1. Computers in Civil Engineering 53:081 Spring 2003 Lecture #8 Roots of Equations: Systems of Equations

2. Problem Statement Up to now we have considered finding the root(s) of a single equation: Now consider finding roots of a set of simultaneous equations: That is, finding a set of x values that simultaneously result in all functions being zero.

3. Example Consider the set of equations: This can be written as (think of x,y as x 1, x 2 ): Roots simultaneously make both functions zero:

4. Discussion of Methods lGraphical Representation –Not so easy to plot 2- or 3-dimensional systems. For example: –How does one plot 4-, 5-, and higher dimensional systems? lBracketing Methods –Inefficient as they would imply search over multi- dimensional space lIterative Methods – Approach of choice

5. Newton-Raphson Algorithm Recall single equation derivation: recall i denotes latest estimate; i+1 is new estimate

6. Newton-Raphson 2-D Generalization Denote: Now our system is: 2-D Taylor series expansion (a.k.a. chain rule):

7. Compare with 1-D case:

8. Our objective for i+1 st iteration is: Rearrange: This is a set of linear equations (no products or roots of variables, all variables occur only to first power) with the only unknowns. Since the system is linear, one can use algebraic manipulations to solve for

9. Solution Denominator is called the “determinant of the Jacobian of the system”

10. Generalization to n-dimensions As before, at the location of the root:

11. and rearrange:Set This is a again as set of linear equations with the only unknowns. Since the system is linear, one can use algebraic manipulations solve.

12. Example Recall the following equation from the 2-D generalization of the Newton-Raphson method: In matrix form:

13. Next: Solving Linear Systems Read the Text