Multivariate Newton-Raphson

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Problem Description

A program to solve sets of equations (N equations with N independent variables), using Newton-Raphson method of successive approximation.

Background & Techniques

Sample case:
Circle, Parabola Intersection

Every student who has had a math class that included the word term "derivative", has probably used Newton's method to calculate the square root of 2.    See the excellent Wikipedia article for a refresher. The extension of this method to sets of equations with multiple independent variables is completely analogous to the single variable method  if we replace the single variable derivative with the partial derivatives of each of the functions with respect to each of the variables.  The math gets messier as does defining a user interface to enter the equations, but the concepts are the same.   The best explanation I found for Newton-Raphson is the set of lecture notes by Dr David Keffer at University of Tennessee at Knoxville.  You can check them out at

I was motivated to explore the multivariate Newton-Raphson method by my previous work on Point from 4 Sensors. which defines distance equations for sensors (or satellites) with known locations to define the location of a target (or GPS receiver).  For demo purpose, the equations in the current program are limited to quadratic polynomials with 4 or fewer independent variables although it could certainly be extended to more complex types.     The initial default case solves the equations illustrated above, a parabola intersecting a circle. 

 The program allows you to specify the number of variables, N, from 1 to 4, for which you need to supply the coefficients for N equations and also N initial guesses for variable values which solve the system.   The program handles the business of making successive estimates until the number of iterations have been tried or, preferably,  the estimates converge close enough satisfying the equations.

You can supply the tolerance; the largest difference between the current equation values and zero (the exact solution).   You can also set the maximum number of iterations before stopping the execution.  In addition to the Solve button, there are buttons to Save and Load cases.    A few sample cases are included with the downloads. 

Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download executable version of the program.

Notes for Programmers

There were two distinct problems to solve when writing this program; implmenting the Newton-Raphson methed and how to let the user input problems to be solved.

The more interesting one was implementing the Newton-Raphson method itself.   The NewtonMulti function is contained in unit UNewtonMulti for portability and has this calling sequence:

Function NewtonMulti(
N:integer; {The number of variables and equations}
 MaxIter:integer;  {Maximum number of iterations to run}
 Tolerance:extended;  {The maximum residual values}
var X:TnVector;  {The array of initial variable value guesses on entry, final result values at exit time}
var Iter:integer;   {The number of iterations executed}
memo1:TMemo;  {A Tmemo where results of each iteration are displayed, Nil = no display}
CallbackRJ:TGetResidualsAndJacobian {Calculates residuals and next Jacobian}

The calling program must provide the procedure to calculate the array of Residuals, the result when the current variable estimates are plugged into the functions, and the Jacobian matrix, the array of partial derivative for each function with respect to each variable.   With these two pieces of data, NewtonMulti can calculate the next set of variable estimates, XN+1  = XN   - (JN)-1RN.  I.E. the next estimates are calculated as the previous estimates minus the changes in variable values which would be solutions if the functions were to proceed to the axis crossing in a straight line with the slope defined by the partial derivatives.  (There, I described it in words, which may not be much of an improvement over saying it in math symbols .   There's an excellent paper by an engineering professor at University of Tennessee Knoxville but which currently seems to be unreachable.  I'm trying track it down  and will post a link here when I do.)

The second problem, user interface, I solved by restricting systems analyzed to order 4 or less quadratic polynomials.  That still requires up to 36 inputs for which Iused TEdit  controls in an array with the Tag properties indicating the row and column to which each refers.   (Row = Tag div 10 and Col = Tag mod 10).

With a little fancy footwork turning off the Visible properties for variables and equations not applicable for the current number of variables, it seems to work OK.  All of the coefficient changing takes place in a separat dialog, CoeffDlg, in unit U_CoeffDlg.   Values are transferred to arrays JacX2Coeff and JacXCoeff containing the squared and linear coefficients for each variable of each equation with vector JacConst  containing the constant terms for each.  .The type definition for these arrays and vectors are contained in the included  UMatrix unit.   UMatrix encapsulates the old Borland Turbo Pascal Numeric Toolbox unit  and  contains one or two matrix operations used here as well.

Running/Exploring the Program 

bullet Download source
bullet Download  executable

Suggestions for Further Explorations

bullet More generalized equation inputs, perhaps freeform with a parser.
bullet The Newton-Raphson method is famous (or infamous) for not converging if initial estimates are not sufficiently near to the solution values.  Tests should be able to detect and warn the user when this is the case.


Original Date: January 24, 2009 

Modified: July 29, 2017



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