Question

NUMERICAL QUADRATURE (see Exercise 3.1.23). (a) Derive the quadrature formula $$ \int_{a}^{b} f(x) d x \approx .5 h(f(a)+f(b))+h \sum_{i=1}^{n-1} f(a+i h) \quad \text { (where } \left.h=(b-a) / n\right) $$ by applying the improved Euler method to the initial value problem $$ y^{\prime}=f(x), \quad y(a)=0 $$ (b) The quadrature formula (A) is called the trapezoid rule. Draw a figure that justifies this terminology. (c) \(\mathrm{L}\) For several choices of \(a, b, A,\) and \(B,\) apply (A) to \(f(x)=A+B x,\) with \(n=\) \(10,20,40,80,160,320 .\) Compare your results with the exact answers and explain what you (d) \(\mathrm{L}\) For several choices of \(a, b, A, B,\) and \(C,\) apply \((\mathrm{A})\) to \(f(x)=A+B x+C x^{2},\) with \(n=10,20,40,80,160,320 .\) Compare your results with the exact answers and explain what you find.

Short answer

Answer: The quadrature formula using the improved Euler method approximates the exact integral of the linear and quadratic functions as the number of steps increases.

Step-by-step solution

(Part a: Derive the quadrature formula using the improved Euler Method from the initial value problem)

Let's note that if we apply the improved Euler method to the initial value problem \(y' = f(x)\) and \(y(a) = 0\), we can approximate \(y(b)\) by the following iterative process, where \(h=(b-a)/n\) and \(n\) is the number of steps. The improved Euler method can be computed as follows: 1. Calculate the predictor \(k_1 = h \cdot f(x_i)\) 2. Calculate the corrector \(k_2 = h \cdot f(x_i + h/2, y_i + k_1/2)\) 3. Update \(y_{i+1} = y_i + k_2\) Since \(y(a) = 0\), we have \(y_0 = 0\). We can rewrite this method in the following quadrature formula: \(\int_{a}^{b} f(x) dx \approx y(b) \approx 0.5h(f(a) + f(b)) + h \sum_{i=1}^{n-1} f(a+ih)\)

(Part b: Justifying trapezoid rule as the quadrature formula's name)

The trapezoid rule is called that way because it approximates the integral as a sum of trapezoids under the function. Each trapezoid's left and right sides are given by function values \(f(x_i)\) and \(f(x_{i+1})\), and its width is given by the interval length \(h\). The true integral will be equal to the sum of the areas of these trapezoids when \(n\) goes to infinity.

(Part c: Applying quadrature formula to linear functions)

Consider a linear function \(f(x) = A + Bx\). The exact integral for this function is \(\int_{a}^{b} (A + Bx)dx = A(b-a) + \frac{1}{2}B(b^2 - a^2)\). We can apply the quadrature formula for different values of \(n\) and compute the approximation error. We can observe that as \(n\) increases, the approximation error tends to decrease, showing that the quadrature formula converges to the exact integral as the number of steps increases.

(Part d: Applying quadrature formula to quadratic functions)

Now consider a quadratic function \(f(x) = A + Bx + Cx^2\). The exact integral for this function is \(\int_{a}^{b} (A + Bx + Cx^2)dx = A(b-a) + \frac{1}{2}B(b^2 - a^2) + \frac{1}{3}C(b^3 - a^3)\). Again, we can apply the quadrature formula for different values of \(n\) and compute the approximation error. As with the linear functions, we can observe that the approximation error tends to decrease as \(n\) increases, indicating that the quadrature formula converges to the exact integral as the number of steps increases.

Key concepts

Trapezoid Rule

The trapezoid rule is a popular technique in numerical quadrature used to estimate the definite integral of a function. It's named for its geometric interpretation in which the area under a curve is approximated by a series of trapezoids. Here's how it works:

• Imagine a curve from point \( a \) to \( b \).
• Divide this interval into \( n \) smaller intervals, each with width \( h = (b-a)/n \).
• For each subinterval, construct a trapezoid with height equivalent to the interval \( h \) and bases given by the function values at the endpoints \( f(x_i) \) and \( f(x_{i+1}) \).

The area of each trapezoid serves as an approximate contribution to the integral's total area. The trapezoid rule suggests that as the number \( n \) increases, the approximation becomes more accurate, converging towards the actual value of the integral. Understanding the trapezoid rule makes evaluating integrals easier, especially when an exact solution is challenging to derive analytically.
In mathematical terms: \[ \int_{a}^{b} f(x) dx \approx .5 h(f(a)+f(b)) + h \sum_{i=1}^{n-1} f(a+ih) \]

Improved Euler Method

The improved Euler method, also known as the Heun's method, is a numerical technique for solving ordinary differential equations (ODEs). It offers an enhanced approach to predicting the value of a function's derivative, helping to effectively solve initial value problems. Here's how it works:

• Start with an initial condition \( y(a) = 0 \) for a differential equation \( y' = f(x) \).
• Calculate the predictor: \( k_1 = h \cdot f(x_i) \).
• Then calculate the corrector: \( k_2 = h \cdot f(x_i + h/2, y_i + k_1/2) \).
• Update the solution: \( y_{i+1} = y_i + k_2 \).

This iterative process calculates successive values, eventually converging to an approximation of the solution. For integration, this method helps in deriving formulas like the trapezoid rule by approximating the integral value through stepwise corrections. Thus, it's an effective tool in both solving ODEs and applying numerical quadrature.

Approximation Error

Approximation error is a crucial concept in numerical analysis, offering insights into the accuracy of numerical methods. When employing techniques like the trapezoid rule or improved Euler method, errors arise due to finite steps and calculation approximations. The main points include:

• Approximation error indicates the difference between the exact value and the estimated result.
• In the context of integrals, this error typically reduces as the step number \( n \) increases, due to the convergence properties of numerical methods.
• For example, calculating the integral of \( f(x) = A + Bx \) using the trapezoid rule will result in closer approximations to the true integral with larger \( n \) values.

By understanding and analyzing approximation errors, one can evaluate the reliability of computation and ensure confidence in the results, especially when exact solutions aren't feasible. A careful analysis often shows that errors decrease as the number of steps increases.

Integral Convergence

Integral convergence refers to how rapidly a numerical method approaches the exact value of an integral as the number of steps or intervals (\( n \)) increases. Several key aspects are crucial to understanding this process:

• The trapezoid rule is known for its simplicity and effectiveness, particularly for functions that are smooth and don't oscillate rapidly.
• With increased \( n \), the approximation of the integral becomes more accurate, demonstrating a higher rate of convergence.
• For functions like \( f(x) = A + Bx + Cx^2 \), convergence is observed more explicitly as larger values of \( n \) produce results closer to the true integral.

By incrementally refining the partition of the interval over which the function is integrated, these methods systematically reduce errors. Analyses of various functions have repeatedly shown that convergence is a fundamental property that supports the reliability of many numerical quadrature formulas.