/usr/share/yacas/solve.rep/code.ys is in yacas 1.3.3-2.
This file is owned by root:root, with mode 0o644.
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* Strategy for Solve(expr, x):
*
* 10. Call Solve'System for systems of equations.
* 20. Check arguments.
* 30. Get rid of "==" in 'expr'.
* 40. Special cases.
* 50. If 'expr' is a polynomial in 'x', try to use PSolve.
* 60. If 'expr' is a product, solve for either factor.
* 70. If 'expr' is a quotient, solve for the denominator.
* 80. If 'expr' is a sum and one of the terms is free of 'x',
* try to use Solve'Simple.
* 90. If every occurance of 'x' is in the same context, use this to reduce
* the equation. For example, in 'Cos(x) + Cos(x)^2 == 1', the variable
* 'x' always occurs in the context 'Cos(x)', and hence we can attack
* the equation by first solving 'y + y^2 == 1', and then 'Cos(x) == y'.
* This does not work for 'Exp(x) + Cos(x) == 2'.
* 100. Apply Simplify to 'expr', and try again.
* 110. Give up.
*/
LocalSymbols(res)
[
10 # Solve(expr_IsList, var_IsList) <-- Solve'System(expr, var);
20 # Solve(_expr, _var)_(Not IsAtom(var) Or IsNumber(var) Or IsString(var)) <--
[ Assert("Solve'TypeError", "Second argument, ":(ToString() Write(var)):", is not the name of a variable") False; {}; ];
30 # Solve(_lhs == _rhs, _var) <-- Solve(lhs - rhs, var);
40 # Solve(0, _var) <-- {var == var};
41 # Solve(a_IsConstant, _var) <-- {};
42 # Solve(_expr, _var)_(Not HasExpr(expr,var)) <--
[ Assert("Solve", "expression ":(ToString() Write(expr)):" does not depend on ":ToString() Write(var)) False; {}; ];
50 # Solve(_expr, _var)_((res := Solve'Poly(expr, var)) != Failed) <-- res;
60 # Solve(_e1 * _e2, _var) <-- [
Local(t,u,s);
t := Union(Solve(e1,var), Solve(e2,var));
u := {};
ForEach(s, t) [
Local(v1,v2);
v1 := WithValue(var, s[2], e1);
v2 := WithValue(var, s[2], e2);
If(Not (IsInfinity(v1) Or (v1 = Undefined) Or
IsInfinity(v2) Or (v2 = Undefined)),
DestructiveAppend(u, s));
];
u;
];
70 # Solve(_e1 / _e2, _var) <-- [
Local(tn, t, s);
tn := Solve(e1, var);
t := {};
ForEach(s, tn)
If(Not(IsZero(WithValue(var, s[2], e2))),
DestructiveAppend(t, s)
);
t;
];
80 # Solve(_e1 + _e2, _var)_(Not HasExpr(e2,var) And (res := Solve'Simple(e1,-e2,var)) != Failed) <-- res;
80 # Solve(_e1 + _e2, _var)_(Not HasExpr(e1,var) And (res := Solve'Simple(e2,-e1,var)) != Failed) <-- res;
80 # Solve(_e1 - _e2, _var)_(Not HasExpr(e2,var) And (res := Solve'Simple(e1,e2,var)) != Failed) <-- res;
80 # Solve(_e1 - _e2, _var)_(Not HasExpr(e1,var) And (res := Solve'Simple(e2,e1,var)) != Failed) <-- res;
85 # Solve(_expr, _var)_((res := Solve'Simple(expr, 0, var)) != Failed) <-- res;
90 # Solve(_expr, _var)_((res := Solve'Reduce(expr, var)) != Failed) <-- res;
95 # Solve(_expr, _var)_((res := Solve'Divide(expr, var)) != Failed) <-- res;
100 # Solve(_expr, _var)_((res := Simplify(expr)) != expr) <-- Solve(res, var);
110 # Solve(_expr, _var) <--
[ Assert("Solve'Fails", "cannot solve equation ":(ToString() Write(expr)):" for ":ToString() Write(var)) False; {}; ];
];
/********** Solve'Poly **********/
/* Tries to solve by calling PSolve */
/* Returns Failed if this doesn't work, and the solution otherwise */
/* CanBeUni is not documented, but defined in univar.rep/code.ys */
/* It returns True iff 'expr' is a polynomial in 'var' */
10 # Solve'Poly(_expr, _var)_(Not CanBeUni(var, expr)) <-- Failed;
/* The call to PSolve can have three kind of results
* 1) PSolve returns a single root
* 2) PSolve returns a list of roots
* 3) PSolve remains unevaluated
*/
20 # Solve'Poly(_expr, _var) <--
LocalSymbols(x)
[
Local(roots);
roots := PSolve(expr, var);
If(Type(roots) = "PSolve",
Failed, /* Case 3 */
If(Type(roots) = "List",
MapSingle({{x},var==x}, roots), /* Case 2 */
{var == roots})); /* Case 1 */
];
/********** Solve'Reduce **********/
/* Tries to solve by reduction strategy */
/* Returns Failed if this doesn't work, and the solution otherwise */
10 # Solve'Reduce(_expr, _var) <--
[
Local(context, expr2, var2, res, sol, sol2, i);
context := Solve'Context(expr, var);
If(context = False,
res := Failed,
[
expr2 := Eval(Subst(context, var2) expr);
If(CanBeUni(var2, expr2) And (Degree(expr2, var2) = 0 Or (Degree(expr2, var2) = 1 And Coef(expr2, var2, 1) = 1)),
res := Failed, /* to prevent infinite recursion */
[
sol2 := Solve(expr2, var2);
If(IsError("Solve'Fails"),
[
ClearError("Solve'Fails");
res := Failed;
],
[
res := {};
i := 1;
While(i <= Length(sol2) And res != Failed) [
sol := Solve(context == (var2 Where sol2[i]), var);
If(IsError("Solve'Fails"),
[
ClearError("Solve'Fails");
res := Failed;
],
res := Union(res, sol));
i++;
];
]);
]);
]);
res;
];
/********** Solve'Context **********/
/* Returns the unique context of 'var' in 'expr', */
/* or {} if 'var' does not occur in 'expr', */
/* or False if the context is not unique. */
10 # Solve'Context(expr_IsAtom, _var) <-- If(expr=var, var, {});
20 # Solve'Context(_expr, _var) <--
[
Local(lst, foundVarP, context, i, res);
lst := Listify(expr);
foundVarP := False;
i := 2;
While(i <= Length(lst) And Not foundVarP) [
foundVarP := (lst[i] = var);
i++;
];
If(foundVarP,
context := expr,
[
context := {};
i := 2;
While(i <= Length(lst) And context != False) [
res := Solve'Context(lst[i], var);
If(res != {} And context != {} And res != context, context := False);
If(res != {} And context = {}, context := res);
i++;
];
]);
context;
];
/********** Solve'Simple **********/
/* Simple solver of equations
*
* Returns (possibly empty) list of solutions,
* or Failed if it cannot handle the equation
*
* Calling format: Solve'Simple(lhs, rhs, var)
* to solve 'lhs == rhs'.
*
* Note: 'rhs' should not contain 'var'.
*/
20 # Solve'Simple(_e1 + _e2, _rhs, _var)_(e1 = var And Not HasExpr(e2,var)) <-- { var == rhs-e2 };
20 # Solve'Simple(_e1 + _e2, _rhs, _var)_(e2 = var And Not HasExpr(e1,var)) <-- { var == rhs-e1 };
20 # Solve'Simple(_e1 - _e2, _rhs, _var)_(e1 = var And Not HasExpr(e2,var)) <-- { var == rhs+e2 };
20 # Solve'Simple(_e1 - _e2, _rhs, _var)_(e2 = var And Not HasExpr(e1,var)) <-- { var == e1-rhs };
20 # Solve'Simple(-(_e1), _rhs, _var)_(e1 = var) <-- { var == -rhs };
20 # Solve'Simple(_e1 * _e2, _rhs, _var)_(e1 = var And Not HasExpr(e2,var)) <-- { var == rhs/e2 };
20 # Solve'Simple(_e1 * _e2, _rhs, _var)_(e2 = var And Not HasExpr(e1,var)) <-- { var == rhs/e1 };
20 # Solve'Simple(_e1 / _e2, _rhs, _var)_(e1 = var And Not HasExpr(e2,var)) <-- { var == rhs*e2 };
10 # Solve'Simple(_e1 / _e2, 0, _var)_(e2 = var And Not HasExpr(e1,var)) <-- { };
20 # Solve'Simple(_e1 / _e2, _rhs, _var)_(e2 = var And Not HasExpr(e1,var)) <-- { var == e1/rhs };
LocalSymbols(x)
[
20 # Solve'Simple(_e1 ^ _n, _rhs, _var)_(e1 = var And IsPositiveInteger(n))
<-- MapSingle({{x}, var == rhs^(1/n)*x}, Exp(2*Pi*I*(1 .. n)/n));
20 # Solve'Simple(_e1 ^ _n, _rhs, _var)_(e1 = var And IsNegativeInteger(n))
<-- MapSingle({{x}, var == rhs^(1/n)*x}, Exp(2*Pi*I*(1 .. (-n))/(-n)));
];
20 # Solve'Simple(_e1 ^ _e2, _rhs, _var)
_ (IsPositiveReal(e1) And e1 != 0 And e2 = var And IsPositiveReal(rhs) And rhs != 0)
<-- { var == Ln(rhs)/Ln(e1) };
/* Note: These rules do not take the periodicity of the trig. functions into account */
10 # Solve'Simple(Sin(_e1), 1, _var)_(e1 = var) <-- { var == 1/2*Pi };
10 # Solve'Simple(Sin(_e1), _rhs, _var)_(e1 = var And rhs = -1) <-- { var == 3/2*Pi };
20 # Solve'Simple(Sin(_e1), _rhs, _var)_(e1 = var) <-- { var == ArcSin(rhs), var == Pi-ArcSin(rhs) };
10 # Solve'Simple(Cos(_e1), 1, _var)_(e1 = var) <-- { var == 0 };
10 # Solve'Simple(Cos(_e1), _rhs, _var)_(e1 = var And rhs = -1) <-- { var == Pi };
20 # Solve'Simple(Cos(_e1), _rhs, _var)_(e1 = var) <-- { var == ArcCos(rhs), var == -ArcCos(rhs) };
20 # Solve'Simple(Tan(_e1), _rhs, _var)_(e1 = var) <-- { var == ArcTan(rhs) };
20 # Solve'Simple(ArcSin(_e1), _rhs, _var)_(e1 = var) <-- { var == Sin(rhs) };
20 # Solve'Simple(ArcCos(_e1), _rhs, _var)_(e1 = var) <-- { var == Cos(rhs) };
20 # Solve'Simple(ArcTan(_e1), _rhs, _var)_(e1 = var) <-- { var == Tan(rhs) };
/* Note: Second rule neglects (2*I*Pi)-periodicity of Exp() */
10 # Solve'Simple(Exp(_e1), 0, _var)_(e1 = var) <-- { };
20 # Solve'Simple(Exp(_e1), _rhs, _var)_(e1 = var) <-- { var == Ln(rhs) };
20 # Solve'Simple(Ln(_e1), _rhs, _var)_(e1 = var) <-- { var == Exp(rhs) };
20 # Solve'Simple(_b^_e1, _rhs, _var)_(e1 = var And IsFreeOf(var,b) And Not IsZero(b)) <-- { var == Ln(rhs) / Ln(b) };
/* The range of Sqrt is the set of (complex) numbers with either
* positive real part, together with the pure imaginary numbers with
* nonnegative real part. */
20 # Solve'Simple(Sqrt(_e1), _rhs, _var)_(e1 = var And IsPositiveReal(Re(rhs)) And Re(rhs) != 0) <-- { var == rhs^2 };
20 # Solve'Simple(Sqrt(_e1), _rhs, _var)_(e1 = var And Re(rhs)=0 And IsPositiveReal(Im(rhs))) <-- { var == rhs^2 };
20 # Solve'Simple(Sqrt(_e1), _rhs, _var)_(e1 = var And Re(rhs)=0 And IsNegativeReal(Im(rhs)) And Im(rhs) != 0) <-- { };
20 # Solve'Simple(Sqrt(_e1), _rhs, _var)_(e1 = var And IsNegativeReal(Re(rhs)) And Re(rhs) != 0) <-- { };
30 # Solve'Simple(_lhs, _rhs, _var) <-- Failed;
/********** Solve'Divide **********/
/* For some classes of equations, it may be easier to solve them if we
* divide through by their first term. A simple example of this is the
* equation Sin(x)+Cos(x)==0
* One problem with this is that we may lose roots if the thing we
* are dividing by shares roots with the whole equation.
* The final HasExprs are an attempt to prevent infinite recursion caused by
* the final Simplify step in Solve undoing what we do here. It's conceivable
* though that this won't always work if the recurring loop is more than two
* steps long. I can't think of any ways this can happen though :)
*/
10 # Solve'Divide(_e1 + _e2, _var)_(HasExpr(e1, var) And HasExpr(e2, var)
And Not (HasExpr(Simplify(1 + (e2/e1)), e1)
Or HasExpr(Simplify(1 + (e2/e1)), e2)))
<-- Solve(1 + (e2/e1), var);
10 # Solve'Divide(_e1 - _e2, _var)_(HasExpr(e1, var) And HasExpr(e2, var)
And Not (HasExpr(Simplify(1 - (e2/e1)), e1)
Or HasExpr(Simplify(1 - (e2/e1)), e2)))
<-- Solve(1 - (e2/e1), var);
20 # Solve'Divide(_e, _v) <-- Failed;
/********** Solve'System **********/
// for now, just use a very simple backsubstitution scheme
Solve'System(_eqns, _vars) <-- Solve'SimpleBackSubstitution(eqns,vars);
// Check(False, "Solve'System: not implemented");
10 # Solve'SimpleBackSubstitution'FindAlternativeForms((_lx) == (_rx)) <--
[
Local(newEq);
newEq := (Simplify(lx) == Simplify(rx));
If (newEq != (lx == rx) And newEq != (0==0),DestructiveAppend(eq,newEq));
newEq := (Simplify(lx - rx) == 0);
If (newEq != (lx == rx) And newEq != (0==0),DestructiveAppend(eq,newEq));
];
20 # Solve'SimpleBackSubstitution'FindAlternativeForms(_equation) <--
[
];
UnFence("Solve'SimpleBackSubstitution'FindAlternativeForms",1);
/* Solving sets of equations using simple backsubstitution.
* Solve'SimpleBackSubstitution takes all combinations of equations and
* variables to solve for, and it then uses SuchThat to find an expression
* for this variable, and then if found backsubstitutes it in the other
* equations in the hope that they become simpler, resulting in a final
* set of solutions.
*/
10 # Solve'SimpleBackSubstitution(eq_IsList,var_IsList) <--
[
If(InVerboseMode(), Echo({"Entering Solve'SimpleBackSubstitution"}));
Local(result,i,j,nrvar,nreq,sub,nrSet,origEq);
eq:=FlatCopy(eq);
origEq:=FlatCopy(eq);
nrvar:=Length(var);
result:={FlatCopy(var)};
nrSet := 0;
//Echo("Before: ",eq);
ForEach(equation,origEq)
[
//Echo("equation ",equation);
Solve'SimpleBackSubstitution'FindAlternativeForms(equation);
];
// eq:=Simplify(eq);
//Echo("After: ",eq);
nreq:=Length(eq);
/* Loop over each variable, solving for it */
/* Echo({eq}); */
For(j:=1,j<=nreq And nrSet < nrvar,j++)
[
Local(vlist);
vlist:=VarListAll(eq[j],`Lambda({pt},Contains(@var,pt)));
For(i:=1,i<=nrvar And nrSet < nrvar,i++)
[
//Echo("eq[",j,"] = ",eq[j]);
//Echo("var[",i,"] = ",var[i]);
//Echo("varlist = ",vlist);
//Echo();
If(Count(vlist,var[i]) = 1,
[
sub := Listify(eq[j]);
sub := sub[2]-sub[3];
//Echo("using ",sub);
sub:=SuchThat(sub,var[i]);
If(InVerboseMode(), Echo({"From ",eq[j]," it follows that ",var[i]," = ",sub}));
If(SolveFullSimplify=True,
result:=Simplify(Subst(var[i],sub)result),
result[1][i]:=sub
);
//Echo("result = ",result," i = ",i);
nrSet++;
//Echo("current result is ",result);
Local(k,reset);
reset:=False;
For(k:=1,k<=nreq And nrSet < nrvar,k++)
If(Contains(VarListAll(eq[k],`Lambda({pt},Contains(@var,pt))),var[i]),
[
Local(original);
original:=eq[k];
eq[k]:=Subst(var[i],sub)eq[k];
If(Simplify(Simplify(eq[k])) = (0 == 0),
eq[k] := (0 == 0),
Solve'SimpleBackSubstitution'FindAlternativeForms(eq[k])
);
// eq[k]:=Simplify(eq[k]);
// eq[k]:=Simplify(eq[k]); //@@@??? TODO I found one example where simplifying twice gives a different result from simplifying once!
If(original!=(0==0) And eq[k] = (0 == 0),reset:=True);
If(InVerboseMode(), Echo({" ",original," simplifies to ",eq[k]}));
]);
nreq:=Length(eq);
vlist:=VarListAll(eq[j],`Lambda({pt},Contains(@var,pt)));
i:=nrvar+1;
// restart at the beginning of the variables.
If(reset,j:=1);
]);
];
];
//Echo("Finished finding results ",var," = ",result);
// eq:=origEq;
// nreq := Length(eq);
Local(zeroeq,tested);
tested:={};
// zeroeq:=FillList(0==0,nreq);
ForEach(item,result)
[
/*
Local(eqSimplified);
eqSimplified := eq;
ForEach(map,Transpose({var,item}))
[
eqSimplified := Subst(map[1],map[2])eqSimplified;
];
eqSimplified := Simplify(Simplify(eqSimplified));
Echo(eqSimplified);
If(eqSimplified = zeroeq,
[
DestructiveAppend(tested,Map("==",{var,item}));
]);
*/
DestructiveAppend(tested,Map("==",{var,item}));
];
/* Echo({"tested is ",tested}); */
If(InVerboseMode(), Echo({"Leaving Solve'SimpleBackSubstitution"}));
tested;
];
/********** OldSolve **********/
10 # OldSolve(eq_IsList,var_IsList) <-- Solve'SimpleBackSubstitution(eq,var);
90 # OldSolve((left_IsList) == right_IsList,_var) <--
OldSolve(Map("==",{left,right}),var);
100 # OldSolve(_left == _right,_var) <--
SuchThat(left - right , 0 , var);
/* HoldArg("OldSolve",arg1); */
/* HoldArg("OldSolve",arg2); */
10 # ContainsExpression(_body,_body) <-- True;
15 # ContainsExpression(body_IsAtom,_expr) <-- False;
20 # ContainsExpression(body_IsFunction,_expr) <--
[
Local(result,args);
result:=False;
args:=Tail(Listify(body));
While(args != {})
[
result:=ContainsExpression(Head(args),expr);
args:=Tail(args);
if (result = True) (args:={});
];
result;
];
SuchThat(_function,_var) <-- SuchThat(function,0,var);
10 # SuchThat(_left,_right,_var)_(left = var) <-- right;
/*This interferes a little with the multi-equation solver...
15 # SuchThat(_left,_right,_var)_CanBeUni(var,left-right) <--
PSolve(MakeUni(left-right,var));
*/
20 # SuchThat(left_IsAtom,_right,_var) <-- var;
30 # SuchThat((_x) + (_y),_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right-y , var);
30 # SuchThat((_y) + (_x),_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right-y , var);
30 # SuchThat(Complex(_r,_i),_right,_var)_ContainsExpression(r,var) <--
SuchThat(r , right-I*i , var);
30 # SuchThat(Complex(_r,_i),_right,_var)_ContainsExpression(i,var) <--
SuchThat(i , right+I*r , var);
30 # SuchThat(_x * _y,_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right/y , var);
30 # SuchThat(_y * _x,_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right/y , var);
30 # SuchThat(_x ^ _y,_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right^(1/y) , var);
30 # SuchThat(_x ^ _y,_right,_var)_ContainsExpression(y,var) <--
SuchThat(y , Ln(right)/Ln(x) , var);
30 # SuchThat(Sin(_x),_right,_var) <--
SuchThat(x , ArcSin(right) , var);
30 # SuchThat(ArcSin(_x),_right,_var) <--
SuchThat(x , Sin(right) , var);
30 # SuchThat(Cos(_x),_right,_var) <--
SuchThat(x , ArcCos(right) , var);
30 # SuchThat(ArcCos(_x),_right,_var) <--
SuchThat(x , Cos(right) , var);
30 # SuchThat(Tan(_x),_right,_var) <--
SuchThat(x , ArcTan(right) , var);
30 # SuchThat(ArcTan(_x),_right,_var) <--
SuchThat(x , Tan(right) , var);
30 # SuchThat(Exp(_x),_right,_var) <--
SuchThat(x , Ln(right) , var);
30 # SuchThat(Ln(_x),_right,_var) <--
SuchThat(x , Exp(right) , var);
30 # SuchThat(_x / _y,_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right*y , var);
30 # SuchThat(_y / _x,_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , y/right , var);
30 # SuchThat(- (_x),_right,_var) <--
SuchThat(x , -right , var);
30 # SuchThat((_x) - (_y),_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , right+y , var);
30 # SuchThat((_y) - (_x),_right,_var)_ContainsExpression(x,var) <--
SuchThat(x , y-right , var);
30 # SuchThat(Sqrt(_x),_right,_var) <--
SuchThat(x , right^2 , var);
Function("SolveMatrix",{matrix,vector})
[
Local(perms,indices,inv,det,n);
n:=Length(matrix);
indices:=Table(i,i,1,n,1);
perms:=Permutations(indices);
inv:=ZeroVector(n);
det:=0;
ForEach(item,perms)
[
Local(i,lc);
lc := LeviCivita(item);
det:=det+Factorize(i,1,n,matrix[i][item[i] ])* lc;
For(i:=1,i<=n,i++)
[
inv[i] := inv[i]+
Factorize(j,1,n,
If(item[j] =i,vector[j ],matrix[j][item[j] ]))*lc;
];
];
Check(det != 0, "Zero determinant");
(1/det)*inv;
];
Function("Newton",{function,variable,initial,accuracy})
[ // since we call a function with HoldArg(), we need to evaluate some variables by hand
`Newton(@function,@variable,initial,accuracy,-Infinity,Infinity);
];
Function("Newton",{function,variable,initial,accuracy,min,max})
[
Local(result,adjust,delta,requiredPrec);
MacroLocal(variable);
requiredPrec := Builtin'Precision'Get();
accuracy:=N((accuracy/10)*10); // Making sure accuracy is rounded correctly
Builtin'Precision'Set(requiredPrec+2);
function:=N(function);
adjust:= -function/Apply("D",{variable,function});
delta:=10000;
result:=initial;
While (result > min And result < max
// avoid numerical underflow due to fixed point math, FIXME when have real floating math
And N(Eval( Max(Re(delta), -Re(delta), Im(delta), -Im(delta)) ) ) > accuracy)
[
MacroSet(variable,result);
delta:=N(Eval(adjust));
result:=result+delta;
];
Builtin'Precision'Set(requiredPrec);
result:=N(Eval((result/10)*10)); // making sure result is rounded to correct precision
if (result <= min Or result >= max) [result := Fail;];
result;
];
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