137 | if (cn>gn) |
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138 | { |
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139 | tmp = s[gn..cn-1]; |
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140 | } |
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141 | // "tmp:";tmp; |
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142 | // out = out+tmp+" "+with; |
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143 | out = out+tmp+with; |
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144 | // "out:";out; |
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145 | gn = cn + sw; |
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146 | if (gn>ss) |
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147 | { |
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148 | // ( (gn>ss) || ((sw>1) && (gn >= ss)) ) |
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149 | // no need to append smth |
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150 | return(out); |
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151 | } |
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152 | // if (gn == ss) |
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153 | // { |
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154 | |
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155 | // } |
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156 | cn = find(s,what,gn); |
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157 | } |
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158 | // and now, append the rest of s |
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159 | // out = out + " "+ s[gn..ss]; |
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160 | out = out + s[gn..ss]; |
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161 | return(out); |
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162 | } |
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163 | example |
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164 | { |
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165 | " EXAMPLE:"; echo=2; |
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166 | ring r = (0,dt,theta),Tt,dp; |
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167 | poly p = (Tt*dt+theta+1)^2+2; |
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168 | string s = texfactorize("",p); |
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169 | s; |
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170 | s = replace(s,"Tt","T_t"); s; |
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171 | s = replace(s,"dt","\\tri t"); s; |
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172 | s = replace(s,"theta","\\theta"); s; |
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173 | } |
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174 | |
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175 | proc exp2pt(poly P, list #) |
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176 | "USAGE: exp2pt(P[,L]); P poly, L an optional list of strings |
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177 | RETURN: string |
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178 | PURPOSE: convert a polynomial M into the TeX format, in nodal form |
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179 | ASSUME: coefficients must not be fractional |
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180 | NOTE: an optional list L contains a string, which will replace the default |
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181 | value 'u' for the discretized function |
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182 | EXAMPLE: example exp2pt; shows examples |
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183 | "{ |
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184 | // given poly in vars [now Tx,Tt are fixed], |
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185 | // create Tex expression for points of lattice |
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186 | // coeffs must not be fractional |
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187 | string varnm = "u"; |
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188 | if (size(#) > 0) |
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189 | { |
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190 | if (typeof(#[1])=="string") |
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191 | { |
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192 | varnm = string(#[1]); |
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193 | } |
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194 | } |
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195 | // varnm; |
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196 | string rz,mz; |
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197 | while (P!=0) |
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198 | { |
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199 | mz = mon2pt(P,varnm); |
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200 | if (mz[1]=="-") |
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201 | { |
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202 | rz = rz+mz; |
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203 | } |
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204 | else |
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205 | { |
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206 | rz = rz + "+" + mz; |
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207 | } |
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208 | P = P-lead(P); |
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209 | } |
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210 | rz = rz[2..size(rz)]; |
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211 | return(rz); |
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212 | } |
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213 | example |
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214 | { |
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215 | " EXAMPLE:"; echo=2; |
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216 | ring r = (0,dh,dt),(Tx,Tt),dp; |
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217 | poly M = (4*dh*Tx^2+1)*(Tt-1)^2; |
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218 | print(exp2pt(M)); |
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219 | print(exp2pt(M,"F")); |
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220 | } |
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221 | |
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222 | proc mon2pt(poly M, string V) |
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223 | "USAGE: mon2pt(M,V); M poly, V a string |
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224 | RETURN: string |
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225 | PURPOSE: convert a monomial M into the TeX format, nodal form |
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226 | EXAMPLE: example mon2pt; shows examples |
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227 | "{ |
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228 | // searches for Tx, then Tt |
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229 | // monomial to the lattice point conversion |
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230 | // c*X^a*Y^b --> c*U^{n+a}_{j+b} |
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231 | number cM = leadcoef(M); |
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232 | intvec e = leadexp(M); |
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233 | // int a = e[2]; // convention: first Tx, then Tt |
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234 | // int b = e[1]; |
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235 | int i; |
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236 | int a , b, c = 0,0,0; |
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237 | int ia,ib,ic = 0,0,0; |
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238 | int nv = nvars(basering); |
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239 | string s; |
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240 | for (i=1; i<=nv ; i++) |
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241 | { |
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242 | s = string(var(i)); |
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243 | if (s=="Tt") { a = e[i]; ia = i;} |
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244 | if (s=="Tx") { b = e[i]; ib = i;} |
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245 | if (s=="Ty") { c = e[i]; ic = i;} |
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246 | } |
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247 | // if (ia==0) {"Error:Tt not found!"; return("");} |
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248 | // if (ib==0) {"Error:Tx not found!"; return("");} |
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249 | // if (ic==0) {"Error:Ty not found!"; return("");} |
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250 | // string tc = texobj("",c); // why not texpoly? |
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251 | string tc = texcoef(cM); |
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252 | string rs; |
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253 | if (cM==-1) |
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254 | { |
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255 | rs = "-"; |
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256 | } |
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257 | if (cM^2 != 1) |
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258 | { |
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259 | // we don't need 1 or -1 as coeffs |
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260 | // rs = clTex(tc)+" "; |
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261 | // rs = par2tex(rmDol(tc))+" "; |
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262 | rs = par2tex(tc)+" "; |
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263 | } |
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264 | // a = 0 or b = 0 |
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265 | rs = rs + V +"^{n"; |
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266 | if (a!=0) |
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267 | { |
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268 | rs = rs +"+"+string(a); |
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269 | } |
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270 | rs = rs +"}_{j"; |
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271 | if (b!=0) |
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272 | { |
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273 | rs = rs +"+"+string(b); |
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274 | } |
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275 | if (c!=0) |
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276 | { |
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277 | rs = rs + ",k+"; |
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278 | rs = rs + string(c); |
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279 | } |
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280 | rs = rs +"}"; |
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281 | return(rs); |
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282 | } |
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283 | example |
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284 | { |
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285 | "EXAMPLE:"; echo=2; |
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286 | ring r = (0,dh,dt),(Tx,Tt),dp; |
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287 | poly M = (4*dh^2-dt)*Tx^3*Tt; |
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288 | print(mon2pt(M,"u")); |
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289 | poly N = ((dh-dt)/(dh+dt))*Tx^2*Tt^2; |
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290 | print(mon2pt(N,"f")); |
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291 | ring r2 = (0,dh,dt),(Tx,Ty,Tt),dp; |
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292 | poly M = (4*dh^2-dt)*Tx^3*Ty^2*Tt; |
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293 | print(mon2pt(M,"u")); |
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294 | } |
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295 | |
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296 | proc npar(number n) |
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297 | "USAGE: npar(n); n a number |
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298 | RETURN: int |
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299 | PURPOSE: searches for 'n' among the parameters and returns its number |
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300 | EXAMPLE: example npar; shows examples |
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301 | "{ |
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302 | // searches for n amongst parameters |
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303 | // and returns its number |
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304 | int i,j=0,0; |
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305 | list L = ringlist(basering); |
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306 | list M = L[1][2]; // pars |
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307 | string sn = string(n); |
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308 | sn = sn[2..size(sn)-1]; |
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309 | for (i=1; i<=size(M);i++) |
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310 | { |
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311 | if (M[i] == sn) |
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312 | { |
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313 | j = i; |
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314 | } |
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315 | } |
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316 | if (j==0) |
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317 | { |
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318 | "Incorrect parameter"; |
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319 | } |
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320 | return(j); |
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321 | } |
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322 | example |
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323 | { |
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324 | "EXAMPLE:"; echo=2; |
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325 | ring r = (0,dh,dt,theta,A),t,dp; |
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326 | npar(dh); |
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327 | number T = theta; |
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328 | npar(T); |
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329 | npar(dh^2); |
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330 | } |
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331 | |
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332 | proc decoef(poly P, number n) |
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333 | "USAGE: decoef(P,n); P a poly, n a number |
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334 | RETURN: ideal |
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335 | PURPOSE: decompose poly P into summands with respect |
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336 | to the presence of a number n in the coefficients |
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337 | NOTE: n is usually a parameter with no power |
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338 | EXAMPLE: example decoef; shows examples |
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339 | "{ |
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340 | // decomposes poly into summands |
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341 | // wrt the presence of a number n in coeffs |
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342 | // returns ideal |
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343 | def br = basering; |
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344 | int i,j=0,0; |
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345 | int pos = npar(n); |
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346 | if ((pos==0) || (P==0)) |
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347 | { |
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348 | return(0); |
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349 | } |
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350 | pos = pos + nvars(basering); |
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351 | // map all pars except to vars, provided no things are in denominator |
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352 | number con = content(P); |
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353 | con = numerator(con); |
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354 | P = cleardenom(P); //destroys content! |
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355 | P = con*P; // restore the numerator part of the content |
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356 | list M = ringlist(basering); |
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357 | list L = M[1..4]; |
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358 | list Pars = L[1][2]; |
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359 | list Vars = L[2] + Pars; |
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360 | L[1] = L[1][1]; // characteristic |
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361 | L[2] = Vars; |
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362 | // for non-comm things: don't need nc but graded algebra |
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363 | // list templ; |
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364 | // L[5] = templ; |
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365 | // L[6] = templ; |
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366 | def @R = ring(L); |
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367 | setring @R; |
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368 | poly P = imap(br,P); |
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369 | poly P0 = subst(P,var(pos),0); |
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370 | poly P1 = P - P0; |
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371 | ideal I = P0,P1; |
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372 | setring br; |
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373 | ideal I = imap(@R,I); |
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374 | kill @R; |
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375 | // check: P0+P1==P |
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376 | poly Q = I[1]+I[2]; |
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377 | if (P!=Q) |
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378 | { |
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379 | "Warning: problem in decoef"; |
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380 | } |
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381 | return(I); |
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382 | // substract the pure part from orig and check if n is remained there |
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383 | } |
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384 | example |
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385 | { |
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386 | " EXAMPLE:"; echo=2; |
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387 | ring r = (0,dh,dt),(Tx,Tt),dp; |
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388 | poly P = (4*dh^2-dt)*Tx^3*Tt + dt*dh*Tt^2 + dh*Tt; |
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389 | decoef(P,dt); |
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390 | decoef(P,dh); |
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391 | } |
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392 | |
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393 | proc texcoef(number n) |
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394 | "USAGE: texcoef(n); n a number |
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395 | RETURN: string |
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396 | PURPOSE: converts the number n into TeX format |
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397 | NOTE: if n is a polynomial, texcoef adds extra brackets. |
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398 | @* Performs some space substitutions |
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399 | EXAMPLE: example texcoef; shows examples |
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400 | "{ |
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401 | // makes tex from n |
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402 | // and uses substitutions |
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403 | // if n is a polynomial, adds brackets |
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404 | number D = denominator(n); |
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405 | int DenIsOne = 0; |
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406 | if ( D==number(1) ) |
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407 | { |
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408 | DenIsOne = 1; |
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409 | } |
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410 | string sd = texpoly("",D); |
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411 | sd = rmDol(sd); |
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412 | sd = par2tex(sd); |
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413 | number N = numerator(n); |
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414 | string sn = texpoly("",N); |
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415 | sn = rmDol(sn); |
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416 | sn = par2tex(sn); |
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417 | string sout=""; |
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418 | int i; |
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419 | int NisPoly = 0; |
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420 | if (DenIsOne) |
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421 | { |
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422 | sout = sn; |
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423 | for(i=1; i<=size(sout); i++) |
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424 | { |
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425 | if ( (sout[i]=="+") || (sout[i]=="-") ) |
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426 | { |
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427 | NisPoly = 1; |
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428 | } |
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429 | } |
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430 | if (NisPoly) |
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431 | { |
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432 | sout = "("+sout+")"; |
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433 | } |
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434 | } |
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435 | else |
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436 | { |
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437 | sout = "\\frac{"+sn+"}{"+sd+"}"; |
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438 | } |
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439 | return(sout); |
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440 | } |
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441 | example |
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442 | { |
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443 | " EXAMPLE:"; echo=2; |
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444 | ring r = (0,dh,dt),(Tx,Tt),dp; |
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445 | number n1,n2,n3 = dt/(4*dh^2-dt),(dt+dh)^2, 1/dh; |
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446 | n1; texcoef(n1); |
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447 | n2; texcoef(n2); |
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448 | n3; texcoef(n3); |
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449 | } |
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450 | |
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451 | static proc rmDol(string s) |
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452 | { |
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453 | // removes $s and _no_ (s on appearance |
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454 | int i = size(s); |
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455 | if (s[1] == "$") { s = s[2..i]; i--;} |
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456 | if (s[1] == "(") { s = s[2..i]; i--;} |
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457 | if (s[i] == "$") { s = s[1..i-1]; i--;} |
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458 | if (s[i] == ")") { s = s[1..i-1];} |
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459 | return(s); |
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460 | } |
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461 | |
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462 | proc difpoly2tex(ideal S, list P, list #) |
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463 | "USAGE: difpoly2tex(S,P[,Q]); S an ideal, P and optional Q are lists |
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464 | RETURN: string |
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465 | PURPOSE: present the difference scheme in the nodal form |
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466 | ASSUME: ideal S is the result of 'decoef' procedure |
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467 | NOTE: a list P may be empty or may contain parameters, which will not |
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468 | appear in denominators |
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469 | @* an optional list Q represents the part of the scheme, depending |
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470 | on other function, than the major part |
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471 | EXAMPLE: example difpoly2tex; shows examples |
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472 | " |
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473 | { |
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474 | // S = sum s_i = orig diff poly or |
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475 | // the result of decoef |
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476 | // P = list of pars (numbers) not to be divided with, may be empty |
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477 | // # is an optional list of polys, repr. the part dep. on "f", not on "u" |
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478 | // S = simplify(S,2); // destroys the leadcoef |
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479 | // rescan S and remove 0s from it |
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480 | int i; |
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481 | ideal T; |
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482 | int ss = ncols(S); |
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483 | int j=1; |
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484 | for(i=1; i<=ss; i++) |
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485 | { |
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486 | if (S[i]!=0) |
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487 | { |
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488 | T[j]=S[i]; |
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489 | j++; |
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490 | } |
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491 | } |
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492 | S = T; |
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493 | ss = j-1; |
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494 | int GotF = 1; |
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495 | list F; |
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496 | if (size(#)>0) |
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497 | { |
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498 | F = #; |
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499 | if ( (size(F)==1) && (F[1]==0) ) |
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500 | { |
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501 | GotF = 0; |
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502 | } |
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503 | } |
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504 | else |
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505 | { |
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506 | GotF = 0; |
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507 | } |
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508 | int sf = size(F); |
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509 | |
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510 | ideal SC; |
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511 | int sp = size(P); |
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512 | intvec np; |
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513 | int GotP = 1; |
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514 | if (sp==0) |
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515 | { |
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516 | GotP = 0; |
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517 | } |
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518 | if (sp==1) |
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519 | { |
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520 | if (P[1]==0) |
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521 | { |
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522 | GotP = 0; |
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523 | } |
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524 | } |
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525 | if (GotP) |
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526 | { |
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527 | for (i=1; i<=sp; i++) |
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528 | { |
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529 | np[i] = npar(P[i])+ nvars(basering); |
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530 | } |
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531 | } |
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532 | for (i=1; i<=ss; i++) |
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533 | { |
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534 | SC[i] = leadcoef(S[i]); |
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535 | } |
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536 | if (GotF) |
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537 | { |
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538 | for (i=1; i<=sf; i++) |
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539 | { |
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540 | SC[ss+i] = leadcoef(F[i]); |
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541 | } |
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542 | } |
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543 | def br = basering; |
---|
544 | // map all pars except to vars, provided no things are in denominator |
---|
545 | list M = ringlist(basering); |
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546 | list L = M[1..4]; // erase nc part |
---|
547 | list Pars = L[1][2]; |
---|
548 | list Vars = L[2] + Pars; |
---|
549 | L[1] = L[1][1]; // characteristic |
---|
550 | L[2] = Vars; |
---|
551 | |
---|
552 | def @R = ring(L); |
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553 | setring @R; |
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554 | ideal SC = imap(br,SC); |
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555 | if (GotP) |
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556 | { |
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557 | for (i=1; i<=sp; i++) |
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558 | { |
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559 | SC = subst(SC,var(np[i]),1); |
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560 | } |
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561 | } |
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562 | poly q=1; |
---|
563 | q = lcm(q,SC); |
---|
564 | setring br; |
---|
565 | poly q = imap(@R,q); |
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566 | number lq = leadcoef(q); |
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567 | // lq; |
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568 | number tmp; |
---|
569 | string sout=""; |
---|
570 | string vname = "u"; |
---|
571 | for (i=1; i<=ss; i++) |
---|
572 | { |
---|
573 | tmp = leadcoef(S[i]); |
---|
574 | S[i] = S[i]/tmp; |
---|
575 | tmp = tmp/lq; |
---|
576 | sout = sout +"+ "+texcoef(tmp)+"\\cdot ("+exp2pt(S[i])+")"; |
---|
577 | } |
---|
578 | if (GotF) |
---|
579 | { |
---|
580 | vname = "p"; //"f"; |
---|
581 | for (i=1; i<=sf; i++) |
---|
582 | { |
---|
583 | tmp = leadcoef(F[i]); |
---|
584 | F[i] = F[i]/tmp; |
---|
585 | tmp = tmp/lq; |
---|
586 | sout = sout +"+ "+texcoef(tmp)+"\\cdot ("+exp2pt(F[i],vname)+")"; |
---|
587 | } |
---|
588 | } |
---|
589 | sout = sout[3..size(sout)]; //rm first + |
---|
590 | return(sout); |
---|
591 | } |
---|
592 | example |
---|
593 | { |
---|
594 | "EXAMPLE:"; echo=2; |
---|
595 | ring r = (0,dh,dt,V),(Tx,Tt),dp; |
---|
596 | poly M = (4*dh*Tx+dt)^2*(Tt-1) + V*Tt*Tx; |
---|
597 | ideal I = decoef(M,dt); |
---|
598 | list L; L[1] = V; |
---|
599 | difpoly2tex(I,L); |
---|
600 | poly G = V*dh^2*(Tt-Tx)^2; |
---|
601 | difpoly2tex(I,L,G); |
---|
602 | } |
---|
603 | |
---|
604 | |
---|
605 | proc magnitude(poly P) |
---|
606 | "USAGE: magnitude(P); P a poly |
---|
607 | RETURN: poly |
---|
608 | PURPOSE: compute the square of the magnitude of a complex expression |
---|
609 | ASSUME: i is the variable of a basering |
---|
610 | EXAMPLE: example magnitude; shows examples |
---|
611 | " |
---|
612 | { |
---|
613 | // check whether i is present among the vars TODO |
---|
614 | // assume i^2+1=0; |
---|
615 | poly re = subst(P,i,0); |
---|
616 | poly im = (P - re)/i; |
---|
617 | return(re^2+im^2); |
---|
618 | } |
---|
619 | example |
---|
620 | { |
---|
621 | "EXAMPLE:"; echo=2; |
---|
622 | ring r = (0,d),(g,i,sin,cos),dp; |
---|
623 | poly P = d*i*sin - g*cos +d^2; |
---|
624 | magnitude(P); |
---|
625 | } |
---|
626 | |
---|
627 | |
---|
628 | static proc clTex(string s) |
---|
629 | // removes beginning and ending $'s |
---|
630 | { |
---|
631 | string t; |
---|
632 | if (size(s)>2) |
---|
633 | { |
---|
634 | // why -3? |
---|
635 | t = s[2..(size(s)-3)]; |
---|
636 | } |
---|
637 | return(t); |
---|
638 | } |
---|
639 | |
---|
640 | static proc ElimModComp(module M, int num) |
---|
641 | { |
---|
642 | // experimental, todo |
---|
643 | // check whether a curr.ord is elim-comp |
---|
644 | string s = ordstr(basering); |
---|
645 | int ns = size(s); |
---|
646 | // s[ns] = "c", "C"; |
---|
647 | if (s[1]="c") |
---|
648 | { |
---|
649 | // num last components |
---|
650 | } |
---|
651 | if (s[1]="C") |
---|
652 | { |
---|
653 | // num first components |
---|
654 | } |
---|
655 | } |
---|
656 | |
---|
657 | |
---|
658 | static proc simfrac(poly up, poly down) |
---|
659 | { |
---|
660 | // simplifies a fraction up/down |
---|
661 | // into the form up/down = RT[1] + RT[2]/down |
---|
662 | list LL = division(up,down); |
---|
663 | list RT; |
---|
664 | RT[1] = LL[1][1,1]; // integer part |
---|
665 | RT[2] = L[2][1]; // new numerator |
---|
666 | return(RT); |
---|
667 | } |
---|
668 | |
---|
669 | proc findifs_example() |
---|
670 | "USAGE: findifs_example(); |
---|
671 | RETURN: nothing (demo) |
---|
672 | PURPOSE: demonstration of our approach and this library |
---|
673 | EXAMPLE: example findifs_example; shows examples |
---|
674 | " |
---|
675 | { |
---|
676 | |
---|
677 | "* Equation: u_tt - A^2 u_xx -B^2 u_yy = 0; A,B are constants"; |
---|
678 | "* we employ three central differences"; |
---|
679 | "* the vector we act on is (u_xx, u_yy, u_tt, u)^T"; |
---|
680 | "* Set up the ring: "; |
---|
681 | "ring r = (0,A,B,dt,dx,dy),(Tx,Ty,Tt),(c,dp);"; |
---|
682 | ring r = (0,A,B,dt,dx,dy),(Tx,Ty,Tt),(c,dp); |
---|
683 | "* Set up the matrix with equation and approximations: "; |
---|
684 | "matrix M[4][4] ="; |
---|
685 | " // direct equation:"; |
---|
686 | " -A^2, -B^2, 1, 0,"; |
---|
687 | " // central difference u_tt"; |
---|
688 | " 0, 0, -dt^2*Tt, (Tt-1)^2,"; |
---|
689 | " // central difference u_xx"; |
---|
690 | " -dx^2*Tx, 0, 0, (Tx-1)^2,"; |
---|
691 | " // central difference u_yy"; |
---|
692 | " 0, -dy^2*Ty, 0, (Ty-1)^2;"; |
---|
693 | matrix M[4][4] = |
---|
694 | // direct equation: |
---|
695 | -A^2, -B^2, 1, 0, |
---|
696 | // central difference u_tt |
---|
697 | 0, 0, -dt^2*Tt, (Tt-1)^2, |
---|
698 | // central difference u_xx |
---|
699 | -dx^2*Tx, 0, 0, (Tx-1)^2, |
---|
700 | // central difference u_yy |
---|
701 | 0, -dy^2*Ty, 0, (Ty-1)^2; |
---|
702 | //========================================= |
---|
703 | // CHECK THE CORRECTNESS OF EQUATIONS AS INPUT: |
---|
704 | ring rX = (0,A,B,dt,dx,dy,Tx,Ty,Tt),(Uxx, Uyy,Utt, U),(c,Dp); |
---|
705 | matrix M = imap(r,M); |
---|
706 | vector X = [Uxx, Uyy, Utt, U]; |
---|
707 | "* Print the differential form of equations: "; |
---|
708 | print(M*X); |
---|
709 | // END CHECK |
---|
710 | //========================================= |
---|
711 | setring r; |
---|
712 | "* Perform the elimination of module components:"; |
---|
713 | " module R = transpose(M);"; |
---|
714 | " module S = std(R);"; |
---|
715 | " poly p = S[4,1];" ; |
---|
716 | module R = transpose(M); |
---|
717 | module S = std(R); |
---|
718 | poly p = S[4,1]; // by elimination of module components |
---|
719 | list L; L[1]=A;L[2] = B; |
---|
720 | ideal I = decoef(p,dt); // make splitting wrt the appearance of dt |
---|
721 | "* Create the nodal of the scheme in TeX format: "; |
---|
722 | " ideal I = decoef(p,dt);"; |
---|
723 | " difpoly2tex(I,L);"; |
---|
724 | difpoly2tex(I,L); // the nodal form of the scheme in TeX |
---|
725 | "* Preparations for the semi-factorized form: "; |
---|
726 | poly pi1 = subst(I[2],B,0); |
---|
727 | poly pi2 = I[2] - pi1; |
---|
728 | " poly pi1 = subst(I[2],B,0);"; |
---|
729 | " poly pi2 = I[2] - pi1;"; |
---|
730 | "* Show the semi-factorized form of the scheme: 1st summand"; |
---|
731 | " factorize(I[1]); "; |
---|
732 | factorize(I[1]); // semi-factorized form of the scheme: 1st summand |
---|
733 | "* Show the semi-factorized form of the scheme: 2nd summand"; |
---|
734 | " factorize(pi1);"; |
---|
735 | factorize(pi1); // semi-factorized form of the scheme: 2nd summand |
---|
736 | "* Show the semi-factorized form of the scheme: 3rd summand"; |
---|
737 | " factorize(pi1);"; |
---|
738 | factorize(pi2); // semi-factorized form of the scheme: 3rd summand |
---|
739 | } |
---|
740 | example |
---|
741 | { |
---|
742 | "EXAMPLE:"; echo=1; |
---|
743 | findifs_example(); |
---|
744 | } |
---|