Core Numerical Methods: Algorithms for Computation

Posted on Jul 4, 2025 in Mathematics

Implementing Euler’s Method

1. Start
2. Define function f(x,y)
3. Read values of initial condition (x₀ and y₀), number of steps (n), and calculation point (x_n)
4. Calculate step size (h) = (x_n – x₀) / n
5. Set i = 0
6. Loop:
   • y_n = y₀ + h * f(x₀ + i*h, y₀)
   • y₀ = y_n
   • i = i + 1
   While i < n
7. Display y_n as result
8. Stop

Implementing Runge-Kutta 4th Order Method

1. Start
2. Define function f(x,y)
3. Read values of initial condition (x₀ and y₀), number of steps (n), and calculation point (x_n)
4. Calculate step size (h) = (x_n – x₀) / n
5. Set i = 0
6. Loop:
   • k₁ = h * f(x₀, y₀)
   • k₂ = h * f(x₀ + h/2, y₀ + k₁/2)
   • k₃ = h * f(x₀ + h/2, y₀ + k₂/2)
   • k₄ = h * f(x₀ + h, y₀ + k₃)
   • k = (k₁ + 2*k₂ + 2*k₃ + k₄) / 6
   • y_n = y₀ + k
   • i = i + 1
   • x₀ = x₀ + h
   • y₀ = y_n
   While i < n
7. Display y_n as result
8. Stop

Implementing Lagrange’s Interpolation

1. Start
2. Read number of data points (n)
3. Read data X_i and Y_i for i = 1 to n
4. Read value of independent variable, say x_p, whose corresponding dependent value, say y_p, is to be determined.
5. Initialize: y_p = 0
6. For i = 1 to n:
   • Set p = 1
   • For j = 1 to n:
     • If i ≠ j then:
       • Calculate p = p * (x_p – X_j) / (X_i – X_j)
     • End If
   • Next j
   • Calculate y_p = y_p + p * Y_i
7. Next i
8. Display value of y_p as interpolated value.
9. Stop

Implementing Newton’s Forward Interpolation

1. Start the program
2. Read n (Number of arguments/data points)
3. For i = 0 to n – 1:
   • Read x[i] and y[i][0]
4. End i
5. Construct the Forward Difference Table:
   • For j = 1 to n – 1:
     • For i = 0 to n – 1 – j:
       • y[i][j] = y[i + 1][j – 1] – y[i][j – 1]
     • End i
   • End j
6. Print the Forward Difference Table:
   • For i = 0 to n – 1:
     • For j = 0 to n – 1 – i:
       • Print y[i][j]
     • End j
     • Print a new line
   • End i
7. Read a (Point of Interpolation)
8. Assign h = x[1] – x[0] (Step Length)
9. Assign u = (a – x[0]) / h
10. Assign sum = y[0][0] and p = 1.0
11. For j = 1 to n – 1:
    • p = p * (u – j + 1) / j
    • sum = sum + p * y[0][j]
12. End j
13. Display a and sum
14. Stop

Integration Using the Trapezoidal Rule

1. Start
2. Define function f(x)
3. Read lower limit of integration, upper limit of integration, and number of sub-intervals
4. Calculate: step size (h) = (upper limit – lower limit) / number of sub-intervals
5. Set: integration value = f(lower limit) + f(upper limit)
6. Set: i = 1
7. If i > number of sub-intervals then goto step 12
8. Calculate: k = lower limit + i * h
9. Calculate: Integration value = Integration Value + 2 * f(k)
10. Increment i by 1 (i = i + 1) and go to step 7
11. Calculate: Integration value = Integration value * (step size / 2)
12. Display Integration value as required answer
13. Stop

Integration Using Simpson’s 1/3 Rule

1. Start
2. Define function f(x)
3. Read lower limit of integration, upper limit of integration, and number of sub-intervals
4. Calculate: step size (h) = (upper limit – lower limit) / number of sub-intervals
5. Set: integration value = f(lower limit) + f(upper limit)
6. Set: i = 1
7. If i > number of sub-intervals then goto step 12
8. Calculate: k = lower limit + i * h
9. If i mod 2 == 0 then:
   • Integration value = Integration Value + 2 * f(k)
   Else:
   • Integration Value = Integration Value + 4 * f(k)
10. End If
11. Increment i by 1 (i = i + 1) and go to step 7
12. Calculate: Integration value = Integration value * (step size / 3)
13. Display Integration value as required answer
14. Stop

Integration Using Simpson’s 3/8 Rule

1. Start
2. Define function f(x)
3. Read lower limit of integration, upper limit of integration, and number of sub-intervals
4. Calculate: step size (h) = (upper limit – lower limit) / number of sub-intervals
5. Set: integration value = f(lower limit) + f(upper limit)
6. Set: i = 1
7. If i > number of sub-intervals then goto step 12
8. Calculate: k = lower limit + i * h
9. If i mod 3 == 0 then:
   • Integration value = Integration Value + 2 * f(k)
   Else:
   • Integration Value = Integration Value + 3 * f(k)
10. End If
11. Increment i by 1 (i = i + 1) and go to step 7
12. Calculate: Integration value = Integration value * (step size * 3 / 8)
13. Display Integration value as required answer
14. Stop

Gauss Elimination Method for System Roots

1. Start
2. Read Number of Unknowns: n
3. Read Augmented Matrix (A) of n by n+1 Size
4. Transform Augmented Matrix (A) to Upper Triangular Matrix by Row Operations.
5. Obtain Solution by Back Substitution.
6. Display Result.
7. Stop

Gauss-Jordan Method for System Roots

1. Start
2. Read Number of Unknowns: n
3. Read Augmented Matrix (A) of n by n+1 Size
4. Transform Augmented Matrix (A) to Diagonal Matrix by Row Operations.
5. Obtain Solution by Making All Diagonal Elements to 1.
6. Display Result.
7. Stop

Jacobi Iteration Method

1. Start
2. Arrange given system of linear equations in diagonally dominant form
3. Read tolerable error (e)
4. Convert the first equation in terms of the first variable, the second equation in terms of the second variable, and so on.
5. Set initial guesses for x₀, y₀, z₀, and so on
6. Substitute values of x₀, y₀, z₀ … from step 5 into equations obtained in step 4 to calculate new values x₁, y₁, z₁, and so on
7. If (|x₀ – x₁| > e OR |y₀ – y₁| > e OR |z₀ – z₁| > e and so on for all variables) then goto step 8
8. Set x₀ = x₁, y₀ = y₁, z₀ = z₁ and so on, and goto step 6
9. Print value of x₁, y₁, z₁, and so on
10. Stop

Gauss-Seidel Iterative Method

1. Start
2. Arrange given system of linear equations in diagonally dominant form
3. Read tolerable error (e)
4. Convert the first equation in terms of the first variable, the second equation in terms of the second variable, and so on.
5. Set initial guesses for x₀, y₀, z₀, and so on
6. Substitute values of y₀, z₀ … from step 5 into the first equation obtained from step 4 to calculate new value of x₁. Use x₁, z₀, u₀ …. in the second equation obtained from step 4 to calculate new value of y₁. Similarly, use x₁, y₁, u₀… to find new z₁ and so on.
7. If (|x₀ – x₁| > e OR |y₀ – y₁| > e OR |z₀ – z₁| > e and so on for all variables) then goto step 8
8. Set x₀ = x₁, y₀ = y₁, z₀ = z₁ and so on, and goto step 6
9. Print value of x₁, y₁, z₁, and so on
10. Stop

Fixed-Point Iteration Method

1. Start
2. Define function f(x)
3. Define function g(x) which is obtained from f(x)=0 such that x = g(x) and |g'(x)| < 1
4. Choose initial guess x₀, Tolerable Error e, and Maximum Iteration N
5. Initialize iteration counter: step = 1
6. Calculate x₁ = g(x₀)
7. Increment iteration counter: step = step + 1
8. If step > N then print “Not Convergent” and goto step 12. Otherwise, goto step 10.
9. Set x₀ = x₁ for next iteration
10. If |f(x₁)| > e then goto step 6. Otherwise, goto step 11.
11. Display x₁ as root.
12. Stop

Secant Method Algorithm

1. Start
2. Define function as f(x)
3. Input initial guesses (x₀ and x₁), tolerable error (e), and maximum iteration (N)
4. Initialize iteration counter i = 1
5. If f(x₀) == f(x₁) then print “Mathematical Error” and goto step 11. Otherwise, goto step 6.
6. Calculate x₂ = x₁ – (x₁ – x₀) * f(x₁) / (f(x₁) – f(x₀))
7. Increment iteration counter i = i + 1
8. If i >= N then print “Not Convergent” and goto step 11. Otherwise, goto step 9.
9. If |f(x₂)| > e then set x₀ = x₁, x₁ = x₂ and goto step 5. Otherwise, goto step 10.
10. Print root as x₂
11. Stop

Newton-Raphson (NR) Method

1. Start
2. Define function as f(x)
3. Define first derivative of f(x) as g(x)
4. Input initial guess (x₀), tolerable error (e), and maximum iteration (N)
5. Initialize iteration counter i = 1
6. If g(x₀) == 0 then print “Mathematical Error” and goto step 12. Otherwise, goto step 7.
7. Calculate x₁ = x₀ – f(x₀) / g(x₀)
8. Increment iteration counter i = i + 1
9. If i >= N then print “Not Convergent” and goto step 12. Otherwise, goto step 10.
10. If |f(x₁)| > e then set x₀ = x₁ and goto step 6. Otherwise, goto step 11.
11. Print root as x₁
12. Stop