Question

The Euclidean norm of an n-dimensional vector x is defined by ∥x∥2​=(∑i=1n​xi2​)1/2. How would you avoid overflow and harmful underflow in this computation?

Answer 1

To avoid overflow and underflow in computing the Euclidean norm of a vector, you can normalize the vector, use an iterative algorithm, and employ a precision data type.

Step-by-step explanation:

To avoid overflow and harmful underflow in the computation of the Euclidean norm of an n-dimensional vector, you can use the following techniques:

1. Normalize the vector by dividing each component by the largest absolute value. This ensures that the values are within a reasonable range.
2. Compute the norm using an iterative algorithm, such as the Kahan summation algorithm, which reduces the accumulation of round-off errors.
3. Use a precision data type, such as double precision, to minimize the impact of numerical inaccuracies.