TABLE

In review

This page runs a BASIC program that makes a table for the accumulation function

\[ A(t) = \int_0^t \cos(s^2)\, ds, \]

the solution of \(y' = \cos(t^2)\) with \(y(0) = 0\). The program is rewritten in Python and runs right here in your browser. Edit the code and press Run to run it again; Reset brings back the original program.

There is no formula for \(A(t)\) in terms of familiar functions, but Euler’s method doesn’t need one. The interval from 0 to 4 is split into numberofsteps pieces of width deltat. On each piece the program computes deltay = fnf(t) * deltat, the rate at the start of the step times the width, adds it to accumulation, and moves t ahead. Each line of output shows three numbers: deltay, the running total accumulation, and the new value of t. Watch deltay change sign as \(\cos(t^2)\) swings between positive and negative; when it is negative, the accumulation goes down.

With only \(2^3 = 8\) steps the table is rough. Change numberofsteps to 2 ** 6 or 2 ** 10 and compare the last value of accumulation with the 1.1330 you get here.

QBasic lets a FUNCTION be written at the end of the program, after the code that uses it. Python needs a function to be defined before it is called, so in the Python the def comes first, and return gives back the value that BASIC sets with fnf = .... COS becomes math.cos, and ^ becomes **.

Notes from the textbook

Add notes here.

 1  DEFDBL A-Z
 2  tinitial = 0
 3  tfinal = 4
 4  t = tinitial
 5  numberofsteps = 2 ^ 3
 6  deltat = (tfinal - tinitial) / numberofsteps
 7  accumulation = 0
 8  FOR k = 1 TO numberofsteps
 9          deltay = fnf(t) * deltat
10          accumulation = accumulation + deltay
11          t = t + deltat
12          PRINT deltay, accumulation, t
13  NEXT k
14  FUNCTION fnf (t)
15          fnf = COS(t ^ 2)
16  END FUNCTION

Output

Everything the program prints appears here, one line for each print(...), like BASIC's PRINT. Numbers printed on the same line are lined up in columns.

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Python code

Press Run (or Ctrl/⌘ + Enter) to run the program again. Press Esc, then Tab, to leave the editor.

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