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- ===== Equations System Solving by Substitution === SUBST.DOC === 30.03.00 ===
- Multiple Solutions Supported. Copyright (C) 2000 by Sergey V. Biryukov
- For DERIVE v.5.00 & later. Email: ciprel@cityline.ru Subject: DERIVE
-
- It is a document file for SUBST.MTH utility. See SUBST.DMO for additional info.
-
- The utility is for solving simple systems of nonlinear equations in symbolic
- form. Such systems are widely used in College Mathematics and especially in
- Physics. Dynamics for example, can not do without such systems with at least
- one nonlinear (quadratic) equation.
-
-
- TWO GENERAL PURPOSE USEFULL FUNCTIONS
-
- 1. SBST(u,x,x0) :=ITERATE(u,APPEND(x),APPEND(x0),1) - substitutes x=x0
- in expression u. x & x0 - vectors or 1 column matrices
- of variables and values respectively.
- 2. TRNSP(v) - transposes matrix v even it is a matrix of equations.
-
- Use All Below functions ONLY in EXACT mode !!!
-
- MAIN FUNCTION:
-
- SOLVE_(m,v):= solves matrix of eq. m for variables v by substitution.
- m - 1 column matrix of equations,
- v - vector (or 1 col. matrix) of solve variables.
- Eq. & variables order in m & v define substitution order
- All returned solutions are tested by substitution in the initial system.
-
-
- ADDITIONAL FUNCTIONS for step by step solving in the case of SOLVE_() fail &
- for solving process illustration & teaching:
-
- SOLF(m,v,n) makes forward substitution for n first equations from m &
- variables from v. The result is returned in the form of a solution tree.
-
- SOLB(u) applies back substitution to the above tree & return 1 or more
- solutions without its validity testing.
-
-
- SOLVING INSTRUCTIONS:
-
- 1. Use SOLVE_() ONLY in EXACT mode !!!
- 2. There is no need to check solutions. SOLVE_() do itself.
- 3. If SOLVE_() fail:
- - Try another equations and/or solve variables order
- - Restrict all variables domains as strong as possible. It will
- prevent exotic complex & senseless solutions.
- - Solve eq. system in 3 steps:
- - forward substitution ( SOLF(m,v,n) function)
- - back substitution ( SOLB(u) )
- - solutions test ( SBST(u,x,x0) )
- - Be patient & cheer up! Most of nonlinear systems has no exact solution.
- Try NEWTONS() function for numerical solution in Approximate mode
- (see DERIVE Help & User Manual).
- - Try to visualise the system or its parts
- - Try to look at the problem from the another point of view.
-
- ..........................................................................
- . .
- . Additional Information is in SUBST.DMO .
- . Try it, please ! .
-
- =========================== SUBST.DOC End ==============================
-