Definition of Error

An error is the difference between the observed or approximated value and the true or actual value. When performing calculations, errors inevitably arise from various sources.

Types of Errors

1. Systematic errors result from known causes such as incorrect formulas or improperly calibrated instruments.

2. Random errors arise from unpredictable and unknown sources like environmental conditions.

Sources of Errors

The main sources of errors are:

• Rounding-off errors: Caused by approximating numbers during computations (e.g., π ≈ 3.14)
• Truncation errors: Caused by terminating an infinite mathematical process prematurely
• Data uncertainty: The quantitative measure of error in the data

Absolute and Relative Errors

Absolute error measures how far a measurement is from the true value:

$\varepsilon_a = |x - x^*|$

where $x$ is the true value and $x^*$ is the approximated value.

Relative error shows how large the error is relative to the true value:

$\varepsilon_r = \frac{|x - x^*|}{|x|}$

Percentage relative error:

$\varepsilon_p = \frac{|x - x^*|}{|x|} \times 100\%$

Example 1: Calculating Absolute and Relative Errors

If the exact value is $x = 2.2056 \times 10^{-4}$ and it is approximated by $x^* = 2.1056 \times 10^{-4}$:

Solution

Absolute error: $\varepsilon_a = |2.2056 \times 10^{-4} - 2.1056 \times 10^{-4}| = 1 \times 10^{-5}$

Relative error: $\varepsilon_r = \frac{1 \times 10^{-5}}{2.2056 \times 10^{-4}} = 0.04534$

Percentage error: $\varepsilon_p = 0.04534 \times 100\% = 4.534\%$

The Newton-Raphson method is an iterative technique for finding roots of equations. It uses the tangent line to approximate the root.

Formula

Given $f(x) = 0$, the Newton-Raphson formula is:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

where $x_n$ is the current approximation and $x_{n+1}$ is the next approximation.

Conditions for Application

• The function must be differentiable
• An initial guess $x_0$ must be selected, preferably in an interval where $f(a) \cdot f(b) < 0$

Example 2: Newton-Raphson Method

Find the root of $f(x) = x^2 - 2x - 1$ in the interval [2, 3], starting with $x_0 = 2$, for 3 iterations.

Solution

Given $f(x) = x^2 - 2x - 1$, we have $f'(x) = 2x - 2$

1st iteration: $x_1 = 2 - \frac{f(2)}{f'(2)} = 2 - \frac{-1}{2} = 2.5$

2nd iteration: $x_2 = 2.5 - \frac{f(2.5)}{f'(2.5)} = 2.5 - \frac{0.25}{3} = 2.416667$

3rd iteration: $x_3 = 2.416667 - \frac{0.00695}{2.83333} = 2.414214$

Therefore, the approximate root is $x \approx 2.414$

Applications of Newton-Raphson

Square root approximation: $x_{n+1} = \frac{1}{2}\left(x_n + \frac{N}{x_n}\right)$

rth root of N: $x_{n+1} = \frac{(r-1)x_n^r + N}{rx_n^{r-1}}$

Reciprocal of N: $x_{n+1} = x_n(2 - Nx_n)$

The secant method is similar to Newton-Raphson but does not require the derivative. It uses two initial values and approximates the derivative using a finite difference.

Formula

$x_{n+1} = x_n - \frac{(x_n - x_{n-1})}{f(x_n) - f(x_{n-1})} \cdot f(x_n)$

Example 3: Secant Method

Find the root of $f(x) = x^2 - 2$ using $x_0 = 1.5$ and $x_1 = 1$, for 3 iterations.

Solution

1st iteration ($n = 1$): $x_2 = 1 - \frac{(1 - 1.5)}{(-1) - 0.25} \times (-1) = 1.4$

2nd iteration ($n = 2$): $x_3 = 1.4 - \frac{(1.4 - 1)}{-0.04 - (-1)} \times (-0.04) = 1.4167$

3rd iteration ($n = 3$): $x_4 = 1.4167 - \frac{(1.4167 - 1.4)}{0.00704 - (-0.04)} \times 0.00704 = 1.4142$

Therefore, the approximate root is $x \approx 1.4142$

Numerical integration approximates the value of a definite integral when analytical methods are difficult or impossible.

Trapezoidal Rule

The trapezoidal rule approximates the area under a curve by dividing it into trapezoids rather than rectangles.

Formula:

$\int_a^b f(x)\,dx \approx \frac{h}{2}[(y_0 + y_n) + 2(y_1 + y_2 + \cdots + y_{n-1})]$

where $h = \frac{b-a}{n}$ and $n$ is the number of strips.

Example 4: Trapezoidal Rule

Evaluate $\int_0^1 x^2 e^{-x^3}\,dx$ using five strips.

Solution

Given $a = 0$, $b = 1$, $n = 5$, so $h = 0.2$

Using the trapezoidal rule: $\int_0^1 f(x)\,dx \approx \frac{0.2}{2}[(0 + 0.3679) + 2(0.0397 + 0.1501 + 0.2901 + 0.3835)]$

$\approx 0.2095$

Simpson's Rule

Simpson's rule approximates the integral using quadratic functions (parabolas). It requires an even number of strips.

Formula:

$\int_a^b f(x)\,dx \approx \frac{h}{3}[(y_0 + y_n) + 4(y_1 + y_3 + \cdots + y_{n-1}) + 2(y_2 + y_4 + \cdots + y_{n-2})]$

where $h = \frac{b-a}{n}$ and $n$ must be even.

Example 5: Simpson's Rule

Evaluate $\int_0^2 \sqrt{x^2 + 1}\,dx$ with $n = 4$ subintervals.

Solution

Given $a = 0$, $b = 2$, $n = 4$, so $h = \frac{1}{2}$

The ordinates are:

• $y_0 = f(0) = 1$
• $y_1 = f(0.5) = \sqrt{1.25}$
• $y_2 = f(1) = \sqrt{2}$
• $y_3 = f(1.5) = \sqrt{3.25}$
• $y_4 = f(2) = \sqrt{5}$

Using Simpson's rule: $\int_0^2 f(x)\,dx \approx \frac{0.5}{3}[(y_0 + y_4) + 4(y_1 + y_3) + 2(y_2)]$

$\approx 2.95796$

In Tanzania, numerical methods are used in many practical situations. For example, when a small-scale farmer in Morogoro wants to calculate the area of an irregularly shaped plot of land to determine how much fertilizer is needed, the trapezoidal rule can be used to approximate the area from measured distances along the boundary. Similarly, banks use iterative methods like Newton-Raphson to calculate interest rates on loans, and engineers apply these techniques to analyze structural loads in building construction projects across the country.