Why Is the Area Calculation of a Triangle Using Given Algorithms Unstable?

[MtX]

Consider the following algorithm for computing the area of a triangle with sides of length A, B, and C:

S = (A+B+C) / 2
Area1(A,B,C) = sqrt(S(S-C)(S-B)(S-A)).

A) Explain why this approach is unstable, and illustrate with an example.

B) A student proposes the following less efficient, but more stable, approach:

Area2(A,B,C) = sqrt((A+B+C)(A+B-C)(A-B+C)(-A+B+C)) / 4

Explain why this approach is still unstable and illustrate with an example.

C) Reaarange the formula to produce a more stable version. Explain why it is more stable.

I really have no clue on how to start off but for parts A) and B), I plan to insert arbitrary values of A, B and C, compare the area of the triangle using sides A, B, and C, versus the value of the area after using the Area1 and Area2 equations as examples. I am guessing the compared values will differ greatly and as a result makes this "approach" unstable.

For Part C) however, I have no clue how to begin. Am I suppose to create my own formula? Or modify theirs? What should I do?

Thanks.

 

[mathman]

I am not sure what "stability" is supposed to mean. I'll assume that it means expression without minus signs. With very little difficulty I got:

area=sqrt(S(S³+AB(A+B)+AC(A+C)+BC(B+C)+ABC)/2)

 

[tacman]

A) The approach presented in the algorithm is unstable because it involves subtracting similar numbers from each other, which can lead to a loss of significant digits and result in a less accurate calculation. This can be illustrated with an example where A=10, B=10, and C=10. Using the Area1 equation, we get S = (10+10+10)/2 = 15 and Area1(10,10,10) = sqrt(15(15-10)(15-10)(15-10)) = sqrt(15*5*5*5) = 37.4166. However, if we use the more accurate value of S = (A+B+C)/2 = (10+10+10)/2 = 15.000, we get a significantly different result of Area1(10,10,10) = sqrt(15.000(15.000-10.000)(15.000-10.000)(15.000-10.000)) = sqrt(15.000*5.000*5.000*5.000) = 37.4166. This shows how a small change in the input values can result in a large difference in the calculated area, making this approach unstable.

B) The proposed approach, Area2, is less efficient because it involves more calculations, but it is more stable because it avoids subtracting similar numbers. However, it is still unstable because it involves multiplying and dividing large numbers, which can lead to a loss of significant digits. For example, using the same input values of A=10, B=10, and C=10, we get Area2(10,10,10) = sqrt((10+10+10)(10+10-10)(10-10+10)(-10+10+10))/4 = sqrt(30*10*10*10)/4 = sqrt(30000)/4 = 54.7723. However, if we use the more accurate value of S = (A+B+C)/2 = (10+10+10)/2 = 15.000, we get a significantly different result of Area2(10,10,10) = sqrt((15.000+15.000+15.000)(15.000+15.000-15.000)(15.000-15.000+15.000)(-15.000+15.000+