Answers to Some Questions about Explicit Sinkhorn Limits posed by Mel Nathanson

By Shalosh B. Ekhad and Doron Zeilberger

.pdf   .ps   .tex

(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org )

Posted: Feb. 27, 2019; Last update of this webpage (but not article): March 3, 1019.

At the Jan. 2018 Joint Mathematics Meetings, Avi Wigderson gave a series of three fascinating lectures (here is the first one) whose starting point was the Sinkhorn algorithm. One of the people in the audience was Mel Nathanson, and this lead him to ask some intriguing questions, some of which we answer in this note.

# Maple package

• SINKHORN.txt, a Maple package to compute Sinkhorn limits exactly (as well as approximately)

## Sample Input and Output Files for the Maple package SINKHORN.txt,

• If you want to see the exact Sinkhorn limits for generic symmetric positive 2 by 2 and 3 by 3 matrices done tersely

the input file generates the output file.

• If you want to see the exact Sinkhorn limits for generic symmetric positive 3 by 3 matrix done verbosely

the input file generates the output file.

• If you want to see the exact Sinkhorn limits for five randomly chosed 3 by 3 numeric symmetric matrices,

the input file generates the output file.

• If you want to see the answer to Mel Nathanson's Problem 1 (Section 14, p.26) posed in his article Matrix scaling, explicit Sinkhorn limits, and arithmetic

the input file generates the output file.

• If you want to see the answer to Mel Nathanson's Problem 2, first part (Section 14, p.26) posed in his above-mentioned article

the input file generates the output file.

• If you want to see the answer to Mel Nathanson's Problem 2, second part (Section 14, p.26) posed in his above-mentioned article

the input file generates the output file.

• If you want to see two examples of matrices begged by Mel Nathanson in Problem 5, (Section 14, p.27) his above-mentioned article

the input file generates the output file.

• [Added March 3, 2019] If you want to see two examples of 3 by 3 matrices that terminate after EXACTLY two and EXACTLY three double iterations (row-scaling followed by column-scaling)

the input file generates the output file.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger