Solving equations with several unknowns in other words, solving system of equations online (fix-argument/drop-parameter f i 1) N)))įix-argument/drop-parameter takes a procedure, an argument index i, and aĬonstant value, and returns the procedure that takes one less argument andĬor-responds to the original with the i-th argument substituted by the constant.The solver of systems of linear equations allows to solve equations with several unknowns: system of equations with 2 unknowns, system of equations with 3 unknowns, system with n unknowns. For a zero, xi = ((f(xi) - N) / D) = ((-N) / D) (let* ((N (fix-argument/drop-parameter f i 0)) (find-eqn (cdr S) (cons (car S) eqns)))))) (define (try unk-index) (introduce f inversion unk-index)) (append (reverse eqns) (cdr S))) (aggregate* (*segment n* 0 unk-index) unknown by using the rest, and inserts it that computes the value of the eliminated (inverter (car S) unk-index pred-result)) (n* (-1+ n-unknowns))) => (lambda (pred-result) (let ((inversion Can this equation be inverted to eliminate > Try the next unknown (try (1+ unk-index))) (let ((n-unknowns (proc/arity (car S)))) (define (find-eqn unk-index) (define (equation-seeker predicate inverter) (lambda (S) The backtracking mechanism is not germane to the discussion. Without extending our solutions to encompass discrete parameters, quadratic-eliminate can produce both by backtracking. The corresponding unknown by expressing it in terms of the remaining unknowns.Įquation-seekertries all the equations and all the argument indices:ĥ Quadratic equations typically have two solutions. It is given the equationĪnd the index of the unknown as arguments. Whether an unknown can be eliminated by using an equation. It takesĪ predicate procedure and an inversion procedure. (not (independent? f i)))) (lambda (f i ignore)Įquation-seekerconstructs a solution method that the solver can use. Itera-tive-solveis a trivial root nder that can be used to solve uni-dimensional equations These equationsĬan appear after reducing a system with algebraically dependent equations. 5 constant-eli-minateremoves tautologies, i.e. Quadratic-eliminate is similar, but solves quadratic equations. WeĪffine-eliminateembodies the method outlined earlier and is examined below. Our list of methods can contain any procedures that can reduce the system. Rational functions and to perform other similar tasks. Simplify-eqn, used in solve-aux, can be used to remove denominators from Is just the composition of the previous overall partial solution and the local partial Solve-auxon the residual system, and the corresponding overall partial solution that When a method succeedes, try-methodscalls Try-methodstests whether the rst method in the list reduces the system, and if
Tautologies (identically zero) after substitution, the partial solution is the identity When the method eliminatesĮqua-tions without eliminating unknowns, perhaps because the equaequa-tions have become Unknowns in terms of the remaining unknowns. When unknowns are eliminated, the partial solution computes the eliminated So-lution methods typically remove some equations or some unknowns from the original To the residual system into solutions of the system given to the solution method.
As before, a partial solution maps solutions Otherwise solve-auxuses try-methodsĪ solution method is a procedure that takes a system of equations and returns a If there are no equations left to solve, we are done,Īnd the partial solution is the total solution. Solve-aux takes two arguments: the residual system that we still need to solve,Īnd a partial solution, i.e., a function that maps solutions of the residual system into Method eliminated a variable or an equation (solve-aux S1 (compose F f1))))))) (if (and (= (length S1) (length S)) (= (proc/arity (car S))
(with-values (lambda () ((car methods) S)) (lambda (S1 f1) (define (try-methods methods S F) (if (null? methods) (error "solve: Inconsistent number of unknowns" S) (solve-aux S (composition-identity arity)))))) (define (solve-aux S F) (let ((arity (proc/arity (car S)))) (if (there-exists? (cdr S) Solution, i.e., the composition identity of N parameters. Our top-level procedure performs some consistency checking and prepares the null The solver could not solve the system fully, and some other method will have to be The solution procedure is the dimensionality of the solution space. Is empty, the original system is completely solved, and the number of parameters of The residual system to produce solutions of the original system. Represented as a list of procedures, and a solution procedure to invoke on solutions of Procedures representing the equations, and returns two values, a residual system Let us now examine a solver written using these ideas.
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