1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Teaching"; |
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4 | info=" |
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5 | LIBRARY: findiff.lib procedures to compute finite difference schemes for |
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6 | linear differential equations |
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7 | AUTHOR: Christian Dingler |
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8 | |
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9 | OVERVIEW: |
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10 | @texinfo |
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11 | Using @code{qepcad}/@code{qepcadsystem} from this |
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12 | library requires the program @code{qepcad} to be installed. |
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13 | You can download @code{qepcad} from |
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14 | @uref{http://www.usna.edu/Users/cs/qepcad/INSTALL/IQ.html} |
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15 | @end texinfo |
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16 | |
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17 | PROCEDURES: |
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18 | visualize(f); shows a scheme in index-notation |
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19 | u(D[,#]); gives some vector; depends on @derivatives |
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20 | scheme([v1,..,vn]); computes the finite difference scheme defined by v1,..,vn |
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21 | laxfrT(Ut,U,space); Lax-Friedrich-approximation for the time-direction |
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22 | laxfrX(Ux,U,space); Lax-Friedrich-approximation for the space-direction |
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23 | forward(U1,U2,VAR); forward-approximation |
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24 | backward(U1,U2,VAR); backward-approximation |
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25 | central1st(U1,U2,VAR); central-approximation of first order |
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26 | central2nd(U1,U2,VAR); central-approximation of second order |
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27 | trapezoid(U1,U2,VAR); trapezoid-approximation |
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28 | midpoint(U1,U2,VAR); midpoint-approximation |
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29 | pyramid(U1,U2,VAR); pyramid-approximation |
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30 | setinitials(variable,der[,#]); constructs and sets the basering for further computations |
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31 | errormap(f); performs the Fouriertransformation of a poly |
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32 | matrixsystem(M,A); gives the scheme of a pde-system as one matrix |
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33 | timestep(M); gives the several timelevels of a scheme derived from a pde-system |
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34 | fouriersystem(M,A); performs the Fouriertransformation of a matrix scheme |
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35 | PartitionVar(f,n); partitions a poly into the var(n)-part and the rest |
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36 | ComplexValue(f); computes the complex value of f, var(1) being the imaginary unit |
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37 | VarToPar(f); substitute var(i) by par(i) |
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38 | ParToVar(f); substitute par(i) by var(i) |
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39 | qepcad(f); ask QEPCAD for equivalent constraints to f<1 |
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40 | qepcadsystem(l); ask QEPCAD for equivalent constraints to all eigenvals of some matrices being <1 |
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41 | "; |
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42 | |
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43 | LIB "ring.lib"; |
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44 | LIB "general.lib"; |
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45 | LIB "standard.lib"; |
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46 | LIB "linalg.lib"; |
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47 | LIB "matrix.lib"; |
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48 | LIB "poly.lib"; |
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49 | LIB "teachstd.lib"; |
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50 | LIB "qhmoduli.lib"; |
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51 | /////////////////////////////////////////////////////////////////////// |
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52 | static proc getit(module M) |
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53 | { |
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54 | int nderiv=pos(U,@derivatives); |
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55 | def M2=groebner(M); |
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56 | module N1=gen(nderiv); |
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57 | def N2=intersect(M2,N1); |
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58 | def S=N2[1][nderiv]; |
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59 | return(S); |
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60 | } |
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61 | /////////////////////////////////////////////////////////////////////// |
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62 | proc visualize(poly f) |
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63 | "USAGE: visualize(f); f of type poly. |
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64 | RETURN: type string; translates the polynomial form of a finite difference scheme into an indexed one as often seen in literature |
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65 | EXAMPLE: example visualize; shows an example |
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66 | " |
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67 | { |
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68 | def n=size(f); |
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69 | string str; |
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70 | intvec v; |
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71 | if (n>0) |
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72 | { |
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73 | int i; |
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74 | int j; |
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75 | for(i=1;i<=n;i++) |
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76 | { |
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77 | intvec w=leadexp(f); |
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78 | for(j=1;j<=size(@variables);j++) |
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79 | { |
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80 | v[j]=w[j+1]; |
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81 | } |
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82 | if(i==1) |
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83 | { |
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84 | str=print(leadcoef(f),"%s")+"*"+"U("+print(v,"%s")+")"; |
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85 | } |
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86 | else |
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87 | { |
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88 | str=str+"+"+print(leadcoef(f),"%s")+"*"+"U("+print(v,"%s")+")"; |
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89 | } |
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90 | kill w; |
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91 | f=f-lead(f); |
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92 | } |
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93 | } |
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94 | return(str); |
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95 | } |
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96 | example |
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97 | { |
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98 | "EXAMPLE:";echo=2; |
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99 | list D="Ux","Ut","U"; |
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100 | list P="a"; |
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101 | list V="t","x"; |
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102 | setinitials(V,D,P); |
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103 | scheme(u(Ut)+a*u(Ux),trapezoid(Ux,U,x),backward(Ut,U,t)); |
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104 | visualize(_); |
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105 | } |
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106 | |
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107 | /////////////////////////////////// |
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108 | static proc imageideal() |
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109 | { |
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110 | |
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111 | def n=size(@variables)-1; |
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112 | ideal IDEAL=var(1),var(2); |
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113 | int j; |
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114 | for(j=1;j<=n;j++) |
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115 | { |
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116 | ideal II=var(2+j+n)+var(1)*var(2+2*n+j); |
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117 | IDEAL=IDEAL+II; |
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118 | kill II; |
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119 | } |
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120 | return(IDEAL); |
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121 | } |
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122 | ///////////////////////////////////// |
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123 | proc u(D,list #) |
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124 | "USAGE: u(D[,#]); D a string that occurs in the list of @derivatives, # an optional list of integers. |
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125 | RETURN: type vector; gives the vector, that corresponds with gen(n)*m, where m is the monomial defined by # |
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126 | EXAMPLE: example u; shows an example |
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127 | " |
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128 | { |
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129 | def n=size(#); |
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130 | def nv=nvars(basering)-1; |
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131 | int nn; |
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132 | if(nv<=n) |
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133 | { |
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134 | nn=nv; |
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135 | } |
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136 | else |
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137 | { |
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138 | nn=n; |
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139 | } |
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140 | int index=pos(D,@derivatives); |
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141 | poly g=1; |
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142 | if(nn>=1) |
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143 | { |
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144 | int j; |
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145 | for(j=1;j<=nn;j++) |
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146 | { |
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147 | int nnn; |
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148 | nnn=int(#[j]); |
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149 | g=var(1+j)**nnn*g; |
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150 | kill nnn; |
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151 | } |
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152 | return(gen(index)*g); |
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153 | } |
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154 | else |
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155 | { |
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156 | return(gen(index)*g); |
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157 | } |
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158 | } |
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159 | example |
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160 | { |
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161 | "EXAMPLE:";echo=2; |
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162 | list D="Ux","Uy","Ut","U"; |
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163 | list P="a","b"; |
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164 | list V="t","x","y"; |
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165 | setinitials(V,D,P); |
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166 | u(Ux); |
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167 | u(Ux,2,3,7); |
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168 | u(Uy)+u(Ut)-u(Ux); |
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169 | u(U)*234-dx*dt*dy*3*u(Uy); |
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170 | } |
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171 | |
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172 | ///////////////////////////////////////////////////////// |
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173 | static proc pos(string D,list L) |
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174 | { |
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175 | int j; |
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176 | int index=-1; |
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177 | def n=size(L); |
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178 | for(j=1;j<=n;j++) |
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179 | { |
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180 | if(D==L[j]) |
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181 | { |
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182 | index=j; |
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183 | } |
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184 | } |
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185 | return(index); |
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186 | } |
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187 | /////////////////////////////////// |
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188 | static proc re(list L) |
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189 | { |
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190 | def n=size(L); |
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191 | int j; |
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192 | for(j=1;j<=n;j++) |
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193 | { |
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194 | string s="string "+print(L[j],"%s")+"="+"nameof("+print(L[j],"%s")+")"+";"; |
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195 | execute(s); |
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196 | kill s; |
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197 | exportto(Top,`L[j]`); |
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198 | } |
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199 | } |
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200 | /////////////////////////////////////////////// |
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201 | proc scheme(list #) |
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202 | "USAGE: scheme([v1,..,vn]); v1,..,vn of type vector |
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203 | RETURN: poly |
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204 | PURPOSE: performs substitutions by the means of Groebner basis computation |
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205 | of the submodule, generated by the input vectors, then intersects the |
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206 | intermediate result with the suitable component in order to get a finite |
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207 | difference scheme |
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208 | NOTE: works only for a single PDE, for the case of a system use @code{matrixsystem} |
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209 | EXAMPLE: example scheme; shows an example |
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210 | " |
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211 | { |
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212 | def N=size(#); |
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213 | if(N==0) |
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214 | { |
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215 | if(defined(M)==1) |
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216 | { |
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217 | kill M; |
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218 | module M; |
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219 | } |
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220 | else |
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221 | { |
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222 | module M; |
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223 | } |
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224 | } |
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225 | else |
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226 | { |
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227 | int j; |
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228 | if(defined(M)==1) |
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229 | { |
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230 | kill M; |
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231 | module M; |
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232 | for(j=1;j<=N;j++) |
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233 | { |
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234 | M=M+#[j]; |
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235 | } |
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236 | } |
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237 | else |
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238 | { |
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239 | module M; |
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240 | for(j=1;j<=N;j++) |
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241 | { |
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242 | M=M+#[j]; |
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243 | } |
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244 | } |
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245 | } |
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246 | def S=getit(M); |
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247 | matrix mat[1][1]=S; |
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248 | list l=timestep(mat); |
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249 | poly f=l[2][1,1]; |
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250 | return(f); |
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251 | } |
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252 | example |
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253 | { |
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254 | "EXAMPLE:";echo=2; |
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255 | list D="Ux","Ut","U"; |
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256 | list P="a"; |
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257 | list V="t","x"; |
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258 | setinitials(V,D,P); |
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259 | def s1=scheme(u(Ut)+a*u(Ux),backward(Ux,U,x),forward(Ut,U,t)); |
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260 | s1; |
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261 | } |
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262 | //////////////////////// |
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263 | static proc diffpar(poly ff) |
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264 | { |
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265 | def gg=print(ff,"%s"); |
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266 | def str="d"+gg; |
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267 | return(`str`); |
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268 | } |
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269 | //////////////////////// |
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270 | proc laxfrT(string Ut, string U, poly space) |
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271 | "USAGE: laxfrT(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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272 | RETURN: type vector; gives a predefined approximation of the Lax-Friedrich-approximation for the derivation in the timevariable as often used in literature; |
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273 | NOTE: see also laxfrX, setinitials, scheme; Warning: laxfrT is not to be interchanged with laxfrX |
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274 | EXAMPLE: example laxfrT; shows an example |
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275 | " |
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276 | { |
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277 | poly time=var(2); |
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278 | poly dtime=diffpar(time); |
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279 | poly dspace=diffpar(space); |
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280 | def v=dtime*space*u(Ut)-time*space*u(U)+1/2*(space**2*u(U)+u(U)); |
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281 | return(v); |
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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 | list D="Ux","Ut","U"; |
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287 | list P="a"; |
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288 | list V="t","x"; |
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289 | setinitials(V,D,P); |
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290 | laxfrT(Ux,U,x); |
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291 | } |
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292 | //////////////////////// |
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293 | proc laxfrX(string Ux, string U, poly space) |
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294 | "USAGE: laxfrX(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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295 | RETURN: type vector; gives a predefined approximation of the Lax-Friedrich-approximation for the derivation in one of the spatial variables as often used in literature; |
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296 | NOTE: see also laxfrT, setinitials, scheme; Warning: laxfrX is not to be interchanged with laxfrT |
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297 | EXAMPLE: example laxfrX; shows an example |
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298 | " |
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299 | { |
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300 | poly dspace = diffpar(space); |
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301 | def v=2*dspace*space*u(Ux)-(space**2-1)*u(U); |
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302 | return(v); |
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303 | } |
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304 | example |
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305 | { |
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306 | "EXAMPLE:";echo=2; |
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307 | list D="Ux","Ut","U"; |
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308 | list P="a"; |
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309 | list V="t","x"; |
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310 | setinitials(V,D,P); |
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311 | laxfrX(Ux,U,x); |
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312 | } |
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313 | //////////////////////// |
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314 | proc forward(string U1,string U2,poly VAR) |
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315 | "USAGE: forward(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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316 | RETURN: type vector; gives a predefined approximation of the forward approximation as often used in literature; |
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317 | NOTE: see also laxfrT,setinitials,scheme; |
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318 | EXAMPLE: example forward; shows an example |
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319 | " |
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320 | { |
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321 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
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322 | { |
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323 | def V1=U1; |
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324 | def V2=U2; |
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325 | } |
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326 | else |
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327 | { |
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328 | def V1=U2; |
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329 | def V2=U1; |
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330 | } |
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331 | def v=diffpar(VAR)*u(V1)+u(V2)-VAR*u(V2); |
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332 | return(v); |
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333 | } |
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334 | example |
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335 | { |
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336 | "EXAMPLE:";echo=2; |
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337 | list D="Ut","Ux","Uy","U"; |
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338 | list V="t","x","y"; |
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339 | list P="a","b"; |
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340 | setinitials(V,D,P); |
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341 | forward(Ux,U,x); |
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342 | forward(Uy,U,y); |
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343 | forward(Ut,U,t); |
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344 | } |
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345 | /////////////////////// |
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346 | proc backward(string U1,string U2,poly VAR) |
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347 | "USAGE: backward(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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348 | RETURN: type vector; gives a predefined approximation of the backward approximation as often used in literature; |
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349 | NOTE: see also forward,laxfrT,setinitials,scheme; |
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350 | EXAMPLE: example backward; shows an example |
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351 | " |
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352 | { |
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353 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
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354 | { |
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355 | def V1=U1; |
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356 | def V2=U2; |
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357 | } |
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358 | else |
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359 | { |
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360 | def V1=U2; |
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361 | def V2=U1; |
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362 | } |
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363 | def v=diffpar(VAR)*VAR*u(V1)+u(V2)-VAR*u(V2); |
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364 | return(v); |
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365 | } |
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366 | example |
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367 | { |
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368 | "EXAMPLE:";echo=2; |
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369 | list D="Ut","Ux","Uy","U"; |
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370 | list V="t","x","y"; |
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371 | list P="a","b"; |
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372 | setinitials(V,D,P); |
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373 | backward(Ux,U,x); |
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374 | backward(Uy,U,y); |
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375 | backward(Ut,U,t); |
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376 | } |
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377 | ///////////////////////////// |
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378 | proc central1st(string U1,string U2,poly VAR) |
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379 | "USAGE: central1st(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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380 | RETURN: type vector; gives a predefined approximation of the first-order-central-approximation as often used in literature; |
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381 | NOTE: see also forward,laxfrT,setinitials,scheme; |
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382 | EXAMPLE: example central1st; shows an example |
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383 | " |
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384 | { |
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385 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
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386 | { |
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387 | def V1=U1; |
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388 | def V2=U2; |
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389 | } |
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390 | else |
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391 | { |
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392 | def V1=U2; |
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393 | def V2=U1; |
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394 | } |
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395 | def v=2*diffpar(VAR)*VAR*u(V1)+u(V2)-VAR**2*u(V2); |
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396 | return(v); |
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397 | } |
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398 | example |
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399 | { |
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400 | "EXAMPLE:";echo=2; |
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401 | list D="Ut","Ux","Uy","U"; |
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402 | list V="t","x","y"; |
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403 | list P="a","b"; |
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404 | setinitials(V,D,P); |
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405 | central1st(Ux,U,x); |
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406 | central1st(Uy,U,y); |
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407 | } |
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408 | //////////////////////////////// |
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409 | proc central2nd(string U1,string U2,poly VAR) |
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410 | "USAGE: central2nd(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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411 | RETURN: type vector; gives a predefined approximation of the second-order-central-approximation as often used in literature; |
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412 | NOTE: see also forward,laxfrT,setinitials,scheme; |
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413 | EXAMPLE: example central2nd; shows an example |
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414 | " |
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415 | { |
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416 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
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417 | { |
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418 | def V1=U1; |
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419 | def V2=U2; |
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420 | } |
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421 | else |
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422 | { |
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423 | def V1=U2; |
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424 | def V2=U1; |
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425 | } |
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426 | def v=diffpar(VAR)**2*VAR*u(V1)-(VAR**2*u(V2)-2*VAR*u(V2)+u(V2)); |
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427 | return(v); |
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428 | } |
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429 | example |
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430 | { |
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431 | "EXAMPLE:";echo=2; |
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432 | list D="Uxx","Ux","Utt","Ut","U"; |
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433 | list P="lambda"; |
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434 | list V="t","x"; |
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435 | setinitials(V,D,P); |
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436 | central2nd(Uxx,U,x); |
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437 | central2nd(Utt,U,t); |
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438 | } |
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439 | ///////////////////////////////// |
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440 | proc trapezoid(string U1,string U2,poly VAR) |
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441 | "USAGE: trapezoid(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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442 | RETURN: type vector; gives a predefined approximation of the trapezoid-approximation as often used in literature; |
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443 | NOTE: see also forward,laxfrT,setinitials,scheme; |
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444 | EXAMPLE: example trapezoid; shows an example |
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445 | " |
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446 | { |
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447 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
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448 | { |
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449 | def V1=U1; |
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450 | def V2=U2; |
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451 | } |
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452 | else |
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453 | { |
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454 | def V1=U2; |
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455 | def V2=U1; |
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456 | } |
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457 | def v=1/2*diffpar(VAR)*(VAR+1)*u(V1)+(1-VAR)*u(V2); |
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458 | return(v); |
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459 | } |
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460 | example |
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461 | { |
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462 | "EXAMPLE:";echo=2; |
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463 | list D="Uxx","Ux","Utt","Ut","U"; |
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464 | list P="lambda"; |
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465 | list V="t","x"; |
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466 | setinitials(V,D,P); |
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467 | trapezoid(Uxx,Ux,x); |
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468 | trapezoid(Ux,U,x); |
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469 | } |
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470 | /////////////////////////////////// |
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471 | proc midpoint(string U1,string U2,poly VAR) |
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472 | "USAGE: midpoint(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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473 | RETURN: type vector; gives a predefined approximation of the midpoint-approximation as often used in literature; |
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474 | NOTE: see also forward,laxfrT,setinitials,scheme; |
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475 | EXAMPLE: example midpoint; shows an example |
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476 | " |
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477 | { |
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478 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
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479 | { |
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480 | def V1=U1; |
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481 | def V2=U2; |
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482 | } |
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483 | else |
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484 | { |
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485 | def V1=U2; |
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486 | def V2=U1; |
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487 | } |
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488 | def v=2*diffpar(VAR)*VAR*u(V1)+(1-VAR**2)*u(V2); |
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489 | return(v); |
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490 | } |
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491 | example |
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492 | { |
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493 | "EXAMPLE:";echo=2; |
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494 | list D="Uxx","Ux","Utt","Ut","U"; |
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495 | list P="lambda"; |
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496 | list V="t","x"; |
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497 | setinitials(V,D,P); |
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498 | midpoint(Ux,U,x); |
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499 | } |
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500 | ////////////////////////////////////// |
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501 | proc pyramid(string U1,string U2,poly VAR) |
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502 | "USAGE: pyramid(U1,U2,var); U1, U2 are the names of occuring derivatives, var is a variable in the basering; |
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503 | RETURN: type vector; gives a predefined approximation of the pyramid-approximation as often used in literature; |
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504 | NOTE: see also forward,laxfrT,setinitials,scheme; |
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505 | EXAMPLE: example pyramid; shows an example |
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506 | " |
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507 | { |
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508 | if(pos(U1,@derivatives)<pos(U2,@derivatives)) |
---|
509 | { |
---|
510 | def V1=U1; |
---|
511 | def V2=U2; |
---|
512 | } |
---|
513 | else |
---|
514 | { |
---|
515 | def V1=U2; |
---|
516 | def V2=U1; |
---|
517 | } |
---|
518 | def v=1/3*diffpar(VAR)*(VAR**2+VAR+1)*u(V1)+(VAR-VAR**3)*u(V2); |
---|
519 | return(v); |
---|
520 | } |
---|
521 | example |
---|
522 | { |
---|
523 | "EXAMPLE:";echo=2; |
---|
524 | list D="Uxx","Ux","Utt","Ut","U"; |
---|
525 | list P="lambda"; |
---|
526 | list V="t","x"; |
---|
527 | setinitials(V,D,P); |
---|
528 | pyramid(Ux,U,x); |
---|
529 | } |
---|
530 | ////////////////////////////////////////////// |
---|
531 | proc setinitials(list variable, list der,list #) |
---|
532 | "USAGE: setinitials(V,D[,P]); V,D,P are lists with strings as elements |
---|
533 | RETURN: no return value: sets the dependence order of the occuring derivatives, |
---|
534 | constructs the suitable ring to compute in containing user chosen parameters, sets new basering |
---|
535 | NOTE: P is optional, used to introduce some additional parameters into the ring. The Sine and |
---|
536 | Cosine values needed for the fourier transformation are symbolically introduced under the names |
---|
537 | string(c)+nameof(variable), i.e. if x is any spatial variable then cx:=cosine(dx*ksi), when |
---|
538 | regarding the fourier transform after ksi (for sine respectively). Artificial parameters I,T,Px,Py |
---|
539 | are introduced for the later eigenvalue analysis. Variables can be transformed into parameters |
---|
540 | of similar name |
---|
541 | EXAMPLE: example setinitials; shows an example |
---|
542 | " |
---|
543 | { |
---|
544 | def LV=variable; |
---|
545 | def @variables=variable; |
---|
546 | def @derivatives=der; |
---|
547 | exportto(Top,@variables); |
---|
548 | exportto(Top,@derivatives); |
---|
549 | re(der); |
---|
550 | int j; |
---|
551 | string dvar="d"+print(LV[1],"%s"); |
---|
552 | dvar=dvar+","+"d"+print(LV[2],"%s"); |
---|
553 | string pvar="P"+print(LV[2],"%s"); |
---|
554 | string COS="C"+print(LV[2],"%s"); |
---|
555 | string SIN="S"+print(LV[2],"%s"); |
---|
556 | for(j=3;j<=size(LV);j++) |
---|
557 | { |
---|
558 | dvar=dvar+","+"d"+print(LV[j],"%s"); |
---|
559 | pvar=pvar+","+"P"+print(LV[j],"%s"); |
---|
560 | COS=COS+","+"C"+print(LV[j],"%s"); |
---|
561 | SIN=SIN+","+"S"+print(LV[j],"%s"); |
---|
562 | } |
---|
563 | string scf="(0,"+"I,"+"T,"+pvar+","+COS+","+SIN+","+print(#,"%s")+","+dvar+")"; //coefficient_field |
---|
564 | string svars="(i"; |
---|
565 | kill j; |
---|
566 | int j; |
---|
567 | for(j=1;j<=size(LV);j++) |
---|
568 | { |
---|
569 | svars=svars+","+print(LV[j],"%s"); |
---|
570 | } |
---|
571 | string cosine; |
---|
572 | string sine; |
---|
573 | kill j; |
---|
574 | int j; |
---|
575 | cosine="c"+print(LV[2],"%s"); |
---|
576 | sine="s"+print(LV[2],"%s"); |
---|
577 | for(j=3;j<=size(LV);j++) |
---|
578 | { |
---|
579 | cosine=cosine+","+"c"+print(LV[j],"%s"); |
---|
580 | sine=sine+","+"s"+print(LV[j],"%s"); |
---|
581 | } |
---|
582 | kill j; |
---|
583 | string strvar=cosine+","+sine+")"; |
---|
584 | svars=svars+","+strvar; ////variables |
---|
585 | string sord="(c,lp)"; ////ordering |
---|
586 | string sring="ring Q="+scf+","+svars+","+sord+";"; |
---|
587 | execute(sring); |
---|
588 | ideal Id=i**2+1; |
---|
589 | int j; |
---|
590 | for(j=1;j<=size(LV)-1;j++) |
---|
591 | { |
---|
592 | ideal II=var(2+j+size(LV)-1)**2+var(2+j+2*(size(LV)-1))**2-1; |
---|
593 | Id=Id+II; |
---|
594 | kill II; |
---|
595 | } |
---|
596 | if(defined(basering)==1) |
---|
597 | { |
---|
598 | kill basering; |
---|
599 | } |
---|
600 | qring R=std(Id); |
---|
601 | setring R; |
---|
602 | // comment by VL: it's better to return this ring! causes many changes |
---|
603 | // across the library |
---|
604 | export(R); |
---|
605 | exportto(Top,basering); |
---|
606 | } |
---|
607 | example |
---|
608 | { |
---|
609 | "EXAMPLE:"; echo = 2; |
---|
610 | list D="Ut","Ux","Uy","U"; |
---|
611 | list V="t","x","y"; |
---|
612 | list P="alpha","beta","gamma"; |
---|
613 | setinitials(V,D,P);////does not show the ring, since there is no output |
---|
614 | basering;///does show the ring |
---|
615 | } |
---|
616 | //////////////////////////// |
---|
617 | proc errormap(poly f) |
---|
618 | "USAGE: errormap(f); f of type poly |
---|
619 | RETURN: type poly; performs the fouriertransformation of a single polynomial |
---|
620 | EXAMPLE: example errormap; shows an example |
---|
621 | " |
---|
622 | { |
---|
623 | ideal Id=imageideal(); |
---|
624 | map phi=R,Id; |
---|
625 | def g=phi(f); |
---|
626 | g=reduce(g,std(0)); |
---|
627 | return(g); |
---|
628 | } |
---|
629 | example |
---|
630 | { |
---|
631 | "EXAMPLE";echo=2; |
---|
632 | list D="Ux","Ut","U"; |
---|
633 | list P="a"; |
---|
634 | list V="t","x"; |
---|
635 | setinitials(V,D,P); |
---|
636 | scheme(u(Ut)+a*u(Ux),central1st(Ux,U,x),backward(Ut,U,t)); |
---|
637 | errormap(_); |
---|
638 | } |
---|
639 | ///////////////////////////////////// |
---|
640 | static proc stepmatrix(int n, poly f) |
---|
641 | { |
---|
642 | int spavars=size(@variables)-1; |
---|
643 | int range=n*spavars; |
---|
644 | if(f==0) |
---|
645 | { |
---|
646 | return(unitmat(range)); |
---|
647 | } |
---|
648 | matrix M[range][range]; |
---|
649 | int length=size(f); |
---|
650 | intvec max=maxexp(f); |
---|
651 | int i; |
---|
652 | intvec shiftback; |
---|
653 | intvec vzero; |
---|
654 | intvec vmax; |
---|
655 | intvec shiftforward; |
---|
656 | for(i=1;i<=size(max);i++) |
---|
657 | { |
---|
658 | shiftback[i]=int(floor(max[i]/2)); |
---|
659 | vzero[i]=0; |
---|
660 | vmax[i]=n-1; |
---|
661 | shiftforward[i]=0; |
---|
662 | } |
---|
663 | kill i; |
---|
664 | int i; |
---|
665 | for(i=1;i<=range;i++) |
---|
666 | { |
---|
667 | poly g=f; |
---|
668 | } |
---|
669 | kill i; |
---|
670 | } |
---|
671 | ////////////////////////////////////// |
---|
672 | static proc floor(n) |
---|
673 | { |
---|
674 | number h1=numerator(n); |
---|
675 | number h2=denominator(n); |
---|
676 | return((h1- (h1 mod h2))/h2); |
---|
677 | } |
---|
678 | ///////////////////////////////////// |
---|
679 | static proc maxexp(poly f) |
---|
680 | { |
---|
681 | int length=size(f); |
---|
682 | poly g=f; |
---|
683 | intvec v; |
---|
684 | int i; |
---|
685 | for(i=1;i<size(@variables);i++) |
---|
686 | { |
---|
687 | v[i]=leadexp(g)[i+2]; |
---|
688 | } |
---|
689 | while(g!=0) |
---|
690 | { |
---|
691 | int j; |
---|
692 | for(j=1;j<size(@variables);j++) |
---|
693 | { |
---|
694 | v[j]=maximal(leadexp(g)[j+2],v[j]); |
---|
695 | g=g-lead(g); |
---|
696 | } |
---|
697 | kill j; |
---|
698 | } |
---|
699 | return(v); |
---|
700 | } |
---|
701 | //////////////////////////////////// |
---|
702 | static proc maximal(int n1,int n2) |
---|
703 | { |
---|
704 | if(n1>n2) |
---|
705 | { |
---|
706 | return(n1); |
---|
707 | } |
---|
708 | else |
---|
709 | { |
---|
710 | return(n2); |
---|
711 | } |
---|
712 | } |
---|
713 | //////////////////////////////////// |
---|
714 | static proc minimal(int n1, int n2) |
---|
715 | { |
---|
716 | return(-maximal(-n1,-n2)); |
---|
717 | } |
---|
718 | //////////////////////////////////// |
---|
719 | static proc MatrixEntry(int n, intvec v) |
---|
720 | { |
---|
721 | int j; |
---|
722 | int entry; |
---|
723 | int spavar=size(@variables)-1; |
---|
724 | for(j=1;j<=spavar;j++) |
---|
725 | { |
---|
726 | entry=entry+v[j]*n**(spavar-j); |
---|
727 | } |
---|
728 | entry=entry+1; |
---|
729 | return(entry); |
---|
730 | } |
---|
731 | ////////////////////////////////// |
---|
732 | static proc CompareVec(intvec ToTest, intvec Reference)//1 if ToTest>=Reference, 0 else |
---|
733 | { |
---|
734 | int i; |
---|
735 | for(i=1;i<=size(@variables)-1;i++) |
---|
736 | { |
---|
737 | if(ToTest[i+2]<Reference[i]) |
---|
738 | { |
---|
739 | return(0); |
---|
740 | } |
---|
741 | } |
---|
742 | return(1); |
---|
743 | } |
---|
744 | ///////////////////////////////// |
---|
745 | static proc MaxVecZero(intvec ToTest, intvec Reference) //KGV, size=size(input) |
---|
746 | { |
---|
747 | int length=size(ToTest); |
---|
748 | int i; |
---|
749 | intvec Maximum; |
---|
750 | for(i=1;i<=length;i++) |
---|
751 | { |
---|
752 | Maximum[i]=maximal(ToTest[i],Reference[i]); |
---|
753 | } |
---|
754 | return(Maximum); |
---|
755 | } |
---|
756 | ////////////////////////////////// |
---|
757 | proc matrixsystem(list Matrices,list Approx) |
---|
758 | "USAGE: matrixsytem(M,A); where the M=Mt,M1,..,Mn is a list with square matrices of the same dimension as entries, and A=At,A1,..,An gives the corresponding approximations for the several variables (t,x1,..,xn) as vector. Intended to solve Mt*U_t + M1*U_x1+..+Mn*U_xn=0 as a linear sytem of partial differential equations numerically by a finite difference scheme; |
---|
759 | RETURN: type matrix; gives back the matrices B1,B2 that represent the finite difference scheme, partitioned into different time levels in the form: B1*U(t=N)=B2*U(t<N), where N is the maximal occurring degree (timelevel) of t. |
---|
760 | EXAMPLE: example matrixsystem; shows an example |
---|
761 | " |
---|
762 | { |
---|
763 | if(size(Matrices)>size(@variables) or size(Matrices)!=size(Approx)) |
---|
764 | { |
---|
765 | ERROR("Check number of variables: it must hold #(matrices)<= #(spatial variables)+1 !!! "); |
---|
766 | } |
---|
767 | if(size(Matrices)!=size(Approx)) |
---|
768 | { |
---|
769 | ERROR("Every variable needs EXACTLY ONE approximation rule, i.e. #(first argument) =#(second argument) ! "); |
---|
770 | } |
---|
771 | ideal Mon=leadmonomial(Approx[1]); |
---|
772 | int N=size(Matrices); |
---|
773 | int i; |
---|
774 | for(i=2;i<=N;i++) |
---|
775 | { |
---|
776 | Mon=Mon,leadmonomial(Approx[i]); |
---|
777 | } |
---|
778 | kill i; |
---|
779 | poly LCM=lcm(Mon); |
---|
780 | matrix M[nrows(Matrices[1])][ncols(Matrices[1])]; |
---|
781 | int i; |
---|
782 | for(i=1;i<=size(Matrices);i++) |
---|
783 | { |
---|
784 | M=M+(LCM/leadmonomial(Approx[i]))*normalize(Approx[i])[size(@derivatives)]*Matrices[i]; |
---|
785 | } |
---|
786 | kill i; |
---|
787 | return(M); |
---|
788 | } |
---|
789 | example |
---|
790 | { |
---|
791 | "EXAMPLE:";echo=2; |
---|
792 | list D="Ut","Ux","Uy","U"; |
---|
793 | list V="t","x","y"; |
---|
794 | list P="a","b"; |
---|
795 | setinitials(V,D,P); |
---|
796 | list Mat=unitmat(2),unitmat(2); |
---|
797 | list Appr=forward(Ut,U,t),forward(Ux,U,x); |
---|
798 | matrixsystem(Mat,Appr); |
---|
799 | } |
---|
800 | ////////////////////////////////// |
---|
801 | proc timestep(matrix M) |
---|
802 | "USAGE: timestep(M); M a square matrix with polynomials over the basering as entries; |
---|
803 | RETURN: type list; gives two matrices M1,M2 that are the splitting of M with respect to the degree of the variable t in the entries, where the first list-entry M1 consists of the polynomials of the highest timelevel and M2 of the lower levels in the form: M=0 => M1=M2, i.e. M1-M2=M |
---|
804 | NOTE: intended to be used for the finite-difference-scheme-construction and partition into the several time steps |
---|
805 | EXAMPLE: example timestep; shows an example |
---|
806 | " |
---|
807 | { |
---|
808 | int N=nrows(M); |
---|
809 | int i; |
---|
810 | int maxdegT; |
---|
811 | for(i=1;i<=N;i++) |
---|
812 | { |
---|
813 | int j; |
---|
814 | for(j=1;j<=N;j++) |
---|
815 | { |
---|
816 | poly f=M[i,j]; |
---|
817 | int k; |
---|
818 | for(k=1;k<=size(f);k++) |
---|
819 | { |
---|
820 | if(leadexp(M[i,j])[2]>maxdegT) |
---|
821 | { |
---|
822 | maxdegT=leadexp(M[i,j])[2]; |
---|
823 | } |
---|
824 | f=f-lead(f); |
---|
825 | } |
---|
826 | kill f; |
---|
827 | kill k; |
---|
828 | } |
---|
829 | kill j; |
---|
830 | } |
---|
831 | kill i; |
---|
832 | matrix highT[nrows(M)][nrows(M)]; |
---|
833 | vector leftside=0; |
---|
834 | int GenIndex=0; |
---|
835 | int i; |
---|
836 | for(i=1;i<=N;i++) |
---|
837 | { |
---|
838 | int j; |
---|
839 | for(j=1;j<=N;j++) |
---|
840 | { |
---|
841 | poly f=M[i,j]; |
---|
842 | int k; |
---|
843 | for(k=1;k<=size(f)+1;k++) |
---|
844 | { |
---|
845 | if(leadexp(f)[2]==maxdegT) |
---|
846 | { |
---|
847 | GenIndex++; |
---|
848 | highT[i,j]=highT[i,j]+lead(f); |
---|
849 | leftside=leftside+highT[i,j]*gen(GenIndex); |
---|
850 | } |
---|
851 | f=f-lead(f); |
---|
852 | } |
---|
853 | kill k; |
---|
854 | kill f; |
---|
855 | } |
---|
856 | kill j; |
---|
857 | } |
---|
858 | kill i; |
---|
859 | matrix tUpper=highT; |
---|
860 | matrix tLower=-1*(M-tUpper); |
---|
861 | tUpper=tUpper/content(leftside); |
---|
862 | tLower=tLower/content(leftside); |
---|
863 | list L=tUpper,tLower; |
---|
864 | return(L); |
---|
865 | } |
---|
866 | example |
---|
867 | { |
---|
868 | "EXAMPLE:"; echo=2; |
---|
869 | list D="Ut","Ux","Uy","U"; |
---|
870 | list V="t","x","y"; |
---|
871 | list P="a","b"; |
---|
872 | setinitials(V,D,P); |
---|
873 | list Mat=unitmat(2),unitmat(2); |
---|
874 | list Apr=forward(Ut,U,t),forward(Ux,U,x); |
---|
875 | matrixsystem(Mat,Apr); |
---|
876 | timestep(_); |
---|
877 | } |
---|
878 | ////////////////////////////////// |
---|
879 | proc fouriersystem(list Matrices, list Approx) |
---|
880 | "USAGE: fouriersystem(M,A); M a list of matrices, A a list of approximations; |
---|
881 | RETURN: type list; each entry is some matrix obtained by performing the substitution of the single approximations into the system of pde's, partitioning the equation into the several timesteps and fouriertransforming these parts |
---|
882 | EXAMPLE: example fouriersystem; shows an example |
---|
883 | " |
---|
884 | { |
---|
885 | matrix M=matrixsystem(Matrices,Approx); |
---|
886 | matrix T1=timestep(M)[1]; |
---|
887 | matrix T0=timestep(M)[2]; |
---|
888 | int i; |
---|
889 | for(i=1;i<=nrows(M);i++) |
---|
890 | { |
---|
891 | int j; |
---|
892 | for(j=1;j<=nrows(M);j++) |
---|
893 | { |
---|
894 | T1[i,j]=errormap(T1[i,j]); |
---|
895 | T1[i,j]=VarToPar(T1[i,j]); |
---|
896 | T0[i,j]=errormap(T0[i,j]); |
---|
897 | T0[i,j]=VarToPar(T0[i,j]); |
---|
898 | } |
---|
899 | kill j; |
---|
900 | } |
---|
901 | kill i; |
---|
902 | ideal EV1=eigenvals(T1)[1]; |
---|
903 | ideal EV0=eigenvals(T0)[1]; |
---|
904 | list L=list(T1,T0),list(EV1,EV0); |
---|
905 | def N1=size(EV1); |
---|
906 | def N0=size(EV0); |
---|
907 | list CV1; |
---|
908 | list CV0; |
---|
909 | int i; |
---|
910 | for(i=1;i<=N1;i++) |
---|
911 | { |
---|
912 | CV1[i]=VarToPar(EV1[i]); |
---|
913 | if(content(CV1[i])==CV1[i]) |
---|
914 | { |
---|
915 | CV1[i]=content(CV1[i]); |
---|
916 | CV1[i]=VarToPar(ComplexValue(numerator(CV1[i])))/VarToPar(ComplexValue(denominator(CV1[i]))); |
---|
917 | } |
---|
918 | } |
---|
919 | kill i; |
---|
920 | int i; |
---|
921 | for(i=1;i<=N0;i++) |
---|
922 | { |
---|
923 | CV0[i]=VarToPar(EV0[i]); |
---|
924 | if(content(CV0[i])==CV0[i]) |
---|
925 | { |
---|
926 | CV0[i]=content(CV0[i]); |
---|
927 | CV0[i]=VarToPar(ComplexValue(numerator(CV0[i])))/VarToPar(ComplexValue(denominator(CV0[i]))); |
---|
928 | } |
---|
929 | } |
---|
930 | kill i; |
---|
931 | list CV=list(CV1,CV0); |
---|
932 | L=L,CV; |
---|
933 | return(L); |
---|
934 | } |
---|
935 | example |
---|
936 | { |
---|
937 | "EXAMPLE:"; echo = 2; |
---|
938 | list D="Ut","Ux","Uy","U"; |
---|
939 | list V="t","x","y"; |
---|
940 | list P="a","b"; |
---|
941 | setinitials(V,D,P); |
---|
942 | matrix M[2][2]=0,-a,-a,0; |
---|
943 | list Mat=unitmat(2),M; |
---|
944 | list Appr=forward(Ut,U,t),trapezoid(Ux,U,x); |
---|
945 | def s=fouriersystem(Mat,Appr);s; |
---|
946 | } |
---|
947 | ////////////////////////////////// |
---|
948 | proc PartitionVar(poly f,int n) |
---|
949 | "USAGE: PartitionVar(f); f a poly in the basering; |
---|
950 | RETURN: type poly; gives back a list L=f1,f2 obtained by the partition of f into two parts f1,f2 with deg_var_n(f1) >0 deg_var_n(f2)=0 |
---|
951 | EXAMPLE: example PartitionVar; shows an example |
---|
952 | " |
---|
953 | { |
---|
954 | if(n>=nvars(basering)) |
---|
955 | { |
---|
956 | ERROR("this variable does not exist in the current basering"); |
---|
957 | } |
---|
958 | int i; |
---|
959 | poly partition=0; |
---|
960 | poly g=f; |
---|
961 | for(i=1;i<=size(f);i++) |
---|
962 | { |
---|
963 | if(leadexp(g)[n]!=0) |
---|
964 | { |
---|
965 | partition=partition+lead(g); |
---|
966 | } |
---|
967 | g=g-lead(g); |
---|
968 | } |
---|
969 | list L=partition,f-partition; |
---|
970 | return(L); |
---|
971 | } |
---|
972 | example |
---|
973 | { |
---|
974 | "EXAMPLE:"; echo = 2; |
---|
975 | list D="Ut","Ux","Uy","U"; |
---|
976 | list V="t","x","y"; |
---|
977 | list P="a","b"; |
---|
978 | setinitials(V,D,P);////does not show the ring, since there is no output |
---|
979 | basering;///does show the ring |
---|
980 | poly f=t**3*cx**2-cy**2*dt+i**3*sx; |
---|
981 | PartitionVar(f,1); ////i is the first variable |
---|
982 | } |
---|
983 | ////////////////////////////////// |
---|
984 | proc ComplexValue(poly f) |
---|
985 | "USAGE: ComplexValue(f); f a poly in the basering; |
---|
986 | RETURN: type poly; gives back the formal complex-value of f, where var(1) is redarded as the imaginary unit. Does only make sence, if the proc <setinitials> is executed before -> nvars <= npars |
---|
987 | EXAMPLE: example ComplexValue; shows an example |
---|
988 | " |
---|
989 | { |
---|
990 | poly g=ParToVar(f); |
---|
991 | def L=PartitionVar(g,1); |
---|
992 | poly f1=subst(L[1],var(1),1); |
---|
993 | poly f2=L[2]; |
---|
994 | poly result=reduce(f1**2+f2**2,std(0)); |
---|
995 | return(result); |
---|
996 | } |
---|
997 | example |
---|
998 | { |
---|
999 | "EXAMPLE:"; echo = 2; |
---|
1000 | list D="Ut","Ux","Uy","U"; |
---|
1001 | list V="t","x","y"; |
---|
1002 | list P="a","b"; |
---|
1003 | setinitials(V,D,P);////does not show the ring, as there is no output |
---|
1004 | basering;///does show the ring |
---|
1005 | poly f=t**3*cx**2-cy**2*dt+i**3*sx; |
---|
1006 | f; |
---|
1007 | VarToPar(f); |
---|
1008 | } |
---|
1009 | ////////////////////////////////// |
---|
1010 | proc VarToPar(poly f) |
---|
1011 | "USAGE: VarToPar(f); f a poly in the basering; |
---|
1012 | RETURN: type poly; gives back the poly obtained by substituting var(i) by par(i), for all variables. Does only make sence, if the proc <setinitials> is executed before -> nvars <= npars; |
---|
1013 | EXAMPLE: example VarToPar; shows an example |
---|
1014 | " |
---|
1015 | { |
---|
1016 | int N=nvars(basering); |
---|
1017 | int i; |
---|
1018 | def g=f; |
---|
1019 | for(i=1;i<=N;i++) |
---|
1020 | { |
---|
1021 | g=subst(g,var(i),par(i)); |
---|
1022 | } |
---|
1023 | return(g); |
---|
1024 | } |
---|
1025 | example |
---|
1026 | { |
---|
1027 | "EXAMPLE:"; echo = 2; |
---|
1028 | list D="Ut","Ux","Uy","U"; |
---|
1029 | list V="t","x","y"; |
---|
1030 | list P="a","b"; |
---|
1031 | setinitials(V,D,P);////does not show the ring, as there is no output |
---|
1032 | basering;///does show the ring |
---|
1033 | poly f=t**3*cx**2-cy**2*dt+i**3*sx; |
---|
1034 | f; |
---|
1035 | VarToPar(f); |
---|
1036 | } |
---|
1037 | ///////////////////////////////////// |
---|
1038 | proc ParToVar(poly f) |
---|
1039 | "USAGE: ParToVar(f); f a poly in the basering; |
---|
1040 | RETURN: type poly; gives back the poly obtained by substituting par(i) by var(i), for the first nvars(basering parameters. Does only make sence, if setinitials is executed before -> nvars <= npars. Is the opposite action to VarToPar, see example ParToVar; |
---|
1041 | EXAMPLE: example ParToVar; shows an example |
---|
1042 | " |
---|
1043 | { |
---|
1044 | int N=nvars(basering); |
---|
1045 | int i; |
---|
1046 | number g=number(VarToPar(f)); |
---|
1047 | number denom=denominator(g); |
---|
1048 | g=denom*g; |
---|
1049 | def gg=subst(g,par(1),var(1)); |
---|
1050 | for(i=2;i<=N;i++) |
---|
1051 | { |
---|
1052 | gg=subst(gg,par(i),var(i)); |
---|
1053 | } |
---|
1054 | return(gg/denom); |
---|
1055 | } |
---|
1056 | example |
---|
1057 | { |
---|
1058 | "EXAMPLE:"; echo = 2; |
---|
1059 | list D="Ut","Ux","Uy","U"; |
---|
1060 | list V="t","x","y"; |
---|
1061 | list P="a","b"; |
---|
1062 | setinitials(V,D,P);////does not show the ring, as there is no output |
---|
1063 | basering;///does show the ring |
---|
1064 | poly f=t**3*cx**2-cy**2*dt+i**3*sx/dt*dx; |
---|
1065 | f; |
---|
1066 | def g=VarToPar(f); |
---|
1067 | g; |
---|
1068 | def h=ParToVar(g); |
---|
1069 | h==f; |
---|
1070 | } |
---|
1071 | ///////////////////////////////////////// |
---|
1072 | proc qepcad(poly f) |
---|
1073 | "USAGE: qepcad(f); f a poly in the basering; |
---|
1074 | RETURN: type list; gives back some constraints that are equivalent to f<1 (computed by QEPCAD); |
---|
1075 | EXAMPLE: example qepcad; shows an example |
---|
1076 | " |
---|
1077 | { |
---|
1078 | // how to test, whether QEPCAD is installed? |
---|
1079 | createQCfilter(); // creates/overwrites qepcadfilter.pl |
---|
1080 | system("sh","rm -f QEPCAD-out"); |
---|
1081 | system("sh","rm -f QEPCAD-in"); |
---|
1082 | if(denominator(content(f))==1) |
---|
1083 | { |
---|
1084 | poly g=f-1; |
---|
1085 | } |
---|
1086 | else |
---|
1087 | { |
---|
1088 | if(f==content(f)) |
---|
1089 | { |
---|
1090 | poly g=f*denominator(content(f))-1*denominator(content(f)); |
---|
1091 | g=ParToVar(g); |
---|
1092 | g=reduce(g,std(0)); |
---|
1093 | } |
---|
1094 | else |
---|
1095 | { |
---|
1096 | poly g=cleardenom(f)-1/content(f); |
---|
1097 | g=ParToVar(g); |
---|
1098 | g=reduce(g,std(0)); |
---|
1099 | } |
---|
1100 | } |
---|
1101 | string in="QEPCAD-in"; |
---|
1102 | string out="QEPCAD-out"; |
---|
1103 | link l1=in; |
---|
1104 | link l2=out; |
---|
1105 | string s1="[trial]"; //description |
---|
1106 | string s2=varlist(); //the variables |
---|
1107 | string s3=nfreevars(); //number of free variables |
---|
1108 | string s4=aquantor()+"["+writepoly(g)+rel()+"]."; //the input prenex formula |
---|
1109 | string s5=projection(); |
---|
1110 | string s6=projection(); |
---|
1111 | string s7=choice(); |
---|
1112 | string s8=solution(); |
---|
1113 | write(l1,s1,s2,s3,s4,s5,s6,s7,s8); |
---|
1114 | system("sh","qepcad < QEPCAD-in | qepcadfilter.pl > QEPCAD-out"); |
---|
1115 | string output=read(out); |
---|
1116 | print(output,"%s"); |
---|
1117 | if(size(output)==0) |
---|
1118 | { |
---|
1119 | return("Try manually"); //maybe too few cells |
---|
1120 | } |
---|
1121 | if(find(output,"FALSE")!=0) |
---|
1122 | { |
---|
1123 | return("FALSE"); |
---|
1124 | } |
---|
1125 | if(find(output,"WARNING")!=0) |
---|
1126 | { |
---|
1127 | return("WARNING! Try manually"); |
---|
1128 | } |
---|
1129 | else |
---|
1130 | { |
---|
1131 | string strpolys=findthepoly(output); |
---|
1132 | list lpolys=listpolynew(strpolys); |
---|
1133 | return(lpolys); |
---|
1134 | } |
---|
1135 | system("sh","rm -f QEPCAD-out"); |
---|
1136 | system("sh","rm -f QEPCAD-in"); |
---|
1137 | |
---|
1138 | } |
---|
1139 | example |
---|
1140 | { |
---|
1141 | "EXAMPLE:"; echo = 2; |
---|
1142 | list D="Ux","Ut","U"; |
---|
1143 | list P="a"; |
---|
1144 | list V="t","x"; |
---|
1145 | setinitials(V,D,P); |
---|
1146 | def s1=scheme(u(Ut)+a*u(Ux),laxfrX(Ux,U,x),laxfrT(Ut,U,x)); |
---|
1147 | s1; |
---|
1148 | def s2=errormap(s1); |
---|
1149 | s2; |
---|
1150 | def s3=ComplexValue(s2);s3; |
---|
1151 | qepcad(s3); |
---|
1152 | } |
---|
1153 | /////////////////////////////////////////// |
---|
1154 | proc createQCfilter() |
---|
1155 | { |
---|
1156 | // writes the following to the file qepcadfilter.pl |
---|
1157 | // is there already such a file? remove it! |
---|
1158 | system("sh","rm -f qepcadfilter.pl"); |
---|
1159 | link l=":w qepcadfilter.pl"; |
---|
1160 | write(l, "#!/usr/bin/perl"); |
---|
1161 | write(l, "$flag = 0;"); |
---|
1162 | write(l, "$res = \"\";"); |
---|
1163 | write(l,"while(<>)"); |
---|
1164 | write(l,"{ if ($_ =~ /Warning|WARNING|warning|Error|error|ERROR/) { print $_; }"); |
---|
1165 | write(l,"elsif ($_ =~ /An\ equivalent/) { $flag = 1; }"); |
---|
1166 | write(l,"elsif ($flag == 1 && $_ ne \"\n\") { print $_; $flag = 0; } }"); |
---|
1167 | } |
---|
1168 | |
---|
1169 | /////////////////////////////////////////// |
---|
1170 | proc qepcadsystem(list l) |
---|
1171 | "USAGE: qepcadsytem(f); l a list; |
---|
1172 | RETURN: list |
---|
1173 | PURPOSE: gives back some constraints that are equivalent to the |
---|
1174 | eigenvalues of the matrices in the list l being < 1 (computed by QEPCAD) |
---|
1175 | EXAMPLE: example qepcadsystem; shows an example |
---|
1176 | " |
---|
1177 | { |
---|
1178 | // how to test, whether QEPCAD is installed? |
---|
1179 | createQCfilter(); // creates/overwrites qepcadfilter.pl |
---|
1180 | system("sh","rm -f QEPCAD-out"); |
---|
1181 | system("sh","rm -f QEPCAD-in"); |
---|
1182 | string in="QEPCAD-in"; |
---|
1183 | string out="QEPCAD-out"; |
---|
1184 | link l1=in; |
---|
1185 | link l2=out; |
---|
1186 | string s1="[trial]"; //description |
---|
1187 | string s2=varlist(); //the variables |
---|
1188 | string s3=nfreevars(); //number of free variables |
---|
1189 | string thepolys; |
---|
1190 | int n2=size(l[2]); |
---|
1191 | int count; |
---|
1192 | int i; |
---|
1193 | list lpolys; |
---|
1194 | int j; |
---|
1195 | for(j=1;j<=n2;j++) |
---|
1196 | { |
---|
1197 | count++; |
---|
1198 | poly g2=ParToVar(l[2][j]); |
---|
1199 | if(denominator(content(g2))==1) |
---|
1200 | { |
---|
1201 | lpolys[count]=writepoly(ParToVar(reduce(g2-1,std(0))))+rel(); |
---|
1202 | } |
---|
1203 | else |
---|
1204 | { |
---|
1205 | if(g2==content(g2)) |
---|
1206 | { |
---|
1207 | g2=g2*denominator(content(g2))-1*denominator(content(g2)); |
---|
1208 | g2=ParToVar(g2); |
---|
1209 | g2=reduce(g2,std(0)); |
---|
1210 | lpolys[count]=writepoly(g2)+rel(); |
---|
1211 | } |
---|
1212 | else |
---|
1213 | { |
---|
1214 | lpolys[count]=writepoly(reduce(ParToVar(cleardenom(g2)-1/content(g2)),std(0)))+rel(); |
---|
1215 | } |
---|
1216 | } |
---|
1217 | kill g2; |
---|
1218 | } |
---|
1219 | kill j; |
---|
1220 | int k; |
---|
1221 | for(k=1;k<=size(lpolys);k++) |
---|
1222 | { |
---|
1223 | thepolys=thepolys+lpolys[k]; |
---|
1224 | if(k<size(lpolys)) |
---|
1225 | { |
---|
1226 | thepolys=thepolys+print(" /","s%")+print("\\ ","s%"); |
---|
1227 | } |
---|
1228 | } |
---|
1229 | string s4=aquantor()+"["+thepolys+"]."; //the input prenex formula |
---|
1230 | string s5=projection(); |
---|
1231 | string s6=projection(); |
---|
1232 | string s7=choice(); |
---|
1233 | string s8=solution(); |
---|
1234 | write(l1,s1,s2,s3,s4,s5,s6,s7,s8); |
---|
1235 | system("sh","qepcad < QEPCAD-in | qepcadfilter.pl > QEPCAD-out"); |
---|
1236 | string output=read(out); |
---|
1237 | print(output,"%s"); |
---|
1238 | if(size(output)==0) |
---|
1239 | { |
---|
1240 | ERROR("Try manually"); //maybe too few cells |
---|
1241 | } |
---|
1242 | if(find(output,"FALSE")!=0) |
---|
1243 | { |
---|
1244 | ERROR("FALSE"); |
---|
1245 | } |
---|
1246 | if(find(output,"WARNING")!=0) |
---|
1247 | { |
---|
1248 | ERROR("WARNING! Try manually"); |
---|
1249 | } |
---|
1250 | else |
---|
1251 | { |
---|
1252 | string strpolys=findthepoly(output); |
---|
1253 | list llpolys=listpolynew(strpolys); |
---|
1254 | return(llpolys); |
---|
1255 | } |
---|
1256 | system("sh","rm -f QEPCAD-out"); |
---|
1257 | system("sh","rm -f QEPCAD-in"); |
---|
1258 | } |
---|
1259 | example |
---|
1260 | { |
---|
1261 | "EXAMPLE:"; echo = 2; |
---|
1262 | list D="Ut","Ux","Uy","U"; |
---|
1263 | list V="t","x","y"; |
---|
1264 | list P="a","b"; |
---|
1265 | setinitials(V,D,P); |
---|
1266 | matrix M[2][2]=0,-a,-a,0; |
---|
1267 | list Mat=unitmat(2),M; |
---|
1268 | list Appr=forward(Ut,U,t),forward(Ux,U,x); |
---|
1269 | //matrixsystem(Mat,Appr); |
---|
1270 | //timestep(_); |
---|
1271 | fouriersystem(Mat,Appr); |
---|
1272 | qepcadsystem(_[2]); |
---|
1273 | } |
---|
1274 | /////////////////////////////////////////// |
---|
1275 | static proc substbracketstar(string s) |
---|
1276 | { |
---|
1277 | int i; |
---|
1278 | int k; |
---|
1279 | int index=1; |
---|
1280 | string finish=s; |
---|
1281 | for(k=1;k<=size(s);k++) |
---|
1282 | { |
---|
1283 | if(finish[1]=="(" or finish[1]=="*" or finish[1]==" ") |
---|
1284 | { |
---|
1285 | kill finish; |
---|
1286 | index=index+1; |
---|
1287 | string finish=s[index..size(s)]; |
---|
1288 | } |
---|
1289 | } |
---|
1290 | for(i=1;i<=size(finish);i++) |
---|
1291 | { |
---|
1292 | if(finish[i]=="*" or finish[i]=="(" or finish[i]== ")") |
---|
1293 | { |
---|
1294 | finish[i]=" "; |
---|
1295 | } |
---|
1296 | } |
---|
1297 | return(finish); |
---|
1298 | } |
---|
1299 | |
---|
1300 | //////////////////////////////////// |
---|
1301 | static proc distribution(string SUM, string MON) |
---|
1302 | { |
---|
1303 | string sum=substbracketstar(SUM); |
---|
1304 | string mon=substbracketstar(MON); |
---|
1305 | string result; |
---|
1306 | list signs; |
---|
1307 | list p; |
---|
1308 | int i; |
---|
1309 | int j; |
---|
1310 | int init; |
---|
1311 | for(i=2;i<=size(sum);i++) |
---|
1312 | { |
---|
1313 | if(sum[i]=="+" or sum[i]=="-") |
---|
1314 | { |
---|
1315 | j++; |
---|
1316 | p[j]=i; |
---|
1317 | } |
---|
1318 | } |
---|
1319 | if(j==0) |
---|
1320 | { |
---|
1321 | if(sum[1]!="-") |
---|
1322 | { |
---|
1323 | result=sum+" "+" "+mon; |
---|
1324 | result="+"+" "+result; |
---|
1325 | } |
---|
1326 | else |
---|
1327 | { |
---|
1328 | result=sum+" "+mon; |
---|
1329 | } |
---|
1330 | } |
---|
1331 | else |
---|
1332 | { |
---|
1333 | int l; |
---|
1334 | int anfang; |
---|
1335 | if(sum[1]=="-") |
---|
1336 | { |
---|
1337 | result="-"+" "+result; |
---|
1338 | anfang=2; |
---|
1339 | } |
---|
1340 | else |
---|
1341 | { |
---|
1342 | result="+"+" "+result; |
---|
1343 | if(sum[1]=="+") |
---|
1344 | { |
---|
1345 | anfang=2; |
---|
1346 | } |
---|
1347 | else |
---|
1348 | { |
---|
1349 | anfang=1; |
---|
1350 | } |
---|
1351 | } |
---|
1352 | for(l=1;l<=j;l++) |
---|
1353 | { |
---|
1354 | string a; |
---|
1355 | int k; |
---|
1356 | for(k=anfang;k<=p[l]-1;k++) |
---|
1357 | { |
---|
1358 | a=a+sum[k]; |
---|
1359 | } |
---|
1360 | result=result+" "+a+" "+mon+" "+sum[p[l]]; |
---|
1361 | anfang=p[l]+1; |
---|
1362 | kill a; |
---|
1363 | kill k; |
---|
1364 | } |
---|
1365 | if(p[j]<size(sum)) |
---|
1366 | { |
---|
1367 | int kk; |
---|
1368 | string aa; |
---|
1369 | for(kk=anfang;kk<=size(sum);kk++) |
---|
1370 | { |
---|
1371 | aa=aa+sum[kk]; |
---|
1372 | } |
---|
1373 | result=result+" "+aa+" "+mon; |
---|
1374 | kill aa; |
---|
1375 | kill kk; |
---|
1376 | } |
---|
1377 | else |
---|
1378 | { |
---|
1379 | int kkk; |
---|
1380 | string aaa; |
---|
1381 | for(kkk=anfang;kkk<size(sum);kkk++) |
---|
1382 | { |
---|
1383 | aaa=aaa+sum[kkk]; |
---|
1384 | } |
---|
1385 | result=result+" "+aaa+" "+mon; |
---|
1386 | kill aaa; |
---|
1387 | kill kkk; |
---|
1388 | } |
---|
1389 | } |
---|
1390 | return(result); |
---|
1391 | } |
---|
1392 | ///////////////////////////////////////////////////////////////// |
---|
1393 | static proc writepoly(poly f) |
---|
1394 | { |
---|
1395 | poly g=f; |
---|
1396 | string lc; |
---|
1397 | string lm; |
---|
1398 | string strpoly; |
---|
1399 | string intermediate; |
---|
1400 | int n=size(f); |
---|
1401 | int i; |
---|
1402 | for(i=1;i<=n;i++) |
---|
1403 | { |
---|
1404 | |
---|
1405 | lc=substbracketstar(string(leadcoef(g))); |
---|
1406 | lm=substbracketstar(string(leadmonom(g))); |
---|
1407 | intermediate=distribution(lc,lm); |
---|
1408 | strpoly=strpoly+" "+intermediate; |
---|
1409 | g=g-lead(g); |
---|
1410 | } |
---|
1411 | return(strpoly); |
---|
1412 | } |
---|
1413 | /////////////////////////////////////////////////////////////// |
---|
1414 | static proc varlist() |
---|
1415 | { |
---|
1416 | poly p1=par(2+size(@variables)); |
---|
1417 | p1=ParToVar(p1); |
---|
1418 | string name=print(p1,"%s"); |
---|
1419 | string p="("+name+","; |
---|
1420 | int i; |
---|
1421 | for(i=3+size(@variables);i<=npars(basering);i++) |
---|
1422 | { |
---|
1423 | p1=ParToVar(par(i)); |
---|
1424 | name=substbracketstar(print(p1,"%s")); |
---|
1425 | p=p+name+","; |
---|
1426 | } |
---|
1427 | p1=ParToVar(par(2)); |
---|
1428 | name=print(p1,"%s"); |
---|
1429 | p=p+name+")"; |
---|
1430 | return(p); |
---|
1431 | } |
---|
1432 | /////////////////////////////////////////////////////// |
---|
1433 | static proc normalization() |
---|
1434 | { |
---|
1435 | int n1=npars(basering)-size(@variables); |
---|
1436 | int n2=npars(basering)-n1; |
---|
1437 | string assumption="assume["+" "+substbracketstar(print(par(n1+1),"%s"))+" >0"; |
---|
1438 | int i; |
---|
1439 | for(i=2;i<=n2;i++) |
---|
1440 | { |
---|
1441 | string s=" /\\"+" "+substbracketstar(print(par(n1+i),"%s"))+" >0"; |
---|
1442 | assumption=assumption+" "+s; |
---|
1443 | kill s; |
---|
1444 | } |
---|
1445 | assumption=assumption+" ]"+newline+"go"; |
---|
1446 | return(assumption); |
---|
1447 | } |
---|
1448 | /////////////////////////////////////////////////////// |
---|
1449 | static proc projection() |
---|
1450 | { |
---|
1451 | string s="go"; |
---|
1452 | return(s); |
---|
1453 | } |
---|
1454 | /////////////////////////////////////////////////////// |
---|
1455 | static proc choice() |
---|
1456 | { |
---|
1457 | string s="go"; |
---|
1458 | return(s); |
---|
1459 | } |
---|
1460 | /////////////////////////////////////////////////////// |
---|
1461 | static proc solution() |
---|
1462 | { |
---|
1463 | string s="go"; |
---|
1464 | return(s); |
---|
1465 | } |
---|
1466 | /////////////////////////////////////////////////////// |
---|
1467 | static proc nfreevars() |
---|
1468 | { |
---|
1469 | int n=npars(basering)-size(@variables)-1; |
---|
1470 | string no=print(n,"%s"); |
---|
1471 | return(no); |
---|
1472 | } |
---|
1473 | ///////////////////////////////////////////////////////// |
---|
1474 | static proc aquantor() |
---|
1475 | { |
---|
1476 | string s="(A "+print(var(2),"%s")+")"; |
---|
1477 | return(s); |
---|
1478 | } |
---|
1479 | //////////////////////////////////////////////////////// |
---|
1480 | static proc rel() |
---|
1481 | { |
---|
1482 | return("<0"); |
---|
1483 | } |
---|
1484 | ///////////////////////////////////////////////////////// |
---|
1485 | static proc rm() |
---|
1486 | { |
---|
1487 | system("sh","rm -f dummy"); |
---|
1488 | system("sh","rm -f QEPCAD-in"); |
---|
1489 | } |
---|
1490 | //////////////////////////////////////////////////////// |
---|
1491 | static proc findthepoly(string word) |
---|
1492 | { |
---|
1493 | int init=1; |
---|
1494 | int index=find(word,newline); |
---|
1495 | if(index==size(word) or index== 0) |
---|
1496 | { |
---|
1497 | return(word); |
---|
1498 | } |
---|
1499 | else |
---|
1500 | { |
---|
1501 | while(index<size(word)) |
---|
1502 | { |
---|
1503 | init=index; |
---|
1504 | index=find(word,newline,index+1); |
---|
1505 | } |
---|
1506 | } |
---|
1507 | string thepoly=word[(init+1)..(size(word)-1)]; |
---|
1508 | return(thepoly); |
---|
1509 | } |
---|
1510 | ////////////////////////////////////////////////// |
---|
1511 | static proc listpolynew(string thepoly) |
---|
1512 | { |
---|
1513 | string p=thepoly; |
---|
1514 | intvec v; |
---|
1515 | p=TestAndDelete(p,"[")[1]; |
---|
1516 | p=TestAndDelete(p,"]")[1]; //remove the brackets |
---|
1517 | string AND="/"+"\\"; |
---|
1518 | string OR="\\"+"/"; |
---|
1519 | list thesigns=AND,OR,"~","<==>","==>","<==";///these can occur in QEPCAD |
---|
1520 | int i; |
---|
1521 | for(i=1;i<=size(thesigns);i++) |
---|
1522 | { |
---|
1523 | list l=TestAndDelete(p,thesigns[i]); |
---|
1524 | p=l[1]; |
---|
1525 | v=addintvec(v,l[2]); |
---|
1526 | kill l; |
---|
1527 | } |
---|
1528 | intvec w=rightorder(v); |
---|
1529 | int N=size(v); |
---|
1530 | list lstrings; |
---|
1531 | if(size(w)==1 and w[1]==0) |
---|
1532 | { |
---|
1533 | N=0; |
---|
1534 | lstrings[1]=p; |
---|
1535 | } |
---|
1536 | else |
---|
1537 | { |
---|
1538 | string s1=p[1..w[1]-1]; |
---|
1539 | lstrings[1]=s1; |
---|
1540 | int j; |
---|
1541 | for(j=1;j<N;j++) |
---|
1542 | { |
---|
1543 | string s2=p[w[j]..w[j+1]-1]; |
---|
1544 | lstrings[j+1]=s2; |
---|
1545 | kill s2; |
---|
1546 | } |
---|
1547 | string s3=p[w[N]..size(p)]; |
---|
1548 | lstrings[N+1]=s3; |
---|
1549 | } |
---|
1550 | list cutstrings; |
---|
1551 | int k; |
---|
1552 | list Id; |
---|
1553 | for(k=1;k<=N+1;k++) |
---|
1554 | { |
---|
1555 | cutstrings[k]=cutoffrel(lstrings[k]); |
---|
1556 | Id[k]=translatenew(cutstrings[k]); |
---|
1557 | } |
---|
1558 | return(Id); |
---|
1559 | |
---|
1560 | |
---|
1561 | } |
---|
1562 | ///////////////////////////////////////////////// |
---|
1563 | static proc translatenew(string word) |
---|
1564 | { |
---|
1565 | int n=size(word); |
---|
1566 | string subword=word[1..n]; |
---|
1567 | string strpoly="poly f="; |
---|
1568 | list linterim; |
---|
1569 | int i; |
---|
1570 | for(i=1;i<=n;i++) |
---|
1571 | { |
---|
1572 | if(i<n and subword[i]==" " and subword[i+1]!="+" and subword[i+1]!="-" and subword[i+1]!=" " and subword[i-1]!="+" and subword[i-1]!="-" and subword[i-1]!=" " and i>1) |
---|
1573 | { |
---|
1574 | linterim[i]="*"; |
---|
1575 | } |
---|
1576 | else |
---|
1577 | { |
---|
1578 | linterim[i]=word[i]; |
---|
1579 | } |
---|
1580 | strpoly=strpoly+print(linterim[i],"%s"); |
---|
1581 | } |
---|
1582 | strpoly=strpoly+";"; |
---|
1583 | execute(strpoly); |
---|
1584 | return(f); |
---|
1585 | } |
---|
1586 | ////////////////////////////////////////////////// |
---|
1587 | static proc rightorder(intvec v) |
---|
1588 | ////////////orders the entries of an intvec from small to bigger |
---|
1589 | { |
---|
1590 | list ldown=intveclist(v); |
---|
1591 | list lup; |
---|
1592 | intvec v1; |
---|
1593 | int counter; |
---|
1594 | int min; |
---|
1595 | int i; |
---|
1596 | for(i=1;i<=size(v);i++) |
---|
1597 | { |
---|
1598 | min=Min(listintvec(ldown)); |
---|
1599 | lup[i]=min; |
---|
1600 | counter=listfind(ldown,min); |
---|
1601 | ldown=delete(ldown,counter); |
---|
1602 | } |
---|
1603 | intvec result=listintvec(lup); |
---|
1604 | return(result); |
---|
1605 | } |
---|
1606 | ///////////////////////////////////////////////// |
---|
1607 | static proc listfind(list l, int n) |
---|
1608 | //////gives the position of n in l, 0 if not found |
---|
1609 | { |
---|
1610 | int i; |
---|
1611 | intvec counter; |
---|
1612 | int j=1; |
---|
1613 | for(i=1;i<=size(l);i++) |
---|
1614 | { |
---|
1615 | if(n==l[i]) |
---|
1616 | { |
---|
1617 | counter[j]=i; |
---|
1618 | j++; |
---|
1619 | } |
---|
1620 | } |
---|
1621 | int result=Min(counter); |
---|
1622 | return(result); |
---|
1623 | } |
---|
1624 | ////////////////////////////////////////////////// |
---|
1625 | static proc cutoffrel(string str) |
---|
1626 | /////////cut off the relations "/=,<= etc." from output-string |
---|
1627 | { |
---|
1628 | list rels="<=",">=","<",">","/=","="; //these are all of qepcad's relations |
---|
1629 | int i; |
---|
1630 | list l; |
---|
1631 | for(i=1;i<=size(rels);i++) |
---|
1632 | { |
---|
1633 | l[i]=find(str,rels[i]); |
---|
1634 | } |
---|
1635 | intvec v=listintvec(l); |
---|
1636 | v=addintvec(v,v); |
---|
1637 | int position=Min(v); |
---|
1638 | string result; |
---|
1639 | if(position==0) |
---|
1640 | { |
---|
1641 | result=str; |
---|
1642 | } |
---|
1643 | else |
---|
1644 | { |
---|
1645 | result=str[1..position-1]; |
---|
1646 | } |
---|
1647 | return(result); |
---|
1648 | } |
---|
1649 | ////////////////////////////////////////////////// |
---|
1650 | static proc addintvec(intvec v1, intvec v2) |
---|
1651 | ///////concatenates two intvecs and deletes occurring zeroentries, output intvec |
---|
1652 | { |
---|
1653 | list lresult; |
---|
1654 | int i; |
---|
1655 | list l1; |
---|
1656 | list l2; |
---|
1657 | for(i=1;i<=size(v1);i++) |
---|
1658 | { |
---|
1659 | l1[i]=v1[i]; |
---|
1660 | } |
---|
1661 | kill i; |
---|
1662 | int k; |
---|
1663 | for(k=1;k<=size(v2);k++) |
---|
1664 | { |
---|
1665 | l2[k]=v2[k]; |
---|
1666 | } |
---|
1667 | lresult=l1+l2; |
---|
1668 | intvec result; |
---|
1669 | int j; |
---|
1670 | int counter=1; |
---|
1671 | for(j=1;j<=size(lresult);j++) |
---|
1672 | { |
---|
1673 | if(lresult[j]!=0) |
---|
1674 | { |
---|
1675 | result[counter]=lresult[j]; |
---|
1676 | counter++; |
---|
1677 | } |
---|
1678 | } |
---|
1679 | return(result); |
---|
1680 | |
---|
1681 | } |
---|
1682 | ///////////////////////////////////////////////// |
---|
1683 | static proc intveclist(intvec v) |
---|
1684 | //////returns the intvec as list |
---|
1685 | { |
---|
1686 | list l; |
---|
1687 | int i; |
---|
1688 | for(i=size(v);i>=1;i--) |
---|
1689 | { |
---|
1690 | l[i]=v[i]; |
---|
1691 | } |
---|
1692 | return(l); |
---|
1693 | } |
---|
1694 | ////////////////////////////////////////////////// |
---|
1695 | static proc listintvec(list l) |
---|
1696 | //////////returns the list as intvec: only integer-entries allowed |
---|
1697 | { |
---|
1698 | intvec v; |
---|
1699 | int i; |
---|
1700 | for(i=size(l);i>=1;i--) |
---|
1701 | { |
---|
1702 | v[i]=l[i]; |
---|
1703 | } |
---|
1704 | return(v); |
---|
1705 | } |
---|
1706 | ////////////////////////////////////////////////// |
---|
1707 | static proc TestAndDelete(string str, string booleansign) |
---|
1708 | { |
---|
1709 | int decide=find(str,booleansign); |
---|
1710 | intvec v; |
---|
1711 | v[1]=decide; |
---|
1712 | while(decide!=0 and decide<size(str))//if booleansign ist not contained in str |
---|
1713 | { |
---|
1714 | int n=size(v); |
---|
1715 | decide=find(str,booleansign,decide+1); |
---|
1716 | if(decide!=0) |
---|
1717 | { |
---|
1718 | v[n+1]=decide; |
---|
1719 | } |
---|
1720 | kill n; |
---|
1721 | } |
---|
1722 | if(size(v)==1 and v[1]==0)/////i.e. booleansign NOT in str |
---|
1723 | { |
---|
1724 | list result=str,v; |
---|
1725 | return(result); |
---|
1726 | } |
---|
1727 | else |
---|
1728 | { |
---|
1729 | list l; |
---|
1730 | string p; |
---|
1731 | int i; |
---|
1732 | int j; |
---|
1733 | for(i=1;i<=size(str);i++)///copy the string str into l |
---|
1734 | { |
---|
1735 | l[i]=str[i]; |
---|
1736 | } |
---|
1737 | kill i; |
---|
1738 | for(j=1;j<=size(v);j++)///replace booleansign by empty char |
---|
1739 | { |
---|
1740 | int k; |
---|
1741 | for(k=0;k<=size(booleansign)-1;k++) |
---|
1742 | { |
---|
1743 | l[v[j]+k]=" "; |
---|
1744 | } |
---|
1745 | kill k; |
---|
1746 | } |
---|
1747 | int s; |
---|
1748 | for(s=1;s<=size(l);s++)///copy back |
---|
1749 | { |
---|
1750 | p=p+string(l[s]); |
---|
1751 | } |
---|
1752 | kill s; |
---|
1753 | kill l; |
---|
1754 | list result=p,v; |
---|
1755 | return(result); |
---|
1756 | } |
---|
1757 | } |
---|
1758 | ////////////////////////////////////////////////// |
---|
1759 | static proc nchoice(int n1,int n2) |
---|
1760 | { |
---|
1761 | int N2=maximal(n1,n2); |
---|
1762 | int N1=minimal(n1,n2); |
---|
1763 | if(N1==0) |
---|
1764 | { |
---|
1765 | return(N2); |
---|
1766 | } |
---|
1767 | else |
---|
1768 | { |
---|
1769 | return(N1); |
---|
1770 | } |
---|
1771 | } |
---|