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* Minified by jsDelivr using Terser v5.37.0.\n * Original file: /npm/loshu@0.0.1/loshu.js\n *\n * Do NOT use SRI with dynamically generated files! More information: https://www.jsdelivr.com/using-sri-with-dynamic-files\n */\n"," /*. @ \n@ @ ooooooooo .' \n `@' ,' o \n o o @ o \n | | o \n o o--O--o o \n | | o \n o @ o o \n @' `. @ o \n @' @ ,' `@ \n@' @' @ `@ \n `. @' O `@ ,' \n @' `@ \n\n \nLOSHU.JS:\nA Linear Algebra Library\nFor Javascript.\n\nLingdong Huang 2019*/\n \n\nvar LOSHU = new function(){\n var that = this;\n \n that.ASSERTION_NONE = 0;\n that.ASSERTION_WARN = 1;\n that.ASSERTION_DEFAULT = 2;\n that.ASSERTION_THROW = 3;\n \n that.SINGULAR = 'singular';\n that.INCONSISTENT = 'inconsistent';\n that.IRREDUCIBLE = 'irreducible';\n that.UNFACTORIZABLE = 'unfactorizable';\n\n var _wasmexist = false;\n \n var _isnode = typeof module !== 'undefined' && module.exports\n if (_isnode) {\n try{\n let {_LOSHUWASM,LOSHUWASM} = require('./loshuwasm');\n that.wasm = LOSHUWASM;\n that._wasm = _LOSHUWASM\n _wasmexist = true;\n }catch(e){/*no wasm*/}\n }else{\n _wasmexist = typeof LOSHUWASM != 'undefined';\n if (_wasmexist){\n /*global describe LOSHUWASM _LOSHUWASM*/\n that.wasm = LOSHUWASM; \n that._wasm = _LOSHUWASM;\n }\n }\n \n var _wasmready = false;\n if (_wasmexist){\n that._wasm.then(function(){\n var msg = \"🚀 web assembly backend initialized.\"\n if (!_isnode) {\n console.log(\"%c \"+msg,\"color:lightgreen; background:black\");\n }else{\n console.log('\\x1b[36m%s\\x1b[0m',msg);\n }\n _wasmready = true;\n });\n }\n function usingwasm(){\n return that.options.usewasm && _wasmexist && _wasmready;\n }\n \n that.options = {\n assertionlvl : that.ASSERTION_DEFAULT, //assertion level\n termcols: 80,\n usewasm: _wasmexist,\n }\n \n \n that.ismat = function(A){\n if (A == undefined){\n return false\n }\n if (typeof A != 'object'){\n return false;\n }\n if (!A.length){\n return false;\n }\n for (var i = 0; i < A.length; i++){\n if (typeof A[i] != 'object'){\n return false;\n }\n if (A[i].length != A[0].length){\n return false;\n }\n if (!A[i] || !A[i].length){\n return false;\n }\n for (var j = 0; j < A[i].length; j++){\n if (typeof A[i][j] != 'number'){\n return false;\n }\n }\n }\n return true;\n }\n \n that.isvec = function(v){\n if (typeof v != 'object'){\n return false;\n }\n for (var i = 0; i < v.length; i++){\n if (typeof v[i] != 'number'){\n return false;\n }\n }\n return true;\n }\n \n that.isnum = function(s){\n return (typeof s == 'number');\n }\n \n that.hasnan = function(A){\n if (typeof A == 'number'){\n return isNaN(A);\n }\n if (A == undefined || A == null || (typeof A != 'object')){\n return true\n }\n for (var i = 0; i < A.length; i++){\n if (that.hasnan(A[i])){\n return true;\n }\n }\n return false;\n }\n \n that.hasinf = function(A){\n if (typeof A == 'number'){\n return A == Infinity || A == -Infinity;\n }\n if (typeof A != 'object'){\n return false;\n }\n for (var i = 0; i < A.length; i++){\n if (that.hasinf(A[i])){\n return true;\n }\n }\n return false;\n }\n \n\n that.issquare = function(A){\n return A.length == A[0].length;\n }\n \n that.isperm = function(A){\n var ep = 1e-10;\n for (var i = 0; i < A.length; i++){\n if (! (A[i].filter(x=>Math.abs(x)Math.abs(x-1)x.filter(y=>y!=0).length).length\n return n;\n }\n \n that.rand = function(m,n){\n assert(m,'isnum','!hasnan');\n if (n == undefined){\n return new Array(m).fill(0).map(x=>Math.random());\n }else{\n return new Array(m).fill([]).map(x=>that.rand(n))\n }\n }\n that.zeros = function(m,n){\n assert(m,'isnum','!hasnan');\n if (n == undefined){\n return new Array(m).fill(0).map(x=>0);\n }else{\n return new Array(m).fill([]).map(x=>that.zeros(n))\n }\n }\n \n that.scale = function(A,s){\n assert(A,'ismat|isvec','!hasnan');\n assert(s,'isnum','!hasnan');\n \n if (typeof A[0] != 'number'){\n return A.map(x=>that.scale(x,s));\n }\n return A.map(x=>x*s);\n }\n that.add = function(A,B){\n assert(A,'ismat|isvec','!hasnan');\n assert(B,'ismat|isvec','!hasnan');\n \n if (typeof A[0] != 'number'){\n var r = []\n for (var i = 0; i < A.length; i++){\n r.push(that.add(A[i],B[i]));\n }\n return r;\n }\n return A.map((x,i)=>(x+B[i]))\n }\n that.sub = function(A,B){\n assert(A,'ismat|isvec','!hasnan');\n assert(B,'ismat|isvec','!hasnan');\n return that.add(A,that.scale(B,-1));\n }\n that.clone = function(A){\n assert(A,'ismat|isvec','!hasnan');\n if (that.isvec(A)){\n return A.slice();\n }else {\n return A.map(x=>x.slice());\n }\n }\n \n that.dot = function(u,v){\n assert(u,'isvec','!hasnan');\n assert(v,'isvec','!hasnan');\n var r = 0;\n for (var i = 0; i < u.length; i++){\n r += u[i]*v[i];\n }\n return r;\n }\n \n that.cross = function(u,v){\n assert(u,'isvec','!hasnan','nrows=3|nrows=7');\n assert(v,'isvec','!hasnan','nrows=3|nrows=7');\n if (u.length != 7){\n return [\n u[1]*v[2] - u[2]*v[1],\n u[2]*v[0] - u[0]*v[2],\n u[0]*v[1] - u[1]*v[0]\n ];\n }else{\n return [\n u[2]*v[4] - u[4]*v[2] + u[3]*v[7] - u[7]*v[3] + u[5]*v[6] - u[6]*v[5],\n u[3]*v[5] - u[5]*v[3] + u[4]*v[1] - u[1]*v[4] + u[6]*v[7] - u[7]*v[6],\n u[4]*v[6] - u[6]*v[4] + u[5]*v[2] - u[2]*v[5] + u[7]*v[1] - u[1]*v[7],\n u[5]*v[7] - u[7]*v[5] + u[6]*v[3] - u[3]*v[6] + u[1]*v[2] - u[2]*v[1],\n u[6]*v[1] - u[1]*v[6] + u[7]*v[4] - u[4]*v[7] + u[2]*v[3] - u[3]*v[2],\n u[7]*v[2] - u[2]*v[7] + u[1]*v[5] - u[5]*v[1] + u[3]*v[4] - u[4]*v[3],\n u[1]*v[3] - u[3]*v[1] + u[2]*v[6] - u[6]*v[2] + u[4]*v[5] - u[5]*v[4],\n ]\n }\n }\n \n that.iden = function(n){\n assert(n,'isnum');\n var r = [];\n for (var i = 0; i < n; i++){\n r.push([])\n for (var j = 0; j < n; j++){\n r[i].push(i==j?1:0)\n }\n }\n return r;\n }\n \n that.proj = function(u,v){\n assert(u,'isvec','!hasnan');\n assert(v,'isvec','!hasnan');\n return that.scale(u,that.dot(u,v)/that.dot(u,u));\n }\n \n that.norm = function(u,{order}={}){\n assert(u,'isvec|ismat','!hasnan');\n \n if (that.isvec(u)){\n if (order == undefined){order = '2'}\n \n if (order == '1'){\n return u.reduce((x,y)=>(x+y),0);\n }else if (order == '2'){\n var r = 0;\n for (var i = 0; i < u.length; i++){\n r += u[i]*u[i];\n }\n return Math.sqrt(r);\n }else{\n throw TypeError(order);\n }\n }else{\n if (order == undefined){order = 'fro'}\n \n if (order == 'fro'){\n var s = 0;\n for (var i = 0; i < u.length; i++){\n for (var j = 0; j < u[i].length; j++){\n s += u[i][j]*u[i][j]\n }\n }\n return Math.sqrt(s);\n }else if (order == '1'){\n var s = 0;\n for (var i = 0; i < u.length; i++){\n s+= Math.max.apply(null,u[i].map(Math.abs));\n }\n return s;\n }else if (order == 'inf'){\n return that.norm(that.transpose(u),{order:'1'})\n }else{\n throw TypeError(order);\n }\n }\n }\n\n that.normalize = function(u){\n assert(u,'isvec','!hasnan');\n var n = that.norm(u);\n return u.map(x=>x/n);\n }\n \n\n that.cond = function(A){\n var n = that.norm(A,{order:'inf'});\n var ninv = that.norm(that.inv(A),{order:'inf'});\n return n * ninv;\n }\n \n that.matmul = function(A, B, {method=\"naive\"}={}) {\n assert(A,'ismat','!hasnan');\n assert(B,'ismat','!hasnan','nrows='+that.ncols(A));\n \n if (usingwasm()){return that.wasm.matmul(A,B)};\n \n if (method == \"naive\"){\n var result = [];\n for (var i = 0; i < A.length; i++) {\n result[i] = [];\n for (var j = 0; j < B[0].length; j++) {\n var sum = 0;\n for (var k = 0; k < A[0].length; k++) {\n sum += A[i][k] * B[k][j];\n }\n result[i][j] = sum;\n }\n }\n return result;\n }else if (method == \"strassen\"){\n function split4(a){\n var au = a.slice(0,a.length/2);\n var al = a.slice(a.length/2);\n var [a11,a12] = that.hsplit(au,a.length/2);\n var [a21,a22] = that.hsplit(al,a.length/2);\n return [a11,a12,a21,a22];\n }\n function _matmul(a,b){\n if (a.length == 1){\n return [[a[0][0]*b[0][0]]]\n }\n var [a11, a12, a21, a22] = split4(a);\n var [b11, b12, b21, b22] = split4(b);\n var m1 = _matmul(that.add(a11,a22), that.add(b11,b22));\n var m2 = _matmul(that.add(a21,a22), b11);\n var m3 = _matmul(a11, that.sub(b12,b22));\n var m4 = _matmul(a22, that.sub(b21,b11));\n var m5 = _matmul(that.add(a11,a12),b22);\n var m6 = _matmul(that.sub(a21,a11), that.add(b11,b12));\n var m7 = _matmul(that.sub(a12,a22), that.add(b21,b22));\n var c11 = that.add(that.sub(that.add(m1,m4),m5),m7);\n var c12 = that.add(m3,m5);\n var c21 = that.add(m2,m4);\n var c22 = that.add(that.add(that.sub(m1,m2),m3),m6);\n return that.hstack(c11,c12).concat(that.hstack(c21,c22));\n \n }\n function nextpow2(x){\n return Math.pow(2, Math.ceil(Math.log(x)/Math.log(2)));\n }\n \n function pad2sq(a,n){\n var b = [];\n for (var i = 0; i < n; i++){\n b.push([])\n for (var j = 0; j < n; j++){\n if (i < a.length && j < a[0].length){\n b[b.length-1].push(a[i][j]);\n }else{\n b[b.length-1].push(0);\n }\n }\n }\n return b; \n }\n var n = Math.max(nextpow2(A.length),nextpow2(A[0].length),\n nextpow2(B.length),nextpow2(B[0].length));\n var a = pad2sq(A,n);\n var b = pad2sq(B,n);\n var c = _matmul(a,b);\n return that.hsplit(c.slice(0,A.length),B[0].length)[0];\n \n }else if (method == \"auto\"){\n if (A.length < 1e20){\n return that.matmul(A,B,{method:\"naive\"});\n }else{\n return that.matmul(A,B,{method:\"strassen\"});\n }\n }else{\n throw TypeError(method);\n }\n \n }\n \n that.mul = function(...items){\n assert(items[0], 'ismat|isvec|isnum', '!hasnan');\n assert(items[1], 'ismat|isvec|isnum', '!hasnan');\n \n return _mul(Array.from(items));\n \n function _mul(args){\n var a = args[0];\n var b = args[1];\n var c;\n if (that.isnum(a)){\n if (that.isnum(b)){\n c= a * b;\n }else if (that.isvec(b) || that.ismat(b)){\n c= that.scale(b,a);\n }\n }else if (that.isvec(a)){\n if (that.isnum(b)){\n c= that.scale(a,b);\n }else if (that.isvec(b)){\n throw TypeError(\n \"vector multiplication is ambiguous: use `dot`, `cross` or `outer`\"\n );\n }else if (that.ismat(b)){\n c= that.matmul(that.transpose([a]),b);\n }\n }else if (that.ismat(a)){\n if (that.isnum(b)){\n c= that.scale(a,b);\n }else if (that.isvec(b)){\n c= that.matmul(a,that.transpose([b]));\n }else if (that.ismat(b)){\n c= that.matmul(a,b);\n }\n }\n\n var rest = args.slice(2);\n\n if (rest.length){\n return _mul([c].concat(args.slice(2)));\n }\n return c;\n }\n }\n \n that.slice = function(A,ri,rj){\n assert(A,'ismat','!hasnan');\n assert(ri,'isvec','!hasnan');\n assert(rj,'isvec','!hasnan');\n var r = [];\n for (var i = Math.max(0,ri[0]); i < Math.min(ri[1],A.length); i++){\n r.push([])\n for (var j = Math.max(0,rj[0]); j < Math.min(rj[1],A[0].length); j++){\n r[r.length-1].push(A[i][j]);\n }\n }\n return r;\n }\n\n that.blit = function(A,ri,rj,B){\n assert(A,'ismat','!hasnan');\n assert(ri,'isvec','!hasnan');\n assert(rj,'isvec','!hasnan');\n assert(B,'ismat','!hasnan');\n \n var r = [];\n for (var i = 0; i < A.length; i++){\n r.push([])\n for (var j = 0; j < A[0].length; j++){\n if (ri[0] <= i && i < ri[1] && rj[0] <= j && j (x+y),0);\n }\n \n that.approx = function(A,B,{epsilon=1e-5}={}){\n assert(A,'ismat|isvec','!hasnan');\n assert(B,'ismat|isvec','!hasnan');\n \n if (A.length != B.length){\n return false;\n }\n if (that.isvec(A)){\n for (var i = 0; i < A.length; i++){\n if (Math.abs(A[i]-B[i]) > epsilon){\n return false;\n }\n }\n return true;\n }\n for (var i = 0; i < A.length; i++){\n if (!that.approx(A[i],B[i])){\n return false\n }\n }\n return true;\n }\n\n \n that.gaussjordan = function(A){\n assert(A,'ismat','!hasnan','nrows='+(that.ncols(A)-1));\n var [a,aug] = that.hsplit(A,that.ncols(A)-1);\n \n var [a_,aug_] = that.gauss(a,aug,{jordan:true,elementary:false,operations:['s','+*','*']});\n var redo = false;\n for (var i = 0; i < a_.length; i++){\n if (a_[i].filter(x=>(x!=0)).length == 0){\n if (aug_[i][0] == 0){\n aug_[i][0] = 1;\n a_[i][i] = 1;\n redo = true;\n }else{\n throw that.INCONSISTENT\n }\n }\n }\n if (redo){\n [a_,aug_] = that.gauss(a_,aug_,{jordan:true});\n }\n return that.transpose(aug_)[0];\n }\n \n that.gauss = function(A,B,{jordan=false,elementary=false,operations=['s','+*','*']}={}){\n assert(A,'ismat','!hasnan');\n assert(B,'ismat','!hasnan','nrows='+that.nrows(A));\n \n var m = that.hstack(A,B);\n var emats = [];\n \n function iszero(x){\n return Math.abs(x)<0.0001;\n }\n\n function elimrow(m,ind,i0,i1){\n if (!operations.includes('+*')){\n return m;\n }\n var r0 = m[i0];\n var r1 = m[i1];\n var p = -r1[ind]/r0[ind]\n var r = that.add(r1,that.scale(r0,p))\n if (elementary){\n var e = that.iden(m.length);\n e[i1] = that.add(e[i1],that.scale(e[i0],p))\n emats.push(e);\n }\n var M = that.clone(m);\n M[i1] = r;\n return M\n }\n \n function countleadzero(r){\n var counter = 0;\n for (var i = 0; i < r.length; i++){\n if (iszero(r[i])){\n counter += 1;\n }else{\n break;\n }\n }\n return counter;\n }\n \n function bbswaps(m,cmp){\n for (var i = 0; i < m.length; i++){\n for (var j = 0; j < m.length-i-1; j++){\n if (cmp(m[j],m[j+1])>0){\n [m[j],m[j+1]] = [m[j+1],m[j]];\n var e = that.iden(m.length);\n [e[j],e[j+1]] = [e[j+1],e[j]];\n emats.push(e);\n }\n }\n }\n }\n\n function reorder(m){\n if (!operations.includes('s')){\n return m;\n }\n var M = that.clone(m);\n var cmp = (x,y)=>(countleadzero(x)-countleadzero(y));\n if (elementary){\n bbswaps(M,cmp);\n }else{\n M.sort(cmp);\n }\n return M\n }\n function normdiag(m){\n if (!operations.includes('*')){\n return m;\n }\n var M = []\n for (var row = 0; row < m.length; row++){\n var p = m[row][row] == 0 ? 1 : 1.0/m[row][row]\n M.push(that.scale(m[row],p))\n if (elementary && p != 1){\n var e = that.iden(m.length);\n e[row] = that.scale(e[row],p);\n emats.push(e);\n }\n }\n return M;\n }\n function solve(m){\n \n function isech(){\n for (var i = 0; i < m.length; i++){\n if (countleadzero(m[i]) < i){\n return false\n }\n }\n return true\n }\n m = reorder(m)\n \n var lastm = undefined;\n while (!isech()){\n for (var ind = 0; ind < that.ncols(A); ind ++){\n for (var row = 1+ind; row < m.length; row++){\n if (!iszero(m[row][ind]) && !iszero(m[ind][ind])){\n\n m = elimrow(m,ind,ind,row);\n m = reorder(m);\n }\n }\n }\n if (lastm != undefined){\n if (that.approx(lastm,m)){\n throw that.IRREDUCIBLE;\n }\n }\n lastm = m;\n }\n\n if (!jordan){\n\n m = normdiag(m);\n \n var [x,y] = that.hsplit(m,that.ncols(A));\n return elementary ? [x,y,emats] : [x,y];\n }\n for (var ind = that.ncols(A)-1; ind > 0; ind--){\n for (var row = 0; row < ind; row++){\n if (!iszero(m[row][ind]) && !iszero(m[ind][ind])){\n m = elimrow(m,ind,ind,row);\n m = reorder(m);\n }\n }\n }\n \n \n m = normdiag(m);\n \n \n var [x,y] = that.hsplit(m,that.ncols(A));\n return elementary ? [x,y,emats] : [x,y];\n }\n return solve(m);\n }\n \n \n \n that.gramschmidt = function(vs){\n assert(vs,'ismat','!hasnan');\n var us = []\n for (var i = 0; i < vs.length; i++){\n var u = vs[i]\n for (var j = 0; j < i; j++){\n u = that.sub(u,that.proj(us[j],vs[i]))\n }\n us.push(u);\n }\n\n return us.map(x=>that.norm(x)!=0?that.normalize(x):x);\n }\n \n that.qr = function(A,{method=\"householder\"}={}){\n assert(A,'ismat','!hasnan');\n \n if (usingwasm()){return that.wasm.qr(A)};\n \n if (method == \"gramschmidt\"){\n assert(A,'rank='+A.length);\n\n var qt = that.gramschmidt(that.transpose(A));\n var q = that.transpose(qt);\n var r = that.matmul(qt,A);\n return [q,r];\n }else if (method == \"householder\"){\n var m = that.nrows(A);\n var n = that.ncols(A);\n \n function getq(A){\n var x = that.transpose(A)[0];\n var a = that.norm(x);\n var e1 = that.zeros(A.length); e1[0] = 1;\n var u = that.sub(x , that.scale(e1,a))\n var v = that.norm(u) == 0 ? u : that.normalize(u);\n \n var I = that.iden(A.length);\n var q1 = that.sub(I, that.scale(that.matmul(that.transpose([v]),[v]),2))\n return q1\n }\n \n var a = A;\n var Q = that.iden(a.length);\n \n for (var i = 0; i < Math.min(m-1,n); i++){\n var q = getq(a);\n var qp = that.blit(that.iden(m), [m-q.length,Infinity],[m-q.length,Infinity], q);\n\n Q = that.matmul(Q,that.transpose(qp));\n a = that.minor(that.matmul(q,a),0,0);\n \n }\n var R = that.matmul(that.transpose(Q),A);\n return [Q,R];\n \n }else{\n throw TypeError(method);\n }\n }\n \n that.householder = function(v){\n assert(v,'isvec','!hasnan');\n v = that.normalize(v);\n return that.sub(that.iden(v.length),that.scale(that.outer(v,v),2));\n }\n \n \n that.hessenberg = function(A){\n assert(A,'ismat','!hasnan');\n \n if (usingwasm()){return that.wasm.hessenberg(A)};\n \n function vhouse(x){\n x = that.norm(x) == 0? x: that.normalize(x);\n var x1 = x.slice(1);\n \n \n \n var s = that.matmul([x1], that.transpose([x1]))[0][0];\n var v = [1].concat(x1);\n \n var beta;\n if (s == 0){\n beta = 0;\n }else{\n var mu = Math.sqrt(x[0]*x[0]+s);\n \n v[0] = x[0] <= 0 ? (x[0]-mu) : (-s/(x[0]+mu))\n beta = 2 * v[0]*v[0] / (s+v[0]*v[0])\n v = v.map(u=>u/v[0]);\n }\n \n \n return [v,beta];\n }\n var n = Math.max(that.nrows(A),that.ncols(A));\n var Q = that.iden(n);\n var H = A;\n for (var k = 0; k < n-2; k++){\n var [v,beta] = vhouse(that.transpose(that.slice(H,[k+1,n],[k,k+1]))[0]);\n \n var I = that.iden(k+1);\n var N = that.zeros(k+1,n-k-1);\n var m = v.length;\n var R = that.sub(that.iden(m),that.mul(beta,that.transpose([v]),[v]));\n \n var h1 = that.matmul(R,that.slice(H,[k+1,n],[k,n]));\n H = that.blit(H,[k+1,n],[k,n],h1);\n \n var h2 = that.matmul(that.slice(H,[0,n],[k+1,n]),R);\n H = that.blit(H,[0,n],[k+1,n],h2);\n \n var P = that.hstack(I,N).concat(that.hstack(that.transpose(N),R));\n Q = that.matmul(Q,P);\n }\n // that.print(H,Q,that.mul(that.transpose(Q),a,Q));\n return [H,Q];\n \n }\n\n that.eigen = function(A,{fancy=true,maxiter=50}={}){\n assert(A,'ismat','!hasnan');\n \n if (usingwasm()){return that.wasm.eigen(A,{maxiter:maxiter})};\n \n if (!fancy){\n var Q = undefined;\n for (var i = 0; i < maxiter; i++){\n var [q,r] = that.qr(A);\n A = that.matmul(r,q);\n Q = Q?that.matmul(Q,q):q;\n }\n var vals = that.diagonal(A);\n // that.print(Q);\n var Qt = that.transpose(Q);\n var pairs = [];\n for (var i = 0; i < vals.length; i++){\n pairs.push({val:vals[i],vec:Qt[i]});\n }\n return pairs;\n }\n \n \n var n = A.length;\n var eigs = [];\n var I = that.iden(n);\n var Qh;\n [A, Qh] = that.hessenberg(A);\n \n function wilkinson(a,b,c){\n var d = (a-c)/2\n var sgn = d > 0 ? 1 : -1\n return c - sgn * b * b / (Math.abs(d) + Math.sqrt(d*d+b*b))\n }\n\n var Q = undefined;\n for (var k = 0; k < n; k++){\n if (k == n - 1){\n eigs.unshift(A[0][0]);\n continue;\n }\n for (var i = 0; i < maxiter; i++){\n var mu = wilkinson(A[A.length-2][A.length-2], \n A[A.length-2][A.length-1], \n A[A.length-1][A.length-1] );\n var muI = that.scale(I,mu)\n var [q,r] = that.qr(that.sub(A,muI));\n var qp = that.blit(I,[0,that.nrows(q)],[0,that.ncols(q)],q);\n Q = Q?that.matmul(Q,qp):qp;\n A = that.add(that.matmul(r,q), muI);\n if (Math.abs(A[A.length-1][A.length-2]) < 1e-8){\n break;\n }\n }\n eigs.unshift(A[A.length-1][A.length-1]); \n A = that.slice(A, [0, A.length-1], [0, A.length-1]);\n }\n var QQ = that.transpose(that.matmul(Qh,Q));\n var pairs = [];\n for (var i = 0; i < eigs.length; i++){\n pairs.push({val:eigs[i],vec:QQ[i]});\n }\n return pairs;\n }\n \n \n \n that.gerschgorin = function(A){\n assert(A,'ismat','!hasnan','issquare');\n var n = that.nrows(A);\n var disks = [];\n for (var i = 0; i < n; i++){\n var r = 0;\n var o = A[i][i];\n for (var j = 0; j < n; j++){\n if (i != j){\n r += Math.abs(A[i][j]);\n }\n }\n disks.push({o:o,r:r});\n }\n return disks;\n }\n \n that.minor = function(A,i,j){\n assert(A,'ismat','!hasnan');\n assert(i,'isnum');\n assert(j,'isnum');\n var r = []\n for (var a = 0; a < A.length; a++){\n if (a == i){\n continue;\n }\n r.push([]);\n for (var b = 0; b < A[a].length; b++){\n if (b == j){\n continue;\n }\n r[r.length-1].push(A[a][b]);\n }\n }\n return r;\n \n }\n \n that.cofactor = function(A,i,j){\n assert(A,'ismat','!hasnan','issquare');\n assert(i,'isnum');\n assert(j,'isnum')\n \n return Math.pow(-1,i+j) * that.det(that.minor(A,i,j))\n }\n \n \n that.det = function(A){\n assert(A,'ismat','!hasnan','issquare');\n \n if (usingwasm()){return that.wasm.det(A)};\n \n var n = A.length;\n if (n == 1){\n return A[0][0];\n }\n if (n == 2){\n return A[0][0]*A[1][1]-A[0][1]*A[1][0];\n }\n var s = 0;\n for (var j = 0; j < n; j++){\n s += A[0][j] * that.cofactor(A,0,j)\n }\n \n return s;\n }\n \n this.adj = function(A){\n assert(A,'ismat','!hasnan','issquare');\n var comat = that.map(A, (x,i,j)=>(that.cofactor(A,i,j)));\n return that.transpose(comat);\n \n }\n \n that.inv = function(A,{method=\"adjugate\"}={}){\n assert(A,'ismat','!hasnan','issquare');\n \n if (usingwasm()){return that.wasm.inv(A)};\n \n if (method == 'adjugate'){\n var d = that.det(A);\n if (Math.abs(d)<1e-10){\n throw that.SINGULAR;\n }\n return that.scale(that.adj(A),1/d);\n \n }else if (method == 'gaussjordan'){\n var I = that.iden(A.length);\n var [a,aug] = that.gauss(A,I,{jordan:true});\n if (that.approx(a,I)){\n return aug;\n }else{\n throw that.SINGULAR\n }\n }else{\n throw TypeError(method);\n }\n }\n \n that.lup = function(A,{p=false}={}){\n assert(A,'ismat','!hasnan','issquare');\n if (p){\n var [u,_,emats] = that.gauss(A,that.iden(A.length),{\n elementary:true,jordan:false,operations:['s','+*']});\n var l = that.iden(A.length);\n var P = that.iden(A.length);\n // console.log(emats)\n for (var i = 0; i < emats.length; i++){\n \n if (that.isperm(emats[i])){\n \n P = that.matmul(emats[i],P);\n \n }else{\n var einv = that.inv(emats[i],{method:'adjugate'});\n l = that.matmul(l,einv);\n \n }\n }\n return [l,u,P]\n }else{\n \n var [u,_,emats] = that.gauss(A,that.iden(A.length),{\n elementary:true,jordan:false,operations:['+*']});\n\n var l = that.iden(A.length);\n for (var i = 0; i < emats.length; i++){\n var einv = that.inv(emats[i],{method:'adjugate'});\n l = that.matmul(l,einv);\n \n }\n return [l,u]\n }\n }\n \n that.cholesky = function(A){\n assert(A,'ismat','!hasnan','issquare','ishermitian');\n var n = A.length;\n var l = that.zeros(n,n)\n\n for (var i = 0; i < n; i++){\n for (var j = 0; j <= i; j++){\n if (i == j){\n var s = A[j][j];\n for (var k = 0; k < j; k++){\n s -= l[j][k]*l[j][k];\n }\n l[j][j] = Math.sqrt(s);\n }else{\n var s = 0;\n for (var k = 0; k < j; k++){\n s += l[i][k] * l[j][k];\n }\n l[i][j] = (A[i][j]-s)/l[j][j];\n }\n }\n }\n return [l, that.transpose(l)];\n }\n \n that.ldl = function(A){\n assert(A,'ismat','!hasnan','issquare','ishermitian');\n \n var n = A.length;\n var d = that.zeros(n,n);\n var l = that.iden(n);\n \n for (var i = 0; i < n; i++){\n for (var j = 0; j <= i; j++){\n if (i == j){\n var s = A[j][j]\n for (var k = 0; k < j; k++){\n s -= l[j][k]*l[j][k]*d[k][k]\n }\n d[j][j] = s;\n }else{\n var s = A[i][j]\n for (var k = 0; k < j; k++){\n s -= l[i][k]*l[j][k]*d[k][k]\n }\n l[i][j] = s/d[j][j];\n }\n }\n }\n return [l,d,that.transpose(l)];\n }\n \n \n that.polar = function(A,{method='svd'}={}){\n assert(A,'ismat','!hasnan','issquare');\n \n if (method=='direct'){\n var ata = that.matmul(that.transpose(A), A);\n var eigs = that.eigen(ata);\n // console.log(eigs)\n if (eigs.length < ata.length){\n throw that.UNFACTORIZABLE\n }\n var S = undefined;\n var B = that.iden(A.length);\n for (var i = 0; i < eigs.length; i++){\n var sc = 1/eigs[i].vec[0];\n for (var j = 0; j < eigs[i].vec.length; j++){\n eigs[i].vec[j] *= sc;\n }\n var col = that.transpose([eigs[i].vec]);\n if (S == undefined){\n S = col;\n }else{\n S = that.hstack(S,col);\n }\n B[i][i] = eigs[i].val\n }\n var C = that.map(B,Math.sqrt);\n var Sinv = that.inv(S);\n var P = that.matmul(that.matmul(S,C),Sinv);\n var U = that.matmul(A,that.inv(P));\n return [U,P];\n \n }else if (method=='svd'){\n var [u,s,v] = that.svd(A);\n var vt = that.transpose(v)\n var B = that.matmul(u,vt);\n var C = that.matmul(that.matmul(v,s),vt);\n return [B,C]\n \n }else{\n throw TypeError(method);\n }\n }\n \n that.svd = function(A){\n assert(A,'ismat','!hasnan');\n\n if (usingwasm()){return that.wasm.svd(A)};\n \n var ata = that.matmul(that.transpose(A), A);\n \n\n \n var eigs = that.eigen(ata);\n eigs.sort((x,y)=>(y.val-x.val));\n // that.print(eigs)\n if (eigs.length < ata.length){\n throw that.UNFACTORIZABLE\n }\n \n \n var U = undefined;\n var V = undefined;\n var S = that.zeros(eigs.length, eigs.length);\n \n for (var i = 0; i < S.length; i++){\n var sig = (eigs[i].val < 1e-10) ? 0 : Math.sqrt(eigs[i].val)\n\n var u, v;\n if (sig != 0){\n // console.log(sig);\n v = that.transpose([that.normalize(eigs[i].vec)]);\n u = that.scale(that.matmul(A,v),1/sig);\n \n }else{\n v = that.transpose([eigs[i].vec]);\n u = that.matmul(A,v);\n }\n U = (U==undefined)?u:that.hstack(U,u);\n V = (V==undefined)?v:that.hstack(V,v);\n\n // that.print(A,i)\n S[i][i] = sig;\n }\n // console.log(that.approx(a,that.matmul(that.matmul(U,S),that.transpose(V))))\n \n return [U,S,V]\n }\n \n that.pinv = function(A, {method='direct'}={}){\n assert(A,'ismat','!hasnan');\n if (method == 'direct'){\n var at = that.transpose(A);\n return that.matmul(that.inv(that.matmul(at,A)),at);\n }else if (method == 'svd'){\n var [u,s,v] = that.svd(A);\n for (var i = 0; i < Math.min(that.nrows(s),that.ncols(s)); i++){\n if (Math.abs(s[i][i]) > 1e-10){\n s[i][i] = 1/s[i][i];\n }else{\n s[i][i] = 0;\n }\n }\n s = that.transpose(s);\n return that.matmul(that.matmul(v,s),that.transpose(u));\n }else{\n throw TypeError(method);\n }\n }\n \n that.affine = function(typ,a,b){\n assert(a,'isvec|isnum','!hasnan');\n \n if (typ == \"rot2d\"){\n return [\n [Math.cos(a), -Math.sin(a), 0],\n [Math.sin(a), Math.cos(a), 0], [0,0,1]\n ]\n }else if (typ == \"rotx\"){\n return [\n [1, 0, 0, 0], [0, Math.cos(a), -Math.sin(a),0], \n [0, Math.sin(a), Math.cos(a), 0], [0, 0, 0, 1]\n ];\n }else if (typ == \"roty\"){\n return [\n [Math.cos(a), 0, Math.sin(a), 0], [0, 1, 0, 0], \n [-Math.sin(a),0, Math.cos(a), 0], [0, 0, 0, 1]\n ];\n }else if (typ == \"rotz\"){\n return [\n [Math.cos(a), -Math.sin(a),0, 0], \n [Math.sin(a), Math.cos(a), 0, 0], \n [0, 0, 1, 0], [0, 0, 0, 1]\n ];\n }else if (typ == \"rotaxial\"){\n var costh = Math.cos(a);\n var sinth = Math.sin(a);\n return [\n [b[0]*b[0] *(1-costh) +costh,\n b[1]*b[0] *(1-costh) -b[2]*sinth,\n b[2]*b[0] *(1-costh) +b[1]*sinth,\n 0],\n [b[0]*b[1] *(1-costh) +b[2]*sinth,\n b[1]*b[1] *(1-costh) +costh,\n b[2]*b[1] *(1-costh) -b[0]*sinth,\n 0],\n [b[0]*b[2] *(1-costh) -b[1]*sinth,\n b[1]*b[2] *(1-costh) +b[0]*sinth,\n b[2]*b[2] *(1-costh) +costh,\n 0],\n [0,0,0,1]\n ]\n }else if (typ == \"transl2d\"){\n return [\n [1, 0, a[0]], [0, 1, a[1]], [0, 0, 1]\n ];\n }else if (typ == \"transl3d\"){\n return [\n [1, 0 ,0, a[0]], [0, 1, 0, a[1]], \n [0, 0, 1, a[2]], [0, 0, 0, 1]\n ];\n }else if (typ == \"scale2d\"){\n if (typeof a == 'number'){\n a = [a,a];\n }\n return [\n [a[0], 0 ,0], [0, a[1], 0], [0, 0, 1]\n ];\n }else if (typ == \"scale3d\"){\n if (typeof a == 'number'){\n a = [a,a,a];\n }\n return [\n [a[0], 0 ,0, 0], [0, a[1], 0, 0], \n [0, 0, a[2], 0], [0, 0, 0, 1]\n ];\n }else if (typ == \"reflect2d\"){\n a = that.normalize(a);\n return [\n [a[0]*a[0]-a[1]*a[1], 2*a[0]*a[1],0],\n [2*a[0]*a[1], a[1]*a[1]-a[0]*a[0],0],\n [0,0,1],\n ]\n }else if (typ == \"reflect3d\"){\n return [\n [1-2*a[0]*a[0], -2*a[0]*a[1], -2*a[0]*a[2], 0],\n [-2*a[0]*a[1], 1-2*a[1]*a[1], -2*a[1]*a[2], 0],\n [-2*a[0]*a[2], -2*a[1]*a[2], 1-2*a[2]*a[2], 0],\n [0,0,0,1]\n ] \n }else{\n throw TypeError(typ);\n }\n }\n \n that.transform = function(A,v){\n assert(A, 'ismat','!hasnan','ncols='+(v.length+1));\n assert(v, 'isvec','!hasnan');\n var u = that.transpose(that.matmul(A, that.transpose([v.concat(1)])))[0];\n // console.log(u);\n for (var i = 0; i < u.length-1; i++){\n u[i] /= u[u.length-1];\n }\n return u.slice(0,-1);\n }\n\n that.magic = function(n){\n assert(n, 'isnum','!hasnan');\n \n var a = that.zeros(n,n);\n if (n % 2 == 1){\n var y = 0;\n var x = Math.floor(n/2);\n \n for (var i = 0; i < n*n; i++){\n a[y][x] = i+1;\n var ny = (y - 1 + n) % n;\n var nx = (x + 1 + n) % n;\n if (a[ny][nx] != 0){\n [x,y] = [x, (y + 1 + n) % n];\n }else{\n [x,y] = [nx,ny];\n }\n }\n \n }else if (n % 4 == 0){\n for (var i = 0; i < n; i++){\n for (var j = 0; j < n; j++){\n if ((i=n*3/4) ||\n (i>=n*3/4 && j=n*3/4 && j>=n*3/4)||\n (i>=n/4 && i = n/4 && j x.filter(y=>Math.abs(y)>1e-10).length);\n qtb = qtb.slice(0,r.length);\n return that.gaussjordan(that.hstack(r,qtb));\n }else{\n throw TypeError(method);\n }\n }\n \n that.lsqfit = function(data, {order=1}={}){\n assert(data, 'ismat','!hasnan');\n var a = that.zeros(data.length,order+1);\n var b = [];\n for (var i = 0; i < data.length; i++){\n for (var j = 0; j < order+1; j++){\n a[i][j] = Math.pow(data[i][0],j);\n }\n b.push(data[i][1]);\n }\n return that.lsq(a,b,{method:'qr'});\n }\n \n that.dist = function(typ0,a,typ1,b){\n assert(a, 'ismat|isvec','!hasnan');\n assert(b, 'ismat|isvec','!hasnan');\n \n function checkplane(a){\n if (that.norm(a.slice(0,-1))<1e-10){\n return TypeError(\"invalid plane equation\")\n }\n }\n \n \n if (typ0 == 'pt'){\n if (typ1 == 'pt'){\n return that.norm(that.sub(a,b));\n }else if (typ1 == 'ln'){\n if (that.norm(b[1]) < 1e-10){\n return that.dist('pt',a,'pt',b[0]);\n }\n var o = b[0];\n var d = that.normalize(b[1]);\n var oa = that.sub(a,o);\n return that.norm(that.sub(oa, that.scale(d, that.dot(oa, d))));\n }else if (typ1 == 'pl'){\n var d = b[b.length-1];\n var c = b.slice(0,-1);\n return Math.abs(that.dot(a,c)-d)/that.norm(c);\n }else{\n throw TypeError(typ1);\n }\n }else if (typ0 == 'ln'){\n if (typ1 == 'pt'){\n return that.dist(typ1,b,typ0,a);\n }else if (typ1 == 'ln'){\n if (a[0].length == 2){\n return that.dist('ln',that.hstack(a,[[0],[0]]),'ln',that.hstack(b,[[0],[0]]));\n }\n if (a[0].length == 3){\n if (that.norm(a[1]) < 1e-10){\n return that.dist('pt',a[0],'ln',b);\n }\n if (that.norm(b[1]) < 1e-10){\n return that.dist('pt',b[0],'ln',a);\n }\n var n = that.cross(a[1],b[1]);\n var r = that.sub(a[0],b[0]);\n var d = that.dot(n,r)/that.norm(n);\n return d;\n }\n var A = that.T([a[1],that.scale(b[1],-1)])\n var B = that.sub(b[0],a[0]);\n var [t,s] = that.lsq(A,B,{method:'qr'});\n var p = that.add(a[0],that.scale(a[1],t));\n var q = that.add(b[0],that.scale(b[1],s));\n return that.dist('pt',p,'pt',q);\n }else if (typ1 == 'pl'){\n checkplane(b);\n var x = that.dot(a[1],b.slice(0,-1));\n if (Math.abs(x) < 1e-10){\n return that.dist('pt',a[0],'pl',b);\n }else{\n return 0;\n }\n }else{\n throw TypeError(typ1);\n }\n }else if (typ0 == 'pl'){\n if (typ1 == 'pt'){\n return that.dist(typ1,b,typ0,a);\n }else if (typ1 == 'ln'){\n return that.dist(typ1,b,typ0,a);\n }else if (typ1 == 'pl'){\n checkplane(a);\n checkplane(b);\n var x = that.dot(that.normalize(a.slice(0,-1)),that.normalize(b.slice(0,-1)));\n \n if (Math.abs(x-1) < 1e-10 || Math.abs(x+1) < 1e-10){\n var p = new Array(a.length-1).fill(1);\n var pvt = 0;\n for (var i = 0; i < a.length-1; i++){\n if (a[i] != 0){\n pvt = i;\n break;\n }\n }\n p[pvt] = a[a.length-1];\n for (var i = 0; i < a.length-1; i++){\n if (i != pvt){\n p[pvt] -= a[i];\n }\n }\n p[pvt] /= a[pvt];\n that.print(p);\n return that.dist('pt',p, 'pl',b)\n }else{\n return 0;\n }\n }else{\n throw TypeError(typ1);\n }\n }\n \n }\n\n \n}\n\n\nif (typeof module !== 'undefined' && module.exports) {\n module.exports = LOSHU \n}else{\n var lo = LOSHU;\n}"]}