Numerical Method using Apache Open Office: A step-by-step guide on how to implement the Method of Successive Approximation (Picard's Iterative Method) algorithm on Apache Open Office spreadsheet.
Good day everyone! This is my 9th #utopian-io tutorial for Apache Open Office. Today, I will share to you the step on how to implement the Method of Successive Approximation (Picard's Iterative Method) algorithm on Apache Open Office spreadsheet. So to start, let first define and state the Method of Successive Approximation (Picard's Iterative Method).
What is Method of Successive Approximation (Picard's Iterative Method)?
A method of finding solutions of differential equations by proceeding from an initial approximation to a series of repeated trial solutions, each depending upon the immediately preceding approximation, in such a manner that the discrepancy between the newest estimated solution and the true solution is systematically reduced.
In this tutorial lets use the following equation and initial condition:
with y desired interval 0 < x < 1/2 . Approximating the desired solution as y = y(x), where:
and sequence of the function yn is given by y0(x) = 1, and for n > 1,
Step-by-step guide in implementing it on Apache Open Office spreadsheet.
To start open the Open Office application, and select spreadsheet. Then, create a 5 x 7 table with the following label as shown.
Input the formulas for each cell under each column headers. Make sure that the cell are first formatted to allow 2 decimal places. Shown below is the means of formating each cell to allow to decimal places
Let start inputting values to the cells. First is to input the initial condition for x which are 0.0, 0.1, ...up to 0.5. X-values should be equally spaced. To do it, simply click the cell and type in values, as shown.
For y1(x), we would apply the following equation:
To implement that on the Open Office spreadsheet, type in the following formula: "=(1+(B4)-((1/3)*(B4^3)))". Then, press ENTER.
To complete all values at y1(x), follow the procedure as shown.
Now we obtain the second approximation. For y2(x), we would apply the following equation:
To implement that on the Open Office spreadsheet, type in the following formula: "=1+B4+(B42)+((1/6)*(B44))+((2/15)(B4^5))+((1/63)(B4^7))". Then, press ENTER.
To complete all values at y2(x), follow the procedure as shown.
Then,we obtain the third approximation. For y3(x), we would apply the following equation:
To implement that on the Open Office spreadsheet, type in the following formula:
Then, press ENTER.
To complete all values at y3(x), follow the procedure as shown.
The result after three approximation is shown below.
To verify the obtain values after three iteration, lets compare the value of y3(x) to y(x). The y(x) values s shown below.
Thus, the values obtain using the Picard's Iterative Method is approximately correct up to 2 decimal places.
Thank you for reading this tutorial! I hope its helpful to you. And, have a nice day ahead!
Best regards,
@juecoree
(snapshot credit:
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