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* fix: Find eigenvectors of defective matrices Previously, attempting to take the `eigs` of any defective matrix was doomed to fail in an attempt to solve a singular linear system. This PR detects the situation (as best as it can given the inherent numerical instability of the current methods used) and handles it. Note that in such cases, it's not possible to return a square matrix whose columns are the eigenvectors corresponding to the returned eigenvalues. In light of that fact and issue #3014, this PR also changes the return value of `eigs` so that the eigenvectors are passed back in a property `eigenvectors` which is an array of plain objects `{value: e, vector: v}`. Note that this PR makes the ancillary changes of correcting the spelling of the filename which was "realSymetric.js," and replacing the now-unnecessary auxiliary function "createArray" therein with `Array(size).fill(element)`. The rationale for performing these changes not strictly related to the issues at hand is that this file is rarely touched and with the level of maintenance hours we have at hand, it's more efficient to do these small refactorings in parallel with the actual bugfixes, which are orthogonal and so will not be obfuscated by this refactor. Note `git diff` does properly track the file name change. However, it also makes a potentially more pervasive change: in order for the numerically-sensitive algorithm to work, it changes the condition on when two very close (double) numbers are "nearlyEqual" from differing by less than DBL_EPSILON to differing by less than or equal to DBL_EPSILON. Although this may change other behaviors than the ones primarily being addressed, I believe it is an acceptable change because (a) It preserves all tests. (b) DBL_EPSILON is well below the standard config.epsilon anyway (c) I believe there are extant issues noting the odd/inconsistent behavior of nearlyEqual near 0 anyway, so I believe this will be overhauled in the future in any case. If so, the eigenvector computation will make a good test that a future nearlyEqual algorithm is working well. To be clear, the direct motivation for the change is that there are multiple cases in the eigenvector computation in which a coefficient that is "supposed" to be zero comes out to precisely DBL_EPSILON, which is fairly unsurprising given that these coefficients are produced by subtracting an eigenvalue from a diagonal entry of a matrix, which is likely to be essentially equal to that eigenvalue. As many tests of defective matrices as I could readily find by web searching have been added as unit tests (and one more in the typescript type testing). An additional case I found still fails, but in the _eigenvalue_ computation rather than the _eigenvector_ search, so that was deemed beyond the scope of this PR and has been filed as issue #3036. Resolves #2879. Resolves #2927. Resolves #3014. * refactor: remove comma that lint now doesn't like * test: add a test for eigs with a precision argument * feat: Use simple shifts in QR eigenvalue iterations that improve convergence Although we might want to use better shifts in the future, we might just use a library instead. But for now I think this: Resolves #2178. Also responds to the review feedback provided in PR #3037.
# Conflicts: # HISTORY.md
* feat: Add option to eigs() to turn off eigenvector computation For large matrices, the eigenvector computation can be noticeably expensive and so it's worthwhile to have a way to turn it off if the eigenvectors will not be used. Resolves #2180. * fix: Add test for precision in options arg of eigs And also a fix for a small bug that the new test uncovered. * test: check eigs with matrix and options * refactor: remove dead code from complexEigs.js * fix: add new signatures of eigs to typescript * test: ensure eigenvectors property not present with eigenvectors: false option * fix: correct balancing code in complexEigs
* added subtractScaler * added subtractScaler missing entries * added test cases for 2 or more parameters, test for subtractScalar instead fo subtract * replaced subtract with subtractScalar whereever possible --------- Co-authored-by: Jos de Jong <[email protected]>
… type like a function
# Conflicts: # HISTORY.md
…n when using `BigNumber` or `Fraction`
…xport the existing definitions
…nction `derivative`
…#3256) * Implement reduceCallback * Add jsdocs * implement simplifyCallback in other functions * Moved recurse to array.js * Format * Separate transform callback * forEach transform * Renamed applyCallback to simplifyCallback * Simplified index transform * renamed to reducedCallback and simplifiedCallback to simpleCallback * chore: fix linting issue * Added forEach benchmark * renamed simplifyCallback to optimizeCallback --------- Co-authored-by: Jos de Jong <[email protected]>
#3251) * Optimize the map and forEach functions in DenseMatrix.js * Changed index back to Array from Uint32Array and clone it using index.slice(0) instead of [...index] * Fixed merge conflicts with the fast callback optimization * Fixed the documentation for _forEach() * Fixed _forEach comment and made it return an immutable index array * Resolved DenseMatrix unit test suggestions --------- Co-authored-by: Jos de Jong <[email protected]>
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