UPSC 2014 Maths Optional Paper 2 Q7b — Step-by-Step Solution

15 marks · Section B

Algorithms and flowcharts for numerical analysis problems · Numerical Analysis · asked 2× in 13 yrs · Read the full method →

Question

Draw a flowchart for Simpson’s one-third rule.

Technique

Standard composite Simpson’s 1/3 rule implementation; flowchart with input validation, accumulator-loop, and output.

Solution

Algorithm. Composite Simpson’s 1/3 rule approximates $\int_a^b f(x)\,dx$ using $n$ subintervals (with $n$ even) of width $h=(b-a)/n$ . The formula is

\int_a^b f(x)\,dx\approx\dfrac{h}{3}\bigl[f(x_0)+f(x_n)+4\!\sum_{i\text{ odd}}f(x_i)+2\!\sum_{i\text{ even (interior)}}f(x_i)\bigr].

Flowchart (textual ASCII rendering)

┌─────────────┐
│   START     │
└──────┬──────┘
       ▼
┌────────────────────────────────────────────┐
│ READ:                                      │
│   f(x), a (lower limit), b (upper limit),  │
│   n (even integer, number of subintervals) │
└──────────────────┬─────────────────────────┘
                   ▼
       ┌─────────────────────────┐
       │ IS n EVEN?              │
       └────┬───────────────┬────┘
         No │            Yes│
            ▼               ▼
       ┌──────────┐  ┌──────────────────┐
       │ PRINT:   │  │ Compute h=(b-a)/n│
       │ "n must  │  │ Initialize:      │
       │  be even"│  │   sum = f(a)+f(b)│
       │ STOP     │  │   i = 1          │
       └──────────┘  └────────┬─────────┘
                              ▼
       ┌──────────────────────────────────┐
       │ LOOP: WHILE i ≤ n-1              │
       │   x_i = a + i*h                  │
       │   IF i is ODD:                   │
       │       sum = sum + 4*f(x_i)       │
       │   ELSE (i EVEN):                 │
       │       sum = sum + 2*f(x_i)       │
       │   i = i + 1                      │
       └─────────────┬────────────────────┘
                     ▼
       ┌──────────────────────────────┐
       │ integral = (h/3) * sum       │
       └──────────────┬───────────────┘
                      ▼
       ┌──────────────────────────┐
       │ PRINT integral           │
       └──────────────┬───────────┘
                      ▼
                 ┌────────┐
                 │  STOP  │
                 └────────┘

Verbal description of steps

Input: function $f(x)$ , interval $[a,b]$ , even integer $n$ .
Validate $n$ : if odd, abort with error.
Compute step size: $h=(b-a)/n$ .
Initialise accumulator: $\text{sum}=f(a)+f(b)$ (endpoint contributions, weight 1).
Loop through interior points $i=1,2,\ldots,n-1$ :
- Compute $x_i=a+ih$ .
- If $i$ is odd: add $4f(x_i)$ to sum.
- If $i$ is even: add $2f(x_i)$ to sum.
Combine: $\text{integral}=\dfrac{h}{3}\cdot\text{sum}$ .
Output the integral value and stop.