# number of vectors in the decomposition of vector

Let $x \in R^n$ be vector with entries $x_1\geq\cdots \geq x_n$.

I would like to decompose this vector into vectors $x^1=(x_1,\ldots,x_m,0,\ldots0), x^2=(0,\ldots,0,x_{m+1},\ldots,x_1,0,\ldots,0)$..., where $2x_m>{x_1}$; $2x_{m+1}<{x_1}$; $x_{m+1}<2x_l;x_{l+1}>2x_{l+1}$ and so on.

How many of those vectors $x^i$? (My filing it should be $log$ terms, but I would like to see the proof of it. Any good sours will be also helpful).

Thank you.

-

## 1 Answer

It will depend on $x$; you may end up using anywhere from $1$ to $n$ vectors. For example, if $x = (4,4,4,3,3,3,3)$ then you will only have one vector in the decomposition. But if $x = (10000,4000,1000,400,100,40,10)$ then you will need $n = 7$ vectors.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 14, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9292862415313721, "perplexity_flag": "head"}