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Added functions: schur, sylvester, and lyap (#2646)

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* Added matrix functions `schur`, `sylvester` and  `lyap`

* Added docs for  `schur`, `sylvester` and  `lyap`

* Added unit tests for  `schur`, `sylvester` and  `lyap`

* Fixed lint and tests errors

* fixed typescript and added Matrix + Array tests

* lint fixed

* fixed example in sylvester.js doc

* Fixed docs and ci

* fixed definition of lyap and sylvester

* remark on diff defs for lyap

* rm a trailing space

Co-authored-by: Jos de Jong <wjosdejong@gmail.com>
This commit is contained in:
Lucas Egidio 2022-11-17 17:16:21 +01:00 committed by GitHub
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1
AUTHORS
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@ -199,6 +199,7 @@ FelixSelter <55546882+FelixSelter@users.noreply.github.com>
Hansuku <1556207795@qq.com>
Windrill <windrill@outlook.com>
Carl Osterwisch <costerwi@gmail.com>
Lucas Egidio <lucasegidio1@gmail.com>
Isaac B <66892358+isaacbyr@users.noreply.github.com>
Ajinkya Chikhale <86607732+ajinkyac03@users.noreply.github.com>
Abdullah Adeel <64294045+mabdullahadeel@users.noreply.github.com>

6
src/expression/embeddedDocs/embeddedDocs.js
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@ -78,6 +78,9 @@ import { roundDocs } from './function/arithmetic/round.js'
import { signDocs } from './function/arithmetic/sign.js'
import { sqrtDocs } from './function/arithmetic/sqrt.js'
import { sqrtmDocs } from './function/arithmetic/sqrtm.js'
import { sylvesterDocs } from './function/matrix/sylvester.js'
import { schurDocs } from './function/matrix/schur.js'
import { lyapDocs } from './function/matrix/lyap.js'
import { squareDocs } from './function/arithmetic/square.js'
import { subtractDocs } from './function/arithmetic/subtract.js'
import { unaryMinusDocs } from './function/arithmetic/unaryMinus.js'
@ -467,6 +470,9 @@ export const embeddedDocs = {
zeros: zerosDocs,
fft: fftDocs,
ifft: ifftDocs,
sylvester: sylvesterDocs,
schur: schurDocs,
lyap: lyapDocs,
// functions - probability
combinations: combinationsDocs,

15
src/expression/embeddedDocs/function/matrix/lyap.js Normal file
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@ -0,0 +1,15 @@
export const lyapDocs = {
name: 'lyap',
category: 'Matrix',
syntax: [
'lyap(A,Q)'
],
description: 'Solves the Continuous-time Lyapunov equation AP+PA\'+Q=0 for P',
examples: [
'lyap([[-2, 0], [1, -4]], [[3, 1], [1, 3]])',
'lyap(A,Q)'
],
seealso: [
'schur', 'sylvester'
]
}

15
src/expression/embeddedDocs/function/matrix/schur.js Normal file
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@ -0,0 +1,15 @@
export const schurDocs = {
name: 'schur',
category: 'Matrix',
syntax: [
'schur(A)'
],
description: 'Performs a real Schur decomposition of the real matrix A = UTU\'',
examples: [
'schur([[1, 0], [-4, 3]])',
'schur(A)'
],
seealso: [
'lyap', 'sylvester'
]
}

15
src/expression/embeddedDocs/function/matrix/sylvester.js Normal file
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@ -0,0 +1,15 @@
export const sylvesterDocs = {
name: 'sylvester',
category: 'Matrix',
syntax: [
'sylvester(A,B,C)'
],
description: 'Solves the real-valued Sylvester equation AX+XB=C for X',
examples: [
'sylvester([[-1, -2], [1, 1]], [[-2, 1], [-1, 2]], [[-3, 2], [3, 0]])',
'sylvester(A,B,C)'
],
seealso: [
'schur', 'lyap'
]
}

3
src/factoriesAny.js
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@ -220,6 +220,9 @@ export { createPinv } from './function/matrix/pinv.js'
export { createEigs } from './function/matrix/eigs.js'
export { createExpm } from './function/matrix/expm.js'
export { createSqrtm } from './function/matrix/sqrtm.js'
export { createSylvester } from './function/algebra/sylvester.js'
export { createSchur } from './function/algebra/decomposition/schur.js'
export { createLyap } from './function/algebra/lyap.js'
export { createDivide } from './function/arithmetic/divide.js'
export { createDistance } from './function/geometry/distance.js'
export { createIntersect } from './function/geometry/intersect.js'

77
src/function/algebra/decomposition/schur.js Normal file
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@ -0,0 +1,77 @@
import { factory } from '../../../utils/factory.js'
const name = 'schur'
const dependencies = [
'typed',
'matrix',
'identity',
'multiply',
'qr',
'norm',
'subtract'
]
export const createSchur = /* #__PURE__ */ factory(name, dependencies, (
{
typed,
matrix,
identity,
multiply,
qr,
norm,
subtract
}
) => {
/**
*
* Performs a real Schur decomposition of the real matrix A = UTU' where U is orthogonal
* and T is upper quasi-triangular.
* https://en.wikipedia.org/wiki/Schur_decomposition
*
* Syntax:
*
* math.schur(A)
*
* Examples:
*
* const A = [[1, 0], [-4, 3]]
* math.schur(A) // returns {T: [[3, 4], [0, 1]], R: [[0, 1], [-1, 0]]}
*
* See also:
*
* sylvester, lyap, qr
*
* @param {Array | Matrix} A Matrix A
* @return {{U: Array | Matrix, T: Array | Matrix}} Object containing both matrix U and T of the Schur Decomposition A=UTU'
*/
return typed(name, {
Array: function (X) {
const r = _schur(matrix(X))
return {
U: r.U.valueOf(),
T: r.T.valueOf()
}
},
Matrix: function (X) {
return _schur(X)
}
})
function _schur (X) {
const n = X.size()[0]
let A = X
let U = identity(n)
let k = 0
let A0
do {
A0 = A
const QR = qr(A)
const Q = QR.Q
const R = QR.R
A = multiply(R, Q)
U = multiply(U, Q)
if ((k++) > 100) { break }
} while (norm(subtract(A, A0)) > 1e-4)
return { U, T: A }
}
})

61
src/function/algebra/lyap.js Normal file
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@ -0,0 +1,61 @@
import { factory } from '../../utils/factory.js'
const name = 'lyap'
const dependencies = [
'typed',
'matrix',
'sylvester',
'multiply',
'transpose'
]
export const createLyap = /* #__PURE__ */ factory(name, dependencies, (
{
typed,
matrix,
sylvester,
multiply,
transpose
}
) => {
/**
*
* Solves the Continuous-time Lyapunov equation AP+PA'+Q=0 for P, where
* Q is an input matrix. When Q is symmetric, P is also symmetric. Notice
* that different equivalent definitions exist for the Continuous-time
* Lyapunov equation.
* https://en.wikipedia.org/wiki/Lyapunov_equation
*
* Syntax:
*
* math.lyap(A, Q)
*
* Examples:
*
* const A = [[-2, 0], [1, -4]]
* const Q = [[3, 1], [1, 3]]
* const P = math.lyap(A, Q)
*
* See also:
*
* sylvester, schur
*
* @param {Matrix | Array} A Matrix A
* @param {Matrix | Array} Q Matrix Q
* @return {Matrix | Array} Matrix P solution to the Continuous-time Lyapunov equation AP+PA'=Q
*/
return typed(name, {
'Matrix, Matrix': function (A, Q) {
return sylvester(A, transpose(A), multiply(-1, Q))
},
'Array, Matrix': function (A, Q) {
return sylvester(matrix(A), transpose(matrix(A)), multiply(-1, Q))
},
'Matrix, Array': function (A, Q) {
return sylvester(A, transpose(matrix(A)), matrix(multiply(-1, Q)))
},
'Array, Array': function (A, Q) {
return sylvester(matrix(A), transpose(matrix(A)), matrix(multiply(-1, Q))).toArray()
}
})
})

145
src/function/algebra/sylvester.js Normal file
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@ -0,0 +1,145 @@
import { factory } from '../../utils/factory.js'
const name = 'sylvester'
const dependencies = [
'typed',
'schur',
'matrixFromColumns',
'matrix',
'multiply',
'range',
'concat',
'transpose',
'index',
'subset',
'add',
'subtract',
'identity',
'lusolve',
'abs'
]
export const createSylvester = /* #__PURE__ */ factory(name, dependencies, (
{
typed,
schur,
matrixFromColumns,
matrix,
multiply,
range,
concat,
transpose,
index,
subset,
add,
subtract,
identity,
lusolve,
abs
}
) => {
/**
*
* Solves the real-valued Sylvester equation AX+XB=C for X, where A, B and C are
* matrices of appropriate dimensions, being A and B squared. Notice that other
* equivalent definitions for the Sylvester equation exist and this function
* assumes the one presented in the original publication of the the Bartels-
* Stewart algorithm, which is implemented by this function.
* https://en.wikipedia.org/wiki/Sylvester_equation
*
* Syntax:
*
* math.sylvester(A, B, C)
*
* Examples:
*
* const A = [[-1, -2], [1, 1]]
* const B = [[2, -1], [1, -2]]
* const C = [[-3, 2], [3, 0]]
* math.sylvester(A, B, C) // returns DenseMatrix [[-0.25, 0.25], [1.5, -1.25]]
*
* See also:
*
* schur, lyap
*
* @param {Matrix | Array} A Matrix A
* @param {Matrix | Array} B Matrix B
* @param {Matrix | Array} C Matrix C
* @return {Matrix | Array} Matrix X, solving the Sylvester equation
*/
return typed(name, {
'Matrix, Matrix, Matrix': _sylvester,
'Array, Matrix, Matrix': function (A, B, C) {
return _sylvester(matrix(A), B, C)
},
'Array, Array, Matrix': function (A, B, C) {
return _sylvester(matrix(A), matrix(B), C)
},
'Array, Matrix, Array': function (A, B, C) {
return _sylvester(matrix(A), B, matrix(C))
},
'Matrix, Array, Matrix': function (A, B, C) {
return _sylvester(A, matrix(B), C)
},
'Matrix, Array, Array': function (A, B, C) {
return _sylvester(A, matrix(B), matrix(C))
},
'Matrix, Matrix, Array': function (A, B, C) {
return _sylvester(A, B, matrix(C))
},
'Array, Array, Array': function (A, B, C) {
return _sylvester(matrix(A), matrix(B), matrix(C)).toArray()
}
})
function _sylvester (A, B, C) {
const n = B.size()[0]
const m = A.size()[0]
const sA = schur(A)
const F = sA.T
const U = sA.U
const sB = schur(multiply(-1, B))
const G = sB.T
const V = sB.U
const D = multiply(multiply(transpose(U), C), V)
const all = range(0, m)
const y = []
const hc = (a, b) => concat(a, b, 1)
const vc = (a, b) => concat(a, b, 0)
for (let k = 0; k < n; k++) {
if (k < (n - 1) && abs(subset(G, index(k + 1, k))) > 1e-5) {
let RHS = vc(subset(D, index(all, k)), subset(D, index(all, k + 1)))
for (let j = 0; j < k; j++) {
RHS = add(RHS,
vc(multiply(y[j], subset(G, index(j, k))), multiply(y[j], subset(G, index(j, k + 1))))
)
}
const gkk = multiply(identity(m), multiply(-1, subset(G, index(k, k))))
const gmk = multiply(identity(m), multiply(-1, subset(G, index(k + 1, k))))
const gkm = multiply(identity(m), multiply(-1, subset(G, index(k, k + 1))))
const gmm = multiply(identity(m), multiply(-1, subset(G, index(k + 1, k + 1))))
const LHS = vc(
hc(add(F, gkk), gmk),
hc(gkm, add(F, gmm))
)
const yAux = lusolve(LHS, RHS)
y[k] = yAux.subset(index(range(0, m), 0))
y[k + 1] = yAux.subset(index(range(m, 2 * m), 0))
k++
} else {
let RHS = subset(D, index(all, k))
for (let j = 0; j < k; j++) { RHS = add(RHS, multiply(y[j], subset(G, index(j, k)))) }
const gkk = subset(G, index(k, k))
const LHS = subtract(F, multiply(gkk, identity(m)))
y[k] = lusolve(LHS, RHS)
}
}
const Y = matrix(matrixFromColumns(...y))
const X = multiply(U, multiply(Y, transpose(V)))
return X
}
})

2
test/node-tests/doc.test.js
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@ -92,7 +92,7 @@ const knownProblems = new Set([
'mod', 'invmod', 'floor', 'fix', 'expm1', 'exp', 'dotPow', 'dotMultiply',
'dotDivide', 'divide', 'ceil', 'cbrt', 'add', 'usolveAll', 'usolve', 'slu',
'rationalize', 'qr', 'lusolve', 'lup', 'lsolveAll', 'lsolve', 'derivative',
'symbolicEqual', 'map'
'symbolicEqual', 'map', 'schur', 'sylvester'
])
function maybeCheckExpectation (name, expected, expectedFrom, got, gotFrom) {

63
test/unit-tests/function/algebra/decomposition/schur.test.js Normal file
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@ -0,0 +1,63 @@
// test schur decomposition
import assert from 'assert'
import math from '../../../../../src/defaultInstance.js'
describe('schur', function () {
it('should calculate schur decomposition of order 5 Array with numbers', function () {
assert.ok(math.norm(math.subtract(math.schur([
[-5.3, -1.4, -0.2, 0.7, 1.0],
[-0.4, -1.0, -0.1, -1.2, 0.7],
[0.3, 0.7, -2.5, 0.7, -0.3],
[3.6, -0.1, 1.4, -2.4, 0.3],
[2.8, 0.7, 1.4, 0.5, -4.8]
]).T, [
[-6.97747746169558, 0.5046853036888738, -0.5269551218982134, 2.9902479419087253, -2.2914719859941908],
[7.686296479877504e-28, -3.667202530573731, -1.3776362163231233, 0.4680921120934126, -0.374760141366345],
[-4.92421882171775e-28, 0.0859443307361262, -3.798852922420336, 0.6595326982269121, 0.38704916773017245],
[-1.8971150504442668e-71, -1.3481560449785515e-44, -8.960727665382592e-44, -1.3867791434503591, 0.5989746088924175],
[5.3038070596554604e-164, -7.015716167009369e-138, 1.0415178925287816e-136, 3.3306222609902884e-93, -0.16968794185997693]
])) < 1e-3)
assert.ok(math.norm(math.subtract(math.schur([
[-5.3, -1.4, -0.2, 0.7, 1.0],
[-0.4, -1.0, -0.1, -1.2, 0.7],
[0.3, 0.7, -2.5, 0.7, -0.3],
[3.6, -0.1, 1.4, -2.4, 0.3],
[2.8, 0.7, 1.4, 0.5, -4.8]
]).U, [
[0.6039524392527362, -0.11955248228665324, 0.5309978859071411, -0.3239619623824945, 0.48377530651243317],
[0.034196874004165004, -0.3407725193032822, -0.05878741847605931, -0.7490924342670748, -0.5640117270489927],
[-0.024973926752867345, 0.6355774530247909, -0.45835474572889245, -0.5151114453511857, 0.3463938944685342],
[-0.42238909820980175, -0.6350073886392834, -0.26854304887535935, -0.13970317163522103, 0.5715948922294664],
[-0.6745633977804205, 0.24977632323107185, 0.6575567219332444, -0.22147775183161147, 0.03380493473031572]
])) < 1e-3)
})
it('should calculate schur decomposition of order 5 Matrix with numbers', function () {
assert.ok(math.norm(math.subtract(math.schur(math.matrix([
[-5.3, -1.4, -0.2, 0.7, 1.0],
[-0.4, -1.0, -0.1, -1.2, 0.7],
[0.3, 0.7, -2.5, 0.7, -0.3],
[3.6, -0.1, 1.4, -2.4, 0.3],
[2.8, 0.7, 1.4, 0.5, -4.8]
])).T, math.matrix([
[-6.97747746169558, 0.5046853036888738, -0.5269551218982134, 2.9902479419087253, -2.2914719859941908],
[7.686296479877504e-28, -3.667202530573731, -1.3776362163231233, 0.4680921120934126, -0.374760141366345],
[-4.92421882171775e-28, 0.0859443307361262, -3.798852922420336, 0.6595326982269121, 0.38704916773017245],
[-1.8971150504442668e-71, -1.3481560449785515e-44, -8.960727665382592e-44, -1.3867791434503591, 0.5989746088924175],
[5.3038070596554604e-164, -7.015716167009369e-138, 1.0415178925287816e-136, 3.3306222609902884e-93, -0.16968794185997693]
]))) < 1e-3)
assert.ok(math.norm(math.subtract(math.schur(math.matrix([
[-5.3, -1.4, -0.2, 0.7, 1.0],
[-0.4, -1.0, -0.1, -1.2, 0.7],
[0.3, 0.7, -2.5, 0.7, -0.3],
[3.6, -0.1, 1.4, -2.4, 0.3],
[2.8, 0.7, 1.4, 0.5, -4.8]
])).U, math.matrix([
[0.6039524392527362, -0.11955248228665324, 0.5309978859071411, -0.3239619623824945, 0.48377530651243317],
[0.034196874004165004, -0.3407725193032822, -0.05878741847605931, -0.7490924342670748, -0.5640117270489927],
[-0.024973926752867345, 0.6355774530247909, -0.45835474572889245, -0.5151114453511857, 0.3463938944685342],
[-0.42238909820980175, -0.6350073886392834, -0.26854304887535935, -0.13970317163522103, 0.5715948922294664],
[-0.6745633977804205, 0.24977632323107185, 0.6575567219332444, -0.22147775183161147, 0.03380493473031572]
]))) < 1e-3)
})
})

49
test/unit-tests/function/algebra/lyap.test.js Normal file
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@ -0,0 +1,49 @@
// test lyapunov equation solver
import assert from 'assert'
import math from '../../../../src/defaultInstance.js'
describe('lyap', function () {
it('should solve lyapunov equation of order 5 with Matrices', function () {
assert.ok(math.norm(math.subtract(math.lyap(math.matrix([
[-5.3, -1.4, -0.2, 0.7, 1.0],
[-0.4, -1.0, -0.1, -1.2, 0.7],
[0.3, 0.7, -2.5, 0.7, -0.3],
[3.6, -0.1, 1.4, -2.4, 0.3],
[2.8, 0.7, 1.4, 0.5, -4.8]
]), math.matrix([
[-2, 0, 0, 0, 0],
[0, -2, 0, 0, 0],
[0, 0, -2, 0, 0],
[0, 0, 0, -2, 0],
[0, 0, 0, 0, -2]
])), math.matrix([
[-0.8567032819332775, 1.6675327139319873, 0.05193348368861031, -1.2336381488442063, -0.3320481938129234],
[1.6675327139319864, -4.577292046354884, -0.24491693525451969, 2.7566635155267862, 0.533179377606574],
[0.05193348368861139, -0.24491693525452285, -0.4786127989547805, -0.10398529391846956, -0.10706505975853312],
[-1.233638148844205, 2.7566635155267827, -0.10398529391847175, -2.5199807906596963, -0.6187026623519729],
[-0.33204819381292294, 0.5331793776065724, -0.10706505975853345, -0.6187026623519725, -0.4199482902478158]
])), 'fro') < 1e-3)
})
it('should solve lyapunov equation of order 5 with Arrays', function () {
assert.ok(math.norm(math.subtract(math.lyap([
[-5.3, -1.4, -0.2, 0.7, 1.0],
[-0.4, -1.0, -0.1, -1.2, 0.7],
[0.3, 0.7, -2.5, 0.7, -0.3],
[3.6, -0.1, 1.4, -2.4, 0.3],
[2.8, 0.7, 1.4, 0.5, -4.8]
], [
[-2, 0, 0, 0, 0],
[0, -2, 0, 0, 0],
[0, 0, -2, 0, 0],
[0, 0, 0, -2, 0],
[0, 0, 0, 0, -2]
]), [
[-0.8567032819332775, 1.6675327139319873, 0.05193348368861031, -1.2336381488442063, -0.3320481938129234],
[1.6675327139319864, -4.577292046354884, -0.24491693525451969, 2.7566635155267862, 0.533179377606574],
[0.05193348368861139, -0.24491693525452285, -0.4786127989547805, -0.10398529391846956, -0.10706505975853312],
[-1.233638148844205, 2.7566635155267827, -0.10398529391847175, -2.5199807906596963, -0.6187026623519729],
[-0.33204819381292294, 0.5331793776065724, -0.10706505975853345, -0.6187026623519725, -0.4199482902478158]
]), 'fro') < 1e-3)
})
})

55
test/unit-tests/function/algebra/sylvester.test.js Normal file
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@ -0,0 +1,55 @@
// test schur decomposition
import assert from 'assert'
import math from '../../../../src/defaultInstance.js'
describe('sylvester', function () {
it('should solve sylvester equation of order 5 with Arrays', function () {
assert.ok(math.norm(math.subtract(math.sylvester([
[-5.3, -1.4, -0.2, 0.7],
[-0.4, -1.0, -0.1, -1.2],
[0.3, 0.7, -2.5, 0.7],
[3.6, -0.1, 1.4, -2.4]
], [
[1.1, -0.3, -0.9, 0.8, -2.5],
[-0.6, 2.6, 0.2, 0.4, 1.3],
[0.4, -0.5, -0.2, 0.2, -0.1],
[-0.4, -1.9, -0.2, 0.5, 1.4],
[0.4, -1.0, -0.1, -0.8, -1.3]
], [
[1.4, 1.1, -1.9, 0.1, 1.2],
[-1.7, 0.1, -0.4, 2.1, 0.5],
[1.9, 2.3, -0.8, -0.7, 1.0],
[-1.1, 2.8, -0.8, -0.3, 0.9]
]), [
[-0.19393862606643053, -0.17101629636521865, 2.709348263225366, -0.0963000767188319, 0.5244718194343121],
[0.38421326955977486, -0.21159588555260944, -6.544262021555474, -0.15113424769761136, -2.312533293658291],
[-2.2708235174374747, 4.498279916441834, 1.4553799673144823, -0.9300926971755248, 2.5508111398452353],
[-1.0935231387905004, 4.113817086842746, 5.747671819196675, -0.9408309030864932, 2.967655969930743]
]), 'fro') < 1e-3)
})
it('should solve sylvester equation of order 5 with Matrices', function () {
assert.ok(math.norm(math.subtract(math.sylvester(math.matrix([
[-5.3, -1.4, -0.2, 0.7],
[-0.4, -1.0, -0.1, -1.2],
[0.3, 0.7, -2.5, 0.7],
[3.6, -0.1, 1.4, -2.4]
]), math.matrix([
[1.1, -0.3, -0.9, 0.8, -2.5],
[-0.6, 2.6, 0.2, 0.4, 1.3],
[0.4, -0.5, -0.2, 0.2, -0.1],
[-0.4, -1.9, -0.2, 0.5, 1.4],
[0.4, -1.0, -0.1, -0.8, -1.3]
]), math.matrix([
[1.4, 1.1, -1.9, 0.1, 1.2],
[-1.7, 0.1, -0.4, 2.1, 0.5],
[1.9, 2.3, -0.8, -0.7, 1.0],
[-1.1, 2.8, -0.8, -0.3, 0.9]
])), math.matrix([
[-0.19393862606643053, -0.17101629636521865, 2.709348263225366, -0.0963000767188319, 0.5244718194343121],
[0.38421326955977486, -0.21159588555260944, -6.544262021555474, -0.15113424769761136, -2.312533293658291],
[-2.2708235174374747, 4.498279916441834, 1.4553799673144823, -0.9300926971755248, 2.5508111398452353],
[-1.0935231387905004, 4.113817086842746, 5.747671819196675, -0.9408309030864932, 2.967655969930743]
])), 'fro') < 1e-3)
})
})

63
types/index.d.ts vendored
View File

@ -465,6 +465,11 @@ declare namespace math {
R: MathCollection
}
interface SchurDecomposition {
U: MathCollection
T: MathCollection
}
interface FractionDefinition {
a: number
b: number
@ -1807,6 +1812,41 @@ declare namespace math {
*/
expm(x: Matrix): Matrix
/**
* Solves the real-valued Sylvester equation AX-XB=C for X, where A, B and C are
* matrices of appropriate dimensions, being A and B squared. The method used is
* the Bartels-Stewart algorithm.
* https://en.wikipedia.org/wiki/Sylvester_equation
* @param A Matrix A
* @param B Matrix B
* @param C Matrix C
* @returns Matrix X, solving the Sylvester equation
*/
sylvester(
A: Matrix | MathArray,
B: Matrix | MathArray,
C: Matrix | MathArray
): Matrix | MathArray
/**
* Performs a real Schur decomposition of the real matrix A = UTU' where U is orthogonal
* and T is upper quasi-triangular.
* https://en.wikipedia.org/wiki/Schur_decomposition
* @param A Matrix A
* @returns Object containing both matrix U and T of the Schur Decomposition A=UTU'
*/
schur(A: Matrix | MathArray): SchurDecomposition
/**
* Solves the Continuous-time Lyapunov equation AP+PA'=Q for P, where Q is a positive semidefinite
* matrix.
* https://en.wikipedia.org/wiki/Lyapunov_equation
* @param A Matrix A
* @param Q Matrix Q
* @returns Matrix P solution to the Continuous-time Lyapunov equation AP+PA'=Q
*/
lyap(A: Matrix | MathArray, Q: Matrix | MathArray): Matrix | MathArray
/**
* Create a 2-dimensional identity matrix with size m x n or n x n. The
* matrix has ones on the diagonal and zeros elsewhere.
@ -3572,6 +3612,9 @@ declare namespace math {
invDependencies: FactoryFunctionMap
expmDependencies: FactoryFunctionMap
sqrtmDependencies: FactoryFunctionMap
sylvesterDependencies: FactoryFunctionMap
schurDependencies: FactoryFunctionMap
lyapDependencies: FactoryFunctionMap
divideDependencies: FactoryFunctionMap
distanceDependencies: FactoryFunctionMap
intersectDependencies: FactoryFunctionMap
@ -5272,6 +5315,26 @@ declare namespace math {
expm(this: MathJsChain<Matrix>): MathJsChain<Matrix>
/**
* Performs a real Schur decomposition of the real matrix A = UTU' where U is orthogonal
* and T is upper quasi-triangular.
* https://en.wikipedia.org/wiki/Schur_decomposition
* @returns Object containing both matrix U and T of the Schur Decomposition A=UTU'
*/
schur(this: MathJsChain<Matrix | MathArray>): SchurDecomposition
/**
* Solves the Continuous-time Lyapunov equation AP+PA'=Q for P, where Q is a positive semidefinite
* matrix.
* https://en.wikipedia.org/wiki/Lyapunov_equation
* @param Q Matrix Q
* @returns Matrix P solution to the Continuous-time Lyapunov equation AP+PA'=Q
*/
lyap(
this: MathJsChain<Matrix | MathArray>,
Q: Matrix | MathArray
): MathJsChain<Matrix | MathArray>
/**
* Create a 2-dimensional identity matrix with size m x n or n x n. The
* matrix has ones on the diagonal and zeros elsewhere.