Eigenvalues of a general matrix

[lpawela]

I'm trying to visualize the numerical range of a matrix on a website. For this, I need to calculate the eigenvalues of a general matrix with complex elements. This should be available since #1743 was merge. Yet, I get the following errors.

<script src="https://unpkg.com/mathjs@10.1.1/lib/browser/math.js"></script>

<script type="text/javascript">
console.log(math.eigs([[1, 2], [3, 4]]));
console.log(math.eigs([[3, -2], [4, -1]]));
console.log(math.eigs([[3, -2], [math.complex(4, 2), -1]]));
</script>

The first two line work as expected, the last one results in the following error

Uncaught TypeError: No ordering relation is defined for complex numbers
    at Function.Complex, Complex (smaller.js:75:13)
    at smaller (typed-function.js:1086:85)
    at complexEigs.js:111:18
    at complexEigs.js:22:15
    at j (eigs.js:100:12)
    at Function.Array (eigs.js:51:14)
    at Object.eigs (typed-function.js:1085:85)
    at (index):1:14924
Complex, Complex	@	smaller.js:75
smaller	@	typed-function.js:1086
(anonymous)	@	complexEigs.js:111
(anonymous)	@	complexEigs.js:22
j	@	eigs.js:100
Array	@	eigs.js:51
eigs	@	typed-function.js:1085
(anonymous)	@	(index):1

Am I doing something? Did I misunderstand the nature of #1743?

[gwhitney]

This is definitely a bug. Thank you so much for reporting! Someone will look at it as soon as they are able, but as you can see, there is a big backlog; any suggestions as to how to address it or better yet a PR fixing it would definitely be welcome.

[lpawela]

@gwhitney playing around with the console in my browser I think I found the issue. These lines

mathjs/src/function/matrix/eigs/complexEigs.js

Lines 67 to 68 in 38d9466

	const zero = big ? bignumber(0) : cplx ? complex(0) : 0
	const one = big ? bignumber(1) : cplx ? complex(1) : 1

are defined based on the type of the input. In the case I found they are complex numbers. Later, we have

mathjs/src/function/matrix/eigs/complexEigs.js

Lines 90 to 91 in 38d9466

	let colNorm = zero
	let rowNorm = zero

which makes both norms complex. Next, we add to the norms the absolute values of matrix elements

mathjs/src/function/matrix/eigs/complexEigs.js

Lines 96 to 97 in 38d9466

	colNorm = addScalar(colNorm, c)
	rowNorm = addScalar(rowNorm, c)

which keeps the norms complex. This in turn makes the variables

mathjs/src/function/matrix/eigs/complexEigs.js

Lines 108 to 109 in 38d9466

	const rowDivRadix = divideScalar(rowNorm, radix)
	const rowMulRadix = multiplyScalar(rowNorm, radix)

complex as well. Finally, we hit the offending line

mathjs/src/function/matrix/eigs/complexEigs.js

Line 111 in 38d9466

while (smaller(c, rowDivRadix)) {

which results in a comparison between a real c and a complex rowDivRadix. I don't know much about javascript, and I don't have a proper environment to prepare and submit a PR, but my suggestion would be to change zero and one to a real value. Would that work, or will it break something?

[gwhitney]

By Jove that's it. All of the norms and radices in this part of the algorithm are intended to be real even in the case of complex entries. Fixing the types prevented the error from being thrown, and produced the correct eigenvalues, Hopefully we can get the PR above merged reasonably quickly and a new release to fix this serious bug. Thank you so much Łukasz for your help. And @josdejong let me know if there's anything in particular I should do to expedite this fix.

[josdejong]

wow 😳 , thanks for debugging this @lpawela and @gwhitney for working it out in a fix. Will publish the fix today or tomorrow

[josdejong]

The fix is published now in v10.3.0