Error Estimate of Vector Addition

Question

Kahan, in Cross on page 14, shows that the angle between the true cross product and the one calculated using floating point arithmetic degrades as the angle between the two crossed vectors approaches zero or π, this angle measuring the error in the floating point calculation of the normal to the plane spanned by the crossed vectors. As the singularities are approached, the orthogonality of the floating point normal becomes more compromised. According to Kahan, the error is:

|sin(∠(t_true, t_float ))| ≤ u+(2/√3)u|csc(∠(t_1, t_2 ))|

where t_true is true orthogonal vector after taking the cross product, t_1⊗t_2, to infinite precision, whereas t_float is the result of the cross product using floating point arithmetic and u is the unit round off.

My question: is there a similar inequality for the sum of two vectors. That is, how far off from the true direction does the floating point vector sum get as the angle between the vectors to be added approaches zero or π? In my particular application, the vectors will be unit length.

Edit: I should add that the vector that is the sum of the two vectors will be normalized.

Answer 1

Take c to be the computed v-w and e is the difference between c and what you wanted to compute, namely (v+deltav) - (w+deltaw). Here, e accounts for both roundoff error in computation and approximation error in your input.

Take r to be the ratio ||e|| / ||c||. This must be less than one, otherwise we simply have no idea where the true solution is.

By trigonometry, the angle between c and c+e is arccos(c^T (c+e) / sqrt(c^T c (c+e)^T (c+e))) = arccos((1 + alpha r) / sqrt(1 + 2 alpha r + r^2), where -1 <= alpha <= 1 is the ratio of e^T c to sqrt(c^T c e^T e). The cosine of the angle between c and c+e is minimised when alpha is exactly -r, at which point the angle is arccos(sqrt(1 - r^2)) = arcsin(r).

You can get the same result from the law of sines. Given a c and a bound on ||e||/||c||, the points that can reasonably be c+e lie on a circle centred at c of radius ||e||. The worst c+e will be on the boundary and the line from the origin to the worst c+e will make a right angle with the line from c+e to c. Thus the angle opposite c is a right angle. By the law of sines, then, ||e|| / sin(theta) = ||c||, from which the same result follows.