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# linearize(?Expression, -Linear, -Residue)

Split and arithmetic expression into linear and nonlinear parts

Expression
  Arithmetic expression with constants and variables
Linear
  Normalized lists of monomials
Residue
  Residual nonlinear components in the form AuxVar=Expr

## Description

The linear component of expression is normalised into a list (sum) of monomials of the form

    		[C0*1, C1*X1, C2*X2, ...]

where Ci are numbers and Xi are distinct variables. The first (constant) term is always present, Ci (i>=1) are nonzero.

linearize/3 converts a general arithmetic expression into a normalised linear form. The nonlinear parts are extracted using auxiliary variables, and returned as Residue, which is a list of Aux=NonLinExpr constraints.

linearize/3 understands all arithmetic expressions plus

    	    lin(Lin)

where Lin is an already normalised linear expression. All variables within Expression (which are free at linearization time) are taken to be numerical variables. If you intend to have variables which can be bound to symbolic expressions rather than numbers, they must be wrapped into an eval/1 functor.

### Modules

This predicate is sensitive to its module context (tool predicate, see @/2).

## Examples

        ?- linearize(3*X-7*Y+2*(X+Y), L, R).
        X = X
        Y = Y
        L = [0 * 1, 5 * X, -5 * Y]
        R = []
        yes.

        ?- linearize(12*3, L, R).
        L = [36 * 1]
        R = []
        yes.

        ?- linearize(X*(3+7*Y)-X, L, R).
        Y = Y
        X = X
        L = [0 * 1, 1 * _308, 2 * X]
        R = [_308 = X * (7 * Y)]
        yes.

## See Also

polynorm / 3, linrenorm / 2, delinearize / 2