Sharp EL-5230 Using the Solver Function Effectively, Newton’s method, ‘Dead end’ approximations

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CALCULATION

Appendix

Using the Solver Function Effectively

The calculator uses Newton’s method to solve equations. (See page 52.) Because of this, the solution it provides may differ from the true solution, or an error message may be displayed for a soluble equation. This section shows how you can obtain a more acceptable solution or make the equation soluble in such cases.

Newton’s method

Newton’s method is a successive approximation technique that uses tangential lines. The calculator chooses an ‘approximate’ solution then calculates and compares the right-hand and left- hand sides of the equation. Based on the result of this comparison, it chooses another ‘approximate’ solution. It repeats this process until there is hardly any discrepancy between the right-hand and left-hand sides of the equation.

y = f(x)

yTangential lines

Solution

x

Initial value

Newton’s method Intersections of dotted lines with the x-axis give successive approximate solutions found using Newton’s method.

‘Dead end’ approximations

When @ h is pressed for the first

time, the calculator takes the value that is stored in memory, or zero if no value is stored, to be the initial expected value for the unknown variable and tries to solve the equation. If it fails to find an acceptable solution using this expected value, it tries again using up to nine more initial expected values until a solution is found. If none of the values

lead by successive approximation toward an acceptable solution — but rather to a ‘dead end’ — the calculator will abort calculation and display an error message.

Range of expected values

After the stored value (or zero) has been tried, new initial expected values are selected according to the range of expected values for the equation. (See ‘Changing the range of expected values’.) To choose which initial expected values to try, the calculator divides the range into eight subranges of equal width and tries each of the values at the edges of these subranges in turn (starting with the lower limit of the range of expected values, a).

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