#### Overview of this book

Table of Contents (16 chapters)
Mastering SciPy
Credits
About the Author
About the Reviewers
www.PacktPub.com
Preface
Free Chapter
Numerical Linear Algebra
Interpolation and Approximation
Differentiation and Integration
Nonlinear Equations and Optimization
Initial Value Problems for Ordinary Differential Equations
Computational Geometry
Descriptive Statistics
Inference and Data Analysis
Mathematical Imaging
Index

## Non-linear equations and systems

In the solution of linear equations and systems, f(x) = 0, we had the choice of using either direct methods or iterative processes. A direct method in that setting was simply the application of an exact formula involving only the four basic operations: addition, subtraction, multiplication, and division. The issues with this method arise when cancellation occurs, mainly whenever sums and subtractions are present. Iterative methods, rather than computing a solution in a finite number of operations, calculate closer and closer approximations to the said solution, improving the accuracy with each step.

In the case of nonlinear equations, direct methods are seldom a good idea. Even when a formula is available, the presence of nonbasic operations leads to uncomfortable rounding errors. Let's see this using a very basic example.

Consider the quadratic equation ax2 + bx + c = 0, with a = 10–10, b = –(1010 + 1)/1010, and c = 1. These are the coefficients of the expanded...