suml1N

Suml1N is a notation used to denote the truncated L1 norm of a finite sequence, that is, the sum of absolute values of the first N terms. For a sequence (x1, x2, ..., xN) in the real numbers (or a real/complex vector with appropriate magnitude for complex values), suml1N is defined as sum_{i=1}^N |xi|. The concept serves as a practical measure of total magnitude for a finite signal or data block and is commonly used in numerical analysis, optimization, and data processing.

As a function of the sequence, suml1N has several key properties. It is nonnegative and vanishes only when all considered terms are zero. It satisfies the triangle inequality: suml1N(x + y) ≤ suml1N(x) + suml1N(y). It is homogeneous: suml1N(c x) = |c| suml1N(x). For a fixed sequence, suml1N is nondecreasing in N, since adding nonnegative absolute values cannot decrease the total. When working with a vector of length N, suml1N defines a norm on R^N with the standard vector operations.

Examples illustrate its use: for the sequence (3, -4, 0, 2), suml1,3 = |3| + |-4| + |0| = 7, while suml1,4 = 9. In practice, suml1N is viewed as the partial or truncated L1 norm and can be employed as a simple, interpretable measure of signal energy or magnitude in streaming data, iterative optimization, or sparsity-inducing frameworks. It relates directly to the full L1 norm as the finite-length counterpart on the first N components.

See also: L1 norm, vector norm, norm properties.