Course Title: Numerical Methods
Course Code: MATH 233
Solving Nonlinear Equations
Objectives
1. Explain the purpose of numerical methods for solving nonlinear equations and systems.
2. Use major numerical methods to approximate roots of nonlinear equations.
3. Assess convergence and accuracy of iterative solutions.
4. Solve systems of nonlinear equations using appropriate numerical techniques.
1 Introduction
Many problems in science and engineering require solving equations that cannot be solved analytically. An
equation of one variable can be written in the form:
𝑓(𝑥) = 0 3.1
Fig. 1. Illustration of equations with no, one, or several solutions.
A solution (root) is a numerical value of 𝑥 that satisfies the equation. Graphically, the solution corresponds
to the point where the function crosses or touches the 𝑥-axis Fig. 1.
When equations are simple, analytical methods may be used. However, many engineering equations contain
transcendental terms and cannot be solved analytically, requiring numerical methods.
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public
posting without permission is not allowed. For more of these, contact: stuvia
, Illustrative Example
The area of the segment 𝐴𝑠 (Fig. 2), of a circle with radius 𝑟 is
given by
1
𝐴𝑠 = 𝑟 2 (𝜃 − 𝑠𝑖𝑛 𝜃) 3.2
2
To determine the angle 𝜃 if 𝐴𝑠 and 𝑟 are given, Eq. (3.2) has to be
solved for 𝜃. Obviously, 𝜃 cannot be written explicitly in terms of
𝐴𝑠 and 𝑟, and the equation cannot be solved analytically.
Fig. 2. Segment of a circle.
1
𝑓(𝜃) = 0 = 𝐴𝑠 − 𝑟 2 (𝜃 − 𝑠𝑖𝑛 𝜃)
2
A numerical solution of an equation 𝑓(𝑥) = 0 is a value of 𝑥 that
satisfies the equation approximately. This means that when 𝑥 is
substituted in the equation, the value of 𝑓(𝑥) is close to zero, but
not exactly zero. For example, when 𝐴𝑠 = 8 m2, 𝑟 = 3 m,
𝑓(𝜃) = 0 = 8 − 4.5(𝜃 − sin 𝜃)
The graph crosses between 2 and 3 (Fig. 3). We try 𝜃 =
2.4, 𝑓(𝜃) = 0.2396. For 𝜃 = 2.43, 𝑓(𝜃) = 0.003683. It is
Fig. 3. A plot of 𝒇(𝜽) = 𝟖 − 𝟒. 𝟓(𝜽 − possible to determine values of 𝜃 that give values of 𝑓(𝜃) that are
𝒔𝒊𝒏 𝜽). closer to zero, but it is impossible to determine a numerical value
of 𝜃 for which 𝑓(𝜃) is exactly zero.
1.1 Numerical vs Analytical Solutions
An analytical solution is obtained by deriving an expression that gives an exact numerical value of the
unknown. A numerical solution is obtained through a process that:
• starts with an approximate solution,
• applies a numerical procedure,
• improves accuracy iteratively.
A numerical solution satisfies the equation approximately, meaning 𝑓(𝑥) is close to zero but not exactly
zero.
1.2 Initial Estimation of Roots
An initial estimate of a solution to 𝑓(𝑥) = 0 can be obtained by:
• plotting 𝑓(𝑥) versus 𝑥 and looking where the function crosses the 𝑥-axis; or
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public
posting without permission is not allowed. For more of these, contact: stuvia
, • detecting sign changes of 𝑓(𝑥) because a change in sign indicates that a root lies within the
interval. Incremental search methods capitalize on this observation by locating an interval
where the function changes sign.
When multiple roots exist, numerical solutions are obtained one root at a time.
1.3 Classification of Numerical Methods
Numerical root-finding methods are classified into two groups with the following properties
Bracketing Open
1. Bisection 1. Newton’s
2. Regula Falsi 2. Secant
3. Fixed-Point Iteration
• Require an interval [𝑎, 𝑏] that • Use one initial estimate.
contains the solution. • Can either diverge or converge
• Endpoints must satisfy: rapidly, depending on the value of
• 𝑓(𝑎)𝑓(𝑏) < 0 the initial guess
• These methods always convergent • Open method is illustrated below
if assumptions are met.
• Bracketing is illustrated below
Fig. 4. Illustration of bracketing method. Fig. 5. Illustration of an open method.
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public
posting without permission is not allowed. For more of these, contact: stuvia
Course Code: MATH 233
Solving Nonlinear Equations
Objectives
1. Explain the purpose of numerical methods for solving nonlinear equations and systems.
2. Use major numerical methods to approximate roots of nonlinear equations.
3. Assess convergence and accuracy of iterative solutions.
4. Solve systems of nonlinear equations using appropriate numerical techniques.
1 Introduction
Many problems in science and engineering require solving equations that cannot be solved analytically. An
equation of one variable can be written in the form:
𝑓(𝑥) = 0 3.1
Fig. 1. Illustration of equations with no, one, or several solutions.
A solution (root) is a numerical value of 𝑥 that satisfies the equation. Graphically, the solution corresponds
to the point where the function crosses or touches the 𝑥-axis Fig. 1.
When equations are simple, analytical methods may be used. However, many engineering equations contain
transcendental terms and cannot be solved analytically, requiring numerical methods.
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public
posting without permission is not allowed. For more of these, contact: stuvia
, Illustrative Example
The area of the segment 𝐴𝑠 (Fig. 2), of a circle with radius 𝑟 is
given by
1
𝐴𝑠 = 𝑟 2 (𝜃 − 𝑠𝑖𝑛 𝜃) 3.2
2
To determine the angle 𝜃 if 𝐴𝑠 and 𝑟 are given, Eq. (3.2) has to be
solved for 𝜃. Obviously, 𝜃 cannot be written explicitly in terms of
𝐴𝑠 and 𝑟, and the equation cannot be solved analytically.
Fig. 2. Segment of a circle.
1
𝑓(𝜃) = 0 = 𝐴𝑠 − 𝑟 2 (𝜃 − 𝑠𝑖𝑛 𝜃)
2
A numerical solution of an equation 𝑓(𝑥) = 0 is a value of 𝑥 that
satisfies the equation approximately. This means that when 𝑥 is
substituted in the equation, the value of 𝑓(𝑥) is close to zero, but
not exactly zero. For example, when 𝐴𝑠 = 8 m2, 𝑟 = 3 m,
𝑓(𝜃) = 0 = 8 − 4.5(𝜃 − sin 𝜃)
The graph crosses between 2 and 3 (Fig. 3). We try 𝜃 =
2.4, 𝑓(𝜃) = 0.2396. For 𝜃 = 2.43, 𝑓(𝜃) = 0.003683. It is
Fig. 3. A plot of 𝒇(𝜽) = 𝟖 − 𝟒. 𝟓(𝜽 − possible to determine values of 𝜃 that give values of 𝑓(𝜃) that are
𝒔𝒊𝒏 𝜽). closer to zero, but it is impossible to determine a numerical value
of 𝜃 for which 𝑓(𝜃) is exactly zero.
1.1 Numerical vs Analytical Solutions
An analytical solution is obtained by deriving an expression that gives an exact numerical value of the
unknown. A numerical solution is obtained through a process that:
• starts with an approximate solution,
• applies a numerical procedure,
• improves accuracy iteratively.
A numerical solution satisfies the equation approximately, meaning 𝑓(𝑥) is close to zero but not exactly
zero.
1.2 Initial Estimation of Roots
An initial estimate of a solution to 𝑓(𝑥) = 0 can be obtained by:
• plotting 𝑓(𝑥) versus 𝑥 and looking where the function crosses the 𝑥-axis; or
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public
posting without permission is not allowed. For more of these, contact: stuvia
, • detecting sign changes of 𝑓(𝑥) because a change in sign indicates that a root lies within the
interval. Incremental search methods capitalize on this observation by locating an interval
where the function changes sign.
When multiple roots exist, numerical solutions are obtained one root at a time.
1.3 Classification of Numerical Methods
Numerical root-finding methods are classified into two groups with the following properties
Bracketing Open
1. Bisection 1. Newton’s
2. Regula Falsi 2. Secant
3. Fixed-Point Iteration
• Require an interval [𝑎, 𝑏] that • Use one initial estimate.
contains the solution. • Can either diverge or converge
• Endpoints must satisfy: rapidly, depending on the value of
• 𝑓(𝑎)𝑓(𝑏) < 0 the initial guess
• These methods always convergent • Open method is illustrated below
if assumptions are met.
• Bracketing is illustrated below
Fig. 4. Illustration of bracketing method. Fig. 5. Illustration of an open method.
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public
posting without permission is not allowed. For more of these, contact: stuvia
