Derive_v5_05 / Derive / Users / EquationSolving / Substitution

===== 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: [e-mail]  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.            
                                                             
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.                                                                        . 
.               Additional Information is in SUBST.DMO                   . 
.                         Try it, please !                               .

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