The Quadratic Formula

Introduction

The following worksheet illustrates the use of a quadratic equation solution for showing the effect of
significant digits on round-off errors The user will enter the a, b and c values as given by the equation
for the standard form of a quadratic equation : ax^2 + bx + c = 0, as well as the number of significant
digits to be displayed in a table that will be created at the end of the program. Two variations of the
quadratic equation solution will be used :

Initialization

restart : with Statistics :

Section 1: Input

This is the only section where the user interacts with the program.
The quadratic formula is derived from the standard form of a quadratic equation: ax^2 + bx + c = 0

Enter coefficient a

Enter coefficient b

Enter coefficient c

Enter range of significant digits to be used.

This is the end of the user section. All information must be entered before proceeding to the next
section. RE-EXECUTE THE PROGRAM.

Section 2: Simulation

The following calculations will be performed inside a loop so that the number of significant digits
used can be varied as specified by the user. The digits command will be used to control the number
of digits Maple uses when calculating.

Variation 1:

j :=sig_low :
for i from sig_low to sig_high do
Digits :=i;
x1a[j] :=


x2a[j] := evalf

j :=j+1;

end do:

Variation 2:

j :=sig_low :
for i from sig_low to sig_high do
Digits :=i;

x1b[j] :=

x2b[j] :=

end do:

Section 3: Spreadsheet

This table shows the values of x1a, x2a, x1b, and x2b and the number of significant digits used in
their calculation.

n :=1 :
with Spread :
tableoutput :=CreateSpreadsheet "Table of Values" :
SetCellFormula (tableoutput, 1, 1, "Sig Digits") ;
SetCellFormula (tableoutput, 1, 2, "x1a" ;)
SetCellFormula (tableoutput, 1, 3, "x1b" ;)
SetCellFormula (tableoutput, 1, 4, "x2a" ;)
SetCellFormula (tableoutput, 1, 5, "x2b" ;)
for j from sig_low to sig_high do
SetCellFormula (tableoutput, nC1, 1, j );
SetCellFormula (tableoutput, nC1, 2, x1a[j]);
SetCellFormula (tableoutput, nC1, 3, x1b[j]);
SetCellFormula (tableoutput, nC1, 4, x2a[j]);
SetCellFormula (tableoutput, nC1, 5, x2b[j]);
n :=n+1;
end do:

EvaluateSpreadsheet (tableoutput)

Section 4: Graphs

These bar graphs show the values of x1 and x2 for both variations of the quadratic function.

Data1 :=Array ([seq (x1a [i] , i = sig_low ..sig_high)]) :
Data2 :=Array ([seq (x1b [i] , i = sig_low ..sig_high )]):
ColumnGraph ([Data1, Data2] , offset = sig_low, title
= " Value of First Root as a Function of Significant Digits", legend = ["x1a", "x1b" ],
titlefont = [ TIMES , BOLD, 12] , labelfont = [TIMES, ROMAN, 12 ], labels
= ["Number of Significant Digits", "Value of Quadratic Root"] , legendstyle = [location
= right];
Data3 :=Array ([seq x2a [i] , i = sig_low ..sig_high )]):
Data4 :=Array ([seq x2b [i] , i = sig_low ..sig_high )]):
ColumnGraph ([Data3, Data4] , offset = sig_low, title
= "Value of Second Root as a Function of Significant Digits", legend = ["x2a", "x2b" ],
titlefont = [ TIMES , BOLD, 12] , labelfont = [TIMES, ROMAN, 12] , labels
= ["Number of Significant Digits", "Value of Quadratic Root"] , legendstyle = [location
= right ]);

Value of First Root as a Function of Significant Digits

Value of Quadratic Root

Number of Significant Digits

Value of Second Root as a Function of Significant Digits

Value of Quadratic Root

Number of Significant Digit

Conclusion

Subtraction of numbers that are nearly equal can result in unwanted inaccuracies. The number of
significant digits used in calculations plays a large role in the creation of these inaccuracies and the
magnitude of the round-off errors. Hence, when the accuracy of calculations is critical, it is
necessary to understand possible sources of error and how they are best avoided.

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