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calc: Solving Systems of Equations

 
 11.6.2 Solving Systems of Equations
 -----------------------------------
 
 You can also use the commands described above to solve systems of
 simultaneous equations.  Just create a vector of equations, then specify
 a vector of variables for which to solve.  (You can omit the surrounding
 brackets when entering the vector of variables at the prompt.)
 
    For example, putting ‘[x + y = a, x - y = b]’ on the stack and typing
 ‘a S x,y <RET>’ produces the vector of solutions ‘[x = a - (a-b)/2, y =
 (a-b)/2]’.  The result vector will have the same length as the variables
 vector, and the variables will be listed in the same order there.  Note
 that the solutions are not always simplified as far as possible; the
 solution for ‘x’ here could be improved by an application of the ‘a n’
 command.
 
    Calc’s algorithm works by trying to eliminate one variable at a time
 by solving one of the equations for that variable and then substituting
 into the other equations.  Calc will try all the possibilities, but you
 can speed things up by noting that Calc first tries to eliminate the
 first variable with the first equation, then the second variable with
 the second equation, and so on.  It also helps to put the simpler (e.g.,
 more linear) equations toward the front of the list.  Calc’s algorithm
 will solve any system of linear equations, and also many kinds of
 nonlinear systems.
 
    Normally there will be as many variables as equations.  If you give
 fewer variables than equations (an “over-determined” system of
 equations), Calc will find a partial solution.  For example, typing ‘a S
 y <RET>’ with the above system of equations would produce ‘[y = a - x]’.
 There are now several ways to express this solution in terms of the
 original variables; Calc uses the first one that it finds.  You can
 control the choice by adding variable specifiers of the form ‘elim(V)’
 to the variables list.  This says that V should be eliminated from the
 equations; the variable will not appear at all in the solution.  For
 example, typing ‘a S y,elim(x)’ would yield ‘[y = a - (b+a)/2]’.
 
    If the variables list contains only ‘elim’ specifiers, Calc simply
 eliminates those variables from the equations and then returns the
 resulting set of equations.  For example, ‘a S elim(x)’ produces ‘[a - 2
 y = b]’.  Every variable eliminated will reduce the number of equations
 in the system by one.
 
    Again, ‘a S’ gives you one solution to the system of equations.  If
 there are several solutions, you can use ‘H a S’ to get a general family
 of solutions, or, if there is a finite number of solutions, you can use
 ‘a P’ to get a list.  (In the latter case, the result will take the form
 of a matrix where the rows are different solutions and the columns
 correspond to the variables you requested.)
 
    Another way to deal with certain kinds of overdetermined systems of
 equations is the ‘a F’ command, which does least-squares fitting to
 satisfy the equations.  Curve Fitting.
 

Info Catalog calc: Multiple Solutions calc: Solving Equations calc: Decomposing Polynomials