7
Solving Nonlinear Algebraic Equations
As a reader of this book, you might be well into mathematics and often “accused” of
being particularly good at solving equations (a typical comment at family dinners!).
How true is it, however, that you can solve many types of equations with pen and
paper alone? Restricting our attention to algebraic equations in one unknown x, you
can certainly do linear equations: ax + b = 0, and quadratic ones: ax 2 + bx + c =
0. You may also know that there are formulas for the roots of cubic and quartic
equations too. Maybe you can do the special trigonometric equation sin x + cos x =
1 as well, but there it (probably?) stops. Equations that are not reducible to one of
those mentioned, cannot be solved by general analytical techniques, which means
that most algebraic equations arising in applications cannot be treated with pen and
paper!
If we exchange the traditional idea of finding exact solutions to equations with
the idea of rather finding approximate solutions, a whole new world of possibilities
opens up. With such an approach, we can in principle solve any algebraic equation.
Let us start by introducing a common generic form for any algebraic equation:
f (x) = 0 .
© The Author(s) 2020
S. Linge, H. P. Langtangen, Programming for Computations - Python,
Texts in Computational Science and Engineering 15,
https://doi.org/10.1007/978-3-030-16877-3_7
175
Solving Nonlinear Algebraic Equations
As a reader of this book, you might be well into mathematics and often “accused” of
being particularly good at solving equations (a typical comment at family dinners!).
How true is it, however, that you can solve many types of equations with pen and
paper alone? Restricting our attention to algebraic equations in one unknown x, you
can certainly do linear equations: ax + b = 0, and quadratic ones: ax 2 + bx + c =
0. You may also know that there are formulas for the roots of cubic and quartic
equations too. Maybe you can do the special trigonometric equation sin x + cos x =
1 as well, but there it (probably?) stops. Equations that are not reducible to one of
those mentioned, cannot be solved by general analytical techniques, which means
that most algebraic equations arising in applications cannot be treated with pen and
paper!
If we exchange the traditional idea of finding exact solutions to equations with
the idea of rather finding approximate solutions, a whole new world of possibilities
opens up. With such an approach, we can in principle solve any algebraic equation.
Let us start by introducing a common generic form for any algebraic equation:
f (x) = 0 .
© The Author(s) 2020
S. Linge, H. P. Langtangen, Programming for Computations - Python,
Texts in Computational Science and Engineering 15,
https://doi.org/10.1007/978-3-030-16877-3_7
175