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1 : sh002i 1050 #163macros.pl
2 :     #macros for Prills 163 problems
3 :    
4 :     #!/usr/bin/perl -w
5 :     #use strict;
6 :     #use Carp;
7 :     BEGIN {
8 :     be_strict();#all variables must be declared local or global
9 :     }
10 :    
11 :     #my @answer = oldivy(1,2,1,8,4);
12 :     #print ("The old program says:\n");
13 :     #print ($answer[0]);
14 :     #print ("\n");
15 :     #@answer = ivy(1,2,1,8,4);
16 :     #print ("My program says:\n");
17 :     #print ($answer[0]);
18 :     #print ("\n");
19 :    
20 :     #the subroutine is invoked with arguments such as complexmult(2,3,4,5) or
21 :     #complexmult(@data), where @data = (2,3,4,5)
22 :    
23 :     sub complexmult {
24 :     my ($S,$T,$U,$V) = @_;
25 :     #this line defines the input arguments as local variables
26 :     my $R = $S *$U -$T * $V;
27 :     my $I = $S *$V + $T * $U ;
28 :     ($R,$I) ;#this returns ($R,$I) from the subroutine
29 :     }
30 :     ##########
31 :     #sub addtwo adds two strings formally
32 :     #An "indicator" for a string is a
33 :     # number ,e.g. coefficient,which indicates
34 :     #whether the string is to be
35 :     #added or is to be regarded as zero.
36 :     #The non-zero terms are formally added as strings.
37 :     #The input is an array
38 :     #($1staddend, $1stindicator,$2ndaddend,$2ndindicator)
39 :     #The return is an array
40 :     #(formal sum, indicator of formal sum)
41 :    
42 :     sub addtwo {
43 :     my ($A,$a,$B,$b) = @_;
44 :     my $out = "0";
45 :     if ($a != 0) {
46 :     $out= $A;
47 :     }
48 :    
49 :    
50 :     if ($b != 0) {
51 :     if ($a == 0) {
52 :     $out= $B ;
53 :     } else {
54 :     $out = "$A + $B";
55 :     }
56 :     }
57 :     my $ind = abs($a) + abs($b);
58 :     ($out,$ind);
59 :     }
60 :     ########
61 :    
62 :     #sub add generalizes sub addtwo to more addends.
63 :     #It formally adds the nonzero terms.
64 :     # The input is an array of even length
65 :     # consisting of each addend,a string,
66 :     # followed by its indicator.
67 :    
68 :     sub add {
69 :     # this function takes the first two terms, puts them together, and keep repeating until you have emptied the list.
70 :     my @sum = ("0" ,0);
71 :     my @list = @_;
72 :     my $el = @list;
73 :     my $x = "0";
74 :     my $y = 0;
75 :     while ($el > 0) {
76 :     $x = shift(@list);
77 :     $y = shift(@list);
78 :     push(@sum,$x,$y);
79 :     @sum = addtwo(@sum);
80 :     $el = @list;
81 :     }
82 :     @sum ;
83 :     }
84 :     #######
85 :    
86 :     # sub diffop cleans up the typed expression
87 :     #of a diff. operator.
88 :     #input @diffop =($A,$B,$C) is the coefficients.
89 :     #input is given as arguments viz difftop($A,$B,$C);
90 :     #output is the diff. operator as a string $L in TEX
91 :    
92 :     sub diffop
93 :     {
94 :     my ($A,$B,$C) = @_ ;
95 :     my ($LDD, $LD ,$LF) = ($A."y'' ",$B."y' ",$C."y ");
96 :     # re-write 1y'' as y''.
97 :     if ($A==1){
98 :     $LDD = "y''";
99 :     }
100 :     # re-write 1y' as y'
101 :     if ($B==1) {
102 :     $LD = "y'";
103 :     }
104 :     # re-write -1y' as -y'
105 :     elsif ($B==-1) {
106 :     $LD = "-y'";
107 :     }
108 :     # re-write 1y as y
109 :     if ($C==1) {
110 :     $LF = "y";
111 :     }
112 :     # re-write -1y as -y
113 :     elsif ($C==-1) {
114 :     $LF = "-y";
115 :     }
116 :     my ($L,$ind) = add($LDD,$A,$LD,$B,$LF,$C);
117 :     $L;
118 :     }
119 :    
120 :     ########
121 :    
122 :     #sub rad simplifies (a/b)*(sqrt(c))
123 :     #input is given as arguments on rad viz.: rad($a,$b,$c);
124 :     #$a,$b,$c are integers and $c>=0 and $b is not zero.
125 :     #output is an array =(answer as string,new$a,new$b, new$c)
126 :    
127 :     sub rad{
128 :     # initalize primes
129 :     my @p = (2,3,5,7,11,13,17,19,23,29);
130 :     my ($a,$b,$c) = @_;
131 :     my $s = "0" ;
132 :     my $i = 0 ;
133 :     # if a=0 then re-write as (0/1)*(sqrt(1)) = 0.
134 :     if ($c*$a ==0){
135 :     $a = 0;
136 :     $b = 1;
137 :     $c = 1;
138 :     }
139 :    
140 :     # if b<0 then re-write the numerator as negative, denominator as positive
141 :     if ($b < 0){
142 :     $a = - $a ;
143 :     $b = - $b ;
144 :     }
145 :    
146 :     my $j = 1 ;
147 :     while($j == 1){
148 :     # can't reduce sqrt(1).
149 :     if ($c == 1){
150 :     last;
151 :     }
152 :     $j = 0;
153 :     foreach $i (@p){
154 :     # if you can factor a prime out of c, do it.
155 :     # ie, sqrt(75) = 5*sqrt(3)
156 :     if ( $c == $i*$i*int($c/($i*$i))){
157 :     $a = $a *$i;
158 :     $c = $c/($i*$i);
159 :     $j=1;
160 :     }
161 :     }
162 :     }
163 :     $j = 1;
164 :    
165 :     # reduce fraction is lowest terms.
166 :     while($j==1){
167 :     # if the denominator is 1, then we're set.
168 :     if ($b==1){
169 :     last;
170 :     }
171 :     $j = 0;
172 :     foreach $i (@p){
173 :     # if you can factor a prime out of both numerator and denominator, do it.
174 :     if ( abs($a) + $b == $i*int(abs($a) /$i) + $i*int($b/$i) ){
175 :     $a = $a /$i;
176 :     $b = $b /$i;
177 :     $j=1;
178 :     }
179 :     }
180 :     }
181 :    
182 :     # $s = answer string
183 :    
184 :     # if you have ($a/1)*sqrt(1) then the answer is simply "$a".
185 :     if ($c == 1) {
186 :     if ($b == 1){
187 :     $s = "$a";
188 :     }
189 :     # if you have ($a/$b)*sqrt(1) then the answer is "$a/$b".
190 :     else {
191 :     $s = "$a/$b";
192 :     }
193 :     }
194 :    
195 :     if ($c > 1) {
196 :     if ($a != 1){
197 :     if ($b == 1) {
198 :     # if denominator is 1, then answer is "$a*sqrt($c)".
199 :     $s = "$a * sqrt($c)";
200 :     }
201 :     # if denominator is nonzero... answer is all three terms.
202 :     else {
203 :     $s = "$a *sqrt($c) /$b";
204 :     }
205 :     } else {
206 :     # if you have "(1/1)*sqrt($c)" then the answer is "sqrt($c)".
207 :     if ($b == 1) {
208 :     $s = "sqrt($c)";
209 :     }
210 :     # if you have "(1/$b)*sqrt($c)" then answer is "sqrt($c)/$b".
211 :     else {
212 :     $s = "sqrt($c) /$b";
213 :     }
214 :     }
215 :     }
216 :     # return all four variables: answer string, reduced numerator, reduced denominator, squareroot with primes factored out
217 :     my $rh_hash = { displaystr => $s,
218 :     num => $a,
219 :     denom => $b,
220 :     root => $c};
221 :     $rh_hash;
222 :    
223 :     }
224 :    
225 :     #######
226 :     sub frac {
227 :     # use rad subroutine
228 :     my ($a,$b) = @_;
229 :     rad($a,$b,1);
230 :     }
231 :     ##########
232 :    
233 :     ####
234 :     #sub exp simplifies exp($r*t) in form for writing perl
235 :     #or tex.The input is exp($r,$ind); $ind indicates whether
236 :     #we want perl or tex mode.$r is a string that represents
237 :     #a number.
238 :     #If $ind = 0 output is "exp(($r)*t)", simplified if possible.
239 :     #If $ind = 1 output is "exp(($r) t)", simplified if possible.
240 :     sub simpleexp {
241 :     my $r = shift;
242 :     my @rr = @_;
243 :     my %options;
244 :     if ($rr[0] eq 'mode') {
245 :     $options{'mode'} = $rr[1];
246 :     } elsif ( $rr[0] == 1) {
247 :     $options{'mode'} = 'typeset';
248 :     } elsif ($rr[0] == 0 ) {
249 :     $options{'mode'} = 'standard';# means we use * for multiplication
250 :     }
251 :    
252 :     my $y = "0";
253 :     if ($r eq "0"){
254 :     if ($options{'mode'} eq 'standard' ) {
255 :     $y = "1";
256 :     } elsif ($options{'mode'} eq 'typeset' ) {
257 :     $y = ""; # multiplication by 1 is the identity
258 :     } else {
259 :     warn "simpleexp doesn't recognize the mode $options{'mode'}";
260 :     }
261 :    
262 :     # change exp(1t) = exp(t)
263 :     }elsif ($r eq "1"){
264 :     $y = "exp(t)";
265 :     }
266 :     # change exp(-1t) = exp(-t)
267 :     elsif ($r eq "-1"){
268 :     $y = "exp(-t)";
269 :     }
270 :     # in typeset modeset you don't use the *
271 :     # in standard modeset you use *
272 :     else {
273 :     if ($options{'mode'} eq 'typeset') {
274 :     $y = "exp(($r)t)";
275 :     } elsif ($options{'mode'} eq 'standard') {
276 :     $y = "exp(($r)*t)";
277 :     } else {
278 :     warn "simpleexp doesn't recognize the mode $options{'mode'}";
279 :     }
280 :     }
281 :     $y;
282 :     }
283 :    
284 :     sub ivy {
285 :     # $a*y'' + $b*y' + $c*y = 0. y(0)=$m. y'(0)=$n.
286 :     my ($a, $b, $c, $m, $n) = @_;
287 :     my $d = ($b*$b)-(4*$a*$c); # d is the discriminant
288 :     my $c1 = "0";
289 :     my $c2 = "0";
290 :     my $r1 = "0";
291 :     my $r2 = "0";
292 :     my $answer = "";
293 :    
294 :     # c1 = first coefficient, c2 = second coefficient, rr1 = e^r1, rr2 = e^r2.
295 :     # c1*rr1 + c2*rr2 = c1*e^r1 + c2*e^r2
296 :    
297 :     if ($d > 0) {
298 :     # y(t) = [m/2 + sqrt(d)*(2An+Bm)/(2d)]e^[t*(-B+sqrt(d))/(2A)] + [m/2 - sqrt(d)*(2An+Bm)/(2d)]e^[t*(-B-sqrt(d))/(2A)]
299 :     my $piece1 = frac($m,2);
300 :     my $piece2 = rad(2*$a*$n+$b*$m,2*$d,$d);
301 :     $c1 = "$piece1->{displaystr} + $piece2->{displaystr}"; # first coefficient: "m/2 + sqrt(d)*(2An+Bm)/(2d)"
302 :     $c2 = "$piece1->{displaystr} - $piece2->{displaystr}"; # second coefficient: "m/2 - sqrt(d)*(2An+Bm)/(2d)"
303 :     $piece1 = frac(-$b,2*$a); # find (-B/(2A)) in lowest terms
304 :     $piece2 = rad(1,2*$a,$d); # find (sqrt(B*B-4AC)/(2A)) in lowest terms
305 :     $r1 = "$piece1->{displaystr} + $piece2->{displaystr}"; # r1: (-B+sqrt(d))/(2A)
306 :     $r2 = "$piece1->{displaystr} - $piece2->{displaystr}"; # r2: (-B-sqrt(d))/(2A)
307 :     my $rr1 = simpleexp($r1,0); # raise e^r1.
308 :     my $rr2 = simpleexp($r2,0); # raise e^r2.
309 :     $answer = "($c2) *$rr2 + ($c1) *$rr1";
310 :    
311 :     }
312 :    
313 :     if ($d == 0) {
314 :     # y(t) = me^((-B/(2A)t) + [(n+mB)/(2A)]*t*e^((-Bt)/(2*A))
315 :     my $piece1 = frac(-$b,2*$a); # find (-B/(2A)) in lowest terms
316 :     my $piece2 = frac(2*$a*$n+$m*$b,2*$a); # find (2An+Bm)/(2A) in lowest terms
317 :     $c1 = "$m"; # first coefficient: "m"
318 :     $c2 = "$piece2->{displaystr}"; # second coefficient: "(n+mB)/(2A)"
319 :     $r1 = "$piece1->{displaystr}"; # r1: (-B/(2A))
320 :     $r2 = $r1; # r2: (-B/(2A))
321 :     my $rr1 = simpleexp($r1,0); # rr1 = e^r1 = e^(-B/(2A))
322 :     my $rr2 = simpleexp($r2,0); # rr2 = e^r2 = e^(-B/(2A))
323 :     $answer = "($c1) *$rr1 + ($c2)*t *$rr2";
324 :     }
325 :    
326 :     # if the descriminant is negative, then the roots are imaginary.
327 :     # recall, e^x where x=a+ib then e^x = (e^a)*cos(bt) + (e^a)*sin(bt).
328 :     if ($d<0){
329 :     # y(t) = me^(-Bt/(2A))*cos(t*sqrt(4AC-B*B)/(2A))+(2An+Bm)*sqrt(4AC-B*B)/(4AC-B*B)*e^(-Bt/(2A))*sin(t*sqrt(4AC-B*B)/(2A))
330 :     my $piece1 = rad (2*$a*$n+$b*$m,-$d,-$d); # find ((2An+Bm)*sqrt(4AC-B*B))/(4AC-B*B) in lowest terms
331 :     my $piece2 = rad (1,2*$a,-$d); # find (sqrt(4AC-B*B)/(2A)) in lowest terms
332 :     $c1 = "$m"; # first coefficient: "m"
333 :     $c2 = "$piece1->{displaystr}"; # second coefficient: "(2An+Bm)*sqrt(4AC-B*B)/(4AC-B*B)"
334 :     my $cs1 = "cos(($piece2->{displaystr})*t)"; # cos(t*sqrt(4AC-B*B)/(2A))
335 :     my $cs2 = "sin(($piece2->{displaystr})*t)"; # sin(t*sqrt(4AC-B*B)/(2A))
336 :     my $piece3 = frac (-$b,2*$a); # find (-B/(2A)) in lowest terms
337 :     $r1 = "$piece3->{displaystr}"; # r1: (-B/(2A))
338 :     $r2 = $r1; # r2: (-B/(2A))
339 :     my $rr1 = simpleexp($r1,0); # rr1 = e^r1 = e^(-B/(2A))
340 :     my $rr2 = simpleexp($r2,0); # rr2 = e^r2 = e^(-B/(2A))
341 :     $answer = "($c1) *($rr1)*($cs1) + ($c2) *($rr2)*($cs2)";
342 :     }
343 :     $answer;
344 :     }
345 :    
346 :     ############
347 :     #sub ivy solves the initial value problem
348 :     # $a*y'' + $b*y' + $c*y = 0, with y(0) = $m, y'(0) = $n
349 :    
350 :     #The numbers $a,$b,$c,$m,$n should be integers with $a not 0.
351 :     #The inputs are given as arguments viz: ivy ($a,$b,$c,$m,$n).
352 :     #The output is the solution as a string giving a function of t.
353 :    
354 :    
355 : gage 3340
356 : sh002i 1050 sub undeterminedExp {
357 :     my ($A,$B,$C,$r,$q0,$q1,$q2) = @_;
358 :     my $P = "$A*x*x + $B *x + $C "; #characteristic poly.
359 :     my $PP = "$A*2*x + $B ";#derivative of characteristic poly.
360 :     my $exp = simpleexp($r,0);
361 :     #$Pr = $P;
362 :     #$Pr =~ s~~x~~$r~~g;
363 :     #$Pr = PG_answer_eval("$Pr ");
364 :     #$PPr = $PP;
365 :     #$PPr =~ s~~x~~$r~~g;
366 :     #$PPr = PG_answer_eval("$PPr ");
367 :     my $Pr = $A *$r *$r + $B *$r + $C ;
368 :     my $PPr = 2* $A* $r + $B;
369 :     #################
370 :     if ($Pr != 0){
371 :     #$r not a root of $P
372 :     # $v0 = "($q0 /$Pr) ";
373 :     my $n1 = -$q1 * $PPr ;
374 :     my $d1 = $Pr * $Pr;
375 :     # $v1 = " ($q1 /$Pr)* t + ($n1 / $d1) ";
376 :     my $n22 = $Pr *$Pr *$q2;
377 :     my $n21 = (- 2 *$PPr ) *$q2;
378 :     my $n20 = (2*$PPr *$PPr - ($A*2*$Pr)) *$q2 ;
379 :     my $d2 = $Pr **3;
380 :     # $v2 = "($n22/$d2) *t*t + ($n21/$d1) *t + ($n20/$d2) ";
381 :     my $c00n = $q0*$d1 -$q1 *$PPr *$Pr + $n20;
382 :     my $c00d = $d2;
383 :     my $fraca = frac ($c00n ,$c00d );
384 :     my $c00 = $fraca->{displaystr};
385 :     my $c01n = $Pr *$q1 - 2 *$PPr *$q2;
386 :     my $c01d = $d1;
387 :     my $fracb = frac($c01n ,$c01d );
388 :     my $c01 = $fracb->{displaystr};
389 :     $c01 = "($c01) *t";
390 :     my $c02n = $q2;
391 :     my $c02d = $Pr;
392 :     my $fracc = frac($c02n ,$c02d );
393 :     my $c02 = $fracc->{displaystr};
394 :     $c02 = "($c02 ) *t*t";
395 :     my @adda = add ($c00,$c00n,$c01,$c01n,$c02,$c02n);
396 :     my $outa = $adda [0];
397 :     my $y = "( $outa ) * $exp " ;
398 :     #############
399 :     } elsif ($PPr != 0){
400 :     #$r a simple root of $P
401 :     my $q2m = -$q2;
402 :     #$v0 = "( $q0/$PPr)*t ";
403 :     my $q2d = 2*$q2;
404 :     my $d10 = 2*$PPr;
405 :     my $d11 = $PPr **2 ;
406 :     my $q1m = -$q1;
407 :     #$v1 = "($q1 / $d10)*t*t + ($q1m / $d11)*t ";
408 :     my $d23 = 3*$PPr;
409 :     my $d22 = $PPr *$PPr ;
410 :     my $d21 = $PPr *$d22;
411 :     #$v2 = "($q2 / $d23) *t*t*t + ($q2m/$d22) *t*t + ($q2d/ #$d21) *t ";
412 :     ######
413 :     my $c10n = $q0 *$d22 -$q1*$A *$PPr + $A*$A*$q2d;
414 :     my $c10d = $d21;
415 :     my $fracd = frac($c10n ,$c10d );
416 :     my $c10 = $fracd->{displaystr};
417 :     #warn " c10 $c10";
418 :     $c10 = "($c10 )*t";
419 :     my $c11n = $PPr * $q1 - 2 *$A * $q2;
420 :     my $c11d = 2*$PPr*$PPr;
421 :     my $frace = frac($c11n ,$c11d );
422 :     my $c11 = $frace->{displaystr};
423 :     #warn " c11 $c11";
424 :     $c11 = "($c11 )*t*t";
425 :     my $c12n = $q2;
426 :     my $c12d = 3*$PPr;
427 :     my $fracf = frac ($c12n ,$c12d );
428 :     my $c12 = $fracf->{displaystr};
429 :     $c12 = "($c12 ) *t*t*t";
430 :     my @addb = add($c10,$c10n,$c11,$c11n,$c12,$c12n);
431 :     my $outb = $addb[0];
432 :     my $y = "( $outb ) * $exp" ;
433 :     ######
434 :     } else {
435 :     # $v2 = "($q2 /12*$A) *t*t*t*t ";
436 :     #v1 = "($q1 /6*$A) *t*t*t ";
437 :     #$v0 = "($q0 /2*$A) *t*t " ;
438 :     my $c20n = $q0;
439 :     my $c20d = 2*$A;
440 :     my $fracg = frac($q0 ,$c20d );
441 :     my $c20 = $fracg->{displaystr};
442 :     $c20 = "($c20 ) *t*t";
443 :     my $c21n = $q1;
444 :     my $c21d = 6*$A;
445 :     my $frach = frac($c21n ,$c21d );
446 :     my $c21 = $frach->{displaystr};
447 :     $c21 = "($c21) *t*t*t";
448 :     my $c22n = $q2;
449 :     my $c22d = 12*$A;
450 :     my $fraci = frac($c22n ,$c22d );
451 :     my $c22 = $fraci->{displaystr};
452 :     $c22 = "($c22 ) *t*t*t*t";
453 :     my @addc = add($c20,$c20n,$c21,$c21n,$c22,$c22n);
454 :     my $outc = $addc[0];
455 :     my $y = "( $outc ) * $exp" ;
456 :     }
457 :    
458 :     }
459 :     #################
460 :    
461 :     # undeterminedSin is a subroutine to solve
462 :     #undetermined coefficient problems that have
463 :     #sines and cosines.
464 :     #The input is an array ($A,$B,$C,$r,$w,$q1,$q0,$r1,$r0)
465 :     #given as arguments on undeterminedSin
466 :     # $L =$A y'' + $B y' + $C y
467 :     # $rhs = ($q1 t + $q0) cos($w t)exp($r t) +
468 :     # ($r1 t + $r0) sin($w t)exp($r t)
469 :     #The subroutine uses undetermined coefficients
470 :     #to find a solution $y of $L = $rhs .
471 :     #The output \is $y
472 :    
473 :     sub undeterminedSin {
474 :     my ($A,$B,$C,$r,$w,$q1,$q0,$r1,$r0) = @_;
475 :     my $Pr = ($A*$r*$r) + $B *$r + $C;
476 :     my $PPr = (2*$A *$r ) + $B ;
477 :     my $re = $Pr -$A* $w*$w ;
478 :     my $im = $PPr * $w ;
479 :     #If P(x) = A x^2 +Bx +C,
480 :     #P($r+i*$w)=$re+i $im.
481 :     my $D = $re **2 + $im **2 ;
482 :     # $D = |P($r + i*$w)|^2
483 :     my $reprime = $PPr;
484 :     my $imprime = 2*$A*$w;
485 :     #If PP(x) = derivative of P(x),
486 :     #PP($r+i $w)=$reprime+$imprime.
487 :     my $cos = "cos($w *t)";
488 :     my $sin = "sin($w *t)";
489 :     if ($w == 1){
490 :     $cos = "cos(t)";
491 :     $sin = "sin(t)";
492 :     }
493 :    
494 :    
495 :     my $exp = simpleexp($r,0);
496 :    
497 :     ############
498 :     if ($D != 0){
499 :     #We first handle case that$r+i$w not a root of $P(x)
500 :     #This solution is based on:
501 :     #Let L[y] = Ay'' +By'+C,z=r+iw,p=P(z),q=P'(z),S,T complex;p!=0.
502 :     #Then L[((St+T-(Sq/p))/p)*e^{zt}]=(St+T)e^{zt}.
503 :     #Take S=q1-i*r1;T=q0-i*r0.Then take real part.
504 :     my ($renum1 ,$imnum1) = complexmult($q1,-$r1,$re, -$im);
505 :    
506 :     #S*(p conjugate)= $renum1+i imnum1
507 :     my ($renum2 ,$imnum2) = complexmult ($renum1,$imnum1,$reprime,$imprime);
508 :     #The above are real and imag parts of q*S*(p conjugate)
509 :     my $first = ($D *$q0 ) - $renum2 ;
510 :     my $second = (-$r0 *$D ) -$imnum2 ;
511 :     my ($renum3, $imnum3 ) = complexmult( $first,$second,$re,-$im);
512 :     #these are re and im of (DT-qS(p conj))*(p conj.)
513 :     my $n1 = $renum1;
514 :     my $n2 = $renum3;
515 :     my $n3 = -$imnum1;
516 :     my $n4 = -$imnum3;
517 :     my $fraca = frac($n1,$D );
518 :     my $tcosp = $fraca->{displaystr};
519 :     #$tcospart = "($tcosp )*t*$exp *$cos ";
520 :     my $tcospart = "($tcosp)*t*$exp *$cos " ; #####################
521 :     my $DD = $D *$D;
522 :     my $fracb = frac($n2 , $DD );
523 :     my $cospart = $fracb->{displaystr};
524 :     $cospart = "($cospart )*$exp*$cos";
525 :     my $fracc = frac($n3 , $D );
526 :     my $tsinpart = $fracc->{displaystr};
527 :     $tsinpart = "($tsinpart )*t*$exp*$sin";
528 :     my $fracd = frac($n4 , $DD );
529 :     my $sinpart = $fracd->{displaystr};
530 :     $sinpart = "($sinpart )*$exp*$sin";
531 :     my @suma = add($tcospart,$n1,$cospart,$n2,$tsinpart,$n3,$sinpart,$n4 );
532 :     my $out = $suma[0];
533 :     #The solution is $out
534 :     } else{
535 :     #We now handle case that $r+iw is a root of $P
536 :     #In this case $PPr = 0 and $PP($r + i$w) = 2*$A*i*$w
537 :     #Solution based on
538 :     #L[((S/2q)t*t -(AS/q*q)t +(T/q)t)e^{zt}]=
539 :     #(St+T)e^{zt}.Notation as for 1st part.
540 :     my $n3 = $q1 - (2*$w *$r0);
541 :     my $n4 = $r1 + (2*$w *$q0 );
542 :     my $n1 = -$r1 ;
543 :     my $n2 = $q1 ;
544 :     my $T2 = 4*$A *$w ;
545 :     my $T1 = $w * $T2 ;
546 :     my $frace = frac($n1 , $T2 );
547 :     my $t2cospart = $frace->{displaystr};
548 :     $t2cospart = "($t2cospart )*t*t*$exp *$cos ";
549 :     my $fracf = frac($n3 , $T1 );
550 :     my $tcospart = $fracf->{displaystr};
551 :     $tcospart = "($tcospart )*t*$exp *$cos ";
552 :     my $fracg = frac($n2 , $T2 );
553 :     my $t2sinpart = $fracg->{displaystr};
554 :     $t2sinpart = "($t2sinpart )*t*t*$exp*$sin";
555 :     my $frach = frac($n4 , $T1 );
556 :     my $tsinpart = $frach->{displaystr};
557 :     $tsinpart = "($tsinpart )*t*$exp*$sin";
558 :     my @addb = add ($t2cospart,$n1,$tcospart,$n3,$t2sinpart,$n2,$tsinpart,$n4 );
559 :     my $out = $addb[0];
560 :     }
561 :    
562 :     }
563 :    
564 :     sub check_eigenvector {
565 :    
566 :     my $eigenvalue = shift;
567 :     my $matrix = shift;
568 :     my %options = @_;
569 :     assign_option_aliases( \%options, );
570 :    
571 :     set_default_options( \%options,
572 :     'debug' => 0,
573 :     'correct_ans' => undef
574 :     );
575 :    
576 :    
577 :     my @correct_vector = ();
578 :     @correct_vector = @{$options{'correct_ans'}} if defined ($options{'correct_ans'});
579 :    
580 :     my $ans_eval = new AnswerEvaluator;
581 :    
582 :     $ans_eval->{debug} = $options{debug};
583 :     my $corr_ans_points = "( " . join(", ", @correct_vector). " )" ;
584 :     $ans_eval->ans_hash( correct_ans => $corr_ans_points );
585 :     $ans_eval->install_pre_filter(\&is_array);
586 :     $ans_eval->install_pre_filter(\&std_num_array_filter);
587 :    
588 :     $ans_eval->install_evaluator(sub { my $rh_ans = shift;
589 :     my %options = @_;
590 :     my @vector = @{$rh_ans->input()};
591 :     return($rh_ans) unless @correct_vector == @vector;
592 :     # make sure the vectors are the same dimension
593 :    
594 :     my $vec = new Matrix(2,1);
595 :     $vec->assign(1,1, $vector[0]);
596 :     $vec->assign(2,1, $vector[1]);
597 :     my $out_vec = $matrix * $vec;
598 :     my @diff;
599 :     $diff[0] = $out_vec->element(1,1) - $vec->element(1,1)*$eigenvalue;
600 :     $diff[1] = $out_vec->element(2,1) - $vec->element(2,1)*$eigenvalue;
601 :     $rh_ans->{score} = zero_check(\@diff);
602 :     $rh_ans;
603 :    
604 :     });
605 :     $ans_eval->install_post_filter( sub { my $rh_ans= shift;
606 :     if ($rh_ans->error_flag('SYNTAX') ) {
607 :     $rh_ans->{ans_message} = $rh_ans->{error_message};
608 :     $rh_ans->clear_error('SYNTAX');
609 :     $rh_ans;
610 :     }
611 :     });
612 :    
613 :     $ans_eval;
614 :     }
615 :    
616 :    
617 :    
618 :     =pod
619 :    
620 :     rungeKutta4a
621 :    
622 : gage 1831
623 : sh002i 1050 =cut
624 : gage 1831
625 : sh002i 1050
626 :    
627 :     sub rungeKutta4a {
628 :     my $rh_ans = shift;
629 :     my %options = @_;
630 :     my $rf_fun = $rh_ans->{rf_diffeq};
631 :     set_default_options( \%options,
632 :     'initial_t' => 1,
633 :     'initial_y' => 1,
634 :     'dt' => .01,
635 :     'num_of_points' => 10, #number of reported points
636 :     'interior_points' => 5, # number of 'interior' steps between reported points
637 :     'debug' => 1, # remind programmers to always pass the debug parameter
638 :     );
639 :     my $t = $options{initial_t};
640 :     my $y = $options{initial_y};
641 :    
642 :     my $num = $options{'num_of_points'}; # number of points
643 :     my $num2 = $options{'interior_points'}; # number of steps between points.
644 :     my $dt = $options{'dt'};
645 :     my $errors = undef;
646 :     my $rf_rhs = sub { my @in = @_;
647 :     my ( $out, $err) = &$rf_fun(@in);
648 :     $errors .= " $err at ( ".join(" , ", @in) . " )<br>\n" if defined($err);
649 :     $out = 'NaN' if defined($err) and not is_a_number($out);
650 :     $out;
651 :     };
652 :    
653 :     my @output = ([$t, $y]);
654 :     my ($i, $j, $K1,$K2,$K3,$K4);
655 :    
656 :     for ($j=0; $j<$num; $j++) {
657 :     for ($i=0; $i<$num2; $i++) {
658 :     $K1 = $dt*&$rf_rhs($t, $y);
659 :     $K2 = $dt*&$rf_rhs($t+$dt/2,$y+$K1/2);
660 :     $K3 = $dt*&$rf_rhs($t+$dt/2, $y+$K2/2);
661 :     $K4 = $dt*&$rf_rhs($t+$dt, $y+$K3);
662 :     $y = $y + ($K1 + 2*$K2 + 2*$K3 + $K4)/6;
663 :     $t = $t + $dt;
664 :     }
665 :     push(@output, [$t, $y]);
666 :     }
667 :     $rh_ans->{evaluation_points} = \@output;
668 :     $rh_ans->throw_error($errors) if defined($errors);
669 :     $rh_ans;
670 :     }
671 :    
672 :    
673 :     sub level_curve_check {
674 :     my $diffEqRHS = shift; #required differential equation
675 :     my $correctEqn = shift; # required answer in order to check the equation
676 :     my %options = @_;
677 :     assign_option_aliases( \%options,
678 :     'vars' => 'var',
679 :     'numPoints' => 'num_of_points',
680 :     'reltol' => 'relTol',
681 :     );
682 :     set_default_options( \%options,
683 :     'initial_t' => 0,
684 :     'initial_y' => 1,
685 :     'var' => [qw( x y )],
686 :     'num_of_points' => 10,
687 :     'tolType' => (defined($options{tol}) ) ? 'absolute' : 'relative',
688 :     'relTol' => .01,
689 :     'tol' => .01,
690 :     'debug' => 0,
691 :     );
692 :    
693 :     my $initial_t = $options{initial_t};
694 :     my $initial_y = $options{initial_y};
695 :     my $var = $options{var};
696 :     my $numPoints = $options{num_of_points};
697 :     my @VARS = get_var_array( $var );
698 :     my ($tolType, $tol);
699 :     if ($options{tolType} eq 'absolute') {
700 :     $tolType = 'absolute';
701 :     $tol = $options{'tol'};
702 :     delete($options{'relTol'}) if exists( $options{'relTol'} );
703 :     } else {
704 :     $tolType = 'relative';
705 :     $tol = $options{'relTol'};
706 :     delete($options{'tol'}) if exists( $options{'tol'} );
707 :     }
708 :     #prepare the correct answer and check its syntax
709 :     my $rh_correct_ans = new AnswerHash;
710 :     $rh_correct_ans ->{correct_ans} = $correctEqn;
711 :     # check and calculate the function defining the differential equation
712 :     $rh_correct_ans->input( $diffEqRHS );
713 :     $rh_correct_ans = check_syntax($rh_correct_ans);
714 :     warn $rh_correct_ans->{error_message},$rh_correct_ans->pretty_print() if $rh_correct_ans->{error_flag};
715 :    
716 :     $rh_correct_ans->{error_flag} = undef;
717 :    
718 :     $rh_correct_ans = function_from_string2($rh_correct_ans,
719 :     ra_vars => [@VARS],
720 :     store_in =>'rf_diffeq',
721 :     debug=>$options{debug}
722 :     );
723 :     warn "Error in compiling instructor's answer: $diffEqRHS<br> $rh_correct_ans->{error_message}<br>\n$rh_correct_ans->pretty_print()"
724 :     if $rh_correct_ans->{error_flag};
725 :    
726 :    
727 :     # create the test points that should lie on a solution curve of the differential equation
728 :     $rh_correct_ans = rungeKutta4a( $rh_correct_ans,
729 :     initial_t => $initial_t,
730 :     initial_y => $initial_y,
731 :     num_of_points => $numPoints,
732 :     debug=>$options{debug}
733 :     );
734 :     warn "Errors in calculating the solution curve $rh_correct_ans->{student_ans}<BR>\n
735 :     $rh_correct_ans->{error_message}<br>\n",$rh_correct_ans->pretty_print() if $rh_correct_ans->catch_error();
736 :     $rh_correct_ans->clear_error();
737 :    
738 :     # check and compile the correct answer submitted by the instructor.
739 :     my ($check_eval) = fun_cmp('c', vars => [@VARS],
740 :     params => ['c'],
741 :     tolType => $options{tolType},
742 :     relTol => $options{relTol},
743 :     tol => $options{tol},
744 :     debug => $options{debug},
745 :     ); # an evaluator that tests for constants;
746 :     $check_eval->ans_hash(evaluation_points => $rh_correct_ans->{evaluation_points});
747 :     $check_eval->evaluate($rh_correct_ans->{correct_ans});
748 :     if( $check_eval->ans_hash->{score} == 0 or (defined($options{debug}) and $options{debug})) {
749 :     # write error message for professor
750 :     my $out1 = $check_eval->ans_hash->{evaluation_points};
751 :     my $rf_corrEq = $check_eval->ans_hash->{rf_student_ans};
752 :     my $error_string = "This equation $correctEqn is not constant on solution curves of y'(t) = $diffEqRHS\r\n<br>
753 :     starting at ( $initial_t , $initial_y )<br>
754 :     $check_eval->ans_hash->pretty_print()".
755 :     "options<br>\n".pretty_print({ vars => [@VARS],
756 :     params => ['c'],
757 :     tolType => $options{tolType},
758 :     relTol => $options{relTol},
759 :     tol => $options{tol},
760 :     debug => $options{debug},
761 :     });
762 :    
763 :     for (my $i=0; $i<$numPoints;$i++) {
764 :     my ($z, $err) = &$rf_corrEq( $out1->[$i][0], $out1->[$i][1] );
765 :     $z = $err if defined $err;
766 :     $error_string .= "F( ". $out1->[$i][0] . " , ". $out1->[$i][1] . " ) = $z <br>\r\n";
767 :     }
768 :     $error_string .= $rh_correct_ans->error_message();
769 :     warn $error_string, $check_eval->ans_hash->pretty_print;
770 :     }
771 :    
772 :     my ($constant_eval) = fun_cmp('c', vars => [@VARS],
773 :     params => ['c'],
774 :     tolType => $options{tolType},
775 :     relTol => $options{relTol},
776 :     tol => $options{tol},
777 :     debug => $options{debug},
778 :     ); # an evaluator that tests for constants;
779 :     $constant_eval->ans_hash(evaluation_points => $rh_correct_ans->{evaluation_points});
780 :     my $answer_evaluator = new AnswerEvaluator;
781 :     $answer_evaluator->ans_hash( correct_ans => $rh_correct_ans->{correct_ans}, # used for answer only
782 :     rf_correct_ans => sub { my @input = @_; pop(@input); },
783 :     # return the last input which is the constant parameter 'c';
784 :     evaluation_points => $rh_correct_ans->{evaluation_points},
785 :     ra_param_vars => ['c'], # compare with constant function
786 :     ra_vars => [@VARS],
787 :     type => 'level_curve',
788 :     );
789 :     $answer_evaluator->install_evaluator(sub { my $ans_hash = shift;
790 :     my %options = @_;
791 :     $constant_eval->evaluate($ans_hash->{student_ans});
792 :     $constant_eval->ans_hash;
793 :     });
794 :    
795 :     $answer_evaluator->install_post_filter( sub { my $ans_hash = shift; $ans_hash->{correct_ans} = $correctEqn; $ans_hash; } );
796 :     $answer_evaluator->install_post_filter( sub { my $rh_ans= shift;
797 :     my %options = @_;
798 :     if ($rh_ans->catch_error('SYNTAX') ) {
799 :     $rh_ans->{ans_message} = $rh_ans->{error_message};
800 :     $rh_ans->clear_error('SYNTAX');
801 :    
802 :     }
803 :     $rh_ans;
804 :     });
805 :    
806 :     $answer_evaluator;
807 :    
808 :     }
809 :    
810 :    
811 :     1;

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