mathjs
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Math.js is an extensive math library for JavaScript and Node.js. It features a flexible expression parser with support for symbolic computation, comes with a large set of built-in functions and constants, and offers an integrated solution to work with dif
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JavaScript
'use strict'
function factory (type, config, load) {
const csFlip = load(require('./csFlip'))
const csFkeep = load(require('./csFkeep'))
const csTdfs = load(require('./csTdfs'))
const add = load(require('../../arithmetic/add'))
const multiply = load(require('../../arithmetic/multiply'))
const transpose = load(require('../../matrix/transpose'))
/**
* Approximate minimum degree ordering. The minimum degree algorithm is a widely used
* heuristic for finding a permutation P so that P*A*P' has fewer nonzeros in its factorization
* than A. It is a gready method that selects the sparsest pivot row and column during the course
* of a right looking sparse Cholesky factorization.
*
* Reference: http://faculty.cse.tamu.edu/davis/publications.html
*
* @param {Number} order 0: Natural, 1: Cholesky, 2: LU, 3: QR
* @param {Matrix} m Sparse Matrix
*
* Reference: http://faculty.cse.tamu.edu/davis/publications.html
*/
const csAmd = function (order, a) {
// check input parameters
if (!a || order <= 0 || order > 3) { return null }
// a matrix arrays
const asize = a._size
// rows and columns
const m = asize[0]
const n = asize[1]
// initialize vars
let lemax = 0
// dense threshold
let dense = Math.max(16, 10 * Math.sqrt(n))
dense = Math.min(n - 2, dense)
// create target matrix C
const cm = _createTargetMatrix(order, a, m, n, dense)
// drop diagonal entries
csFkeep(cm, _diag, null)
// C matrix arrays
const cindex = cm._index
const cptr = cm._ptr
// number of nonzero elements in C
let cnz = cptr[n]
// allocate result (n+1)
const P = []
// create workspace (8 * (n + 1))
const W = []
const len = 0 // first n + 1 entries
const nv = n + 1 // next n + 1 entries
const next = 2 * (n + 1) // next n + 1 entries
const head = 3 * (n + 1) // next n + 1 entries
const elen = 4 * (n + 1) // next n + 1 entries
const degree = 5 * (n + 1) // next n + 1 entries
const w = 6 * (n + 1) // next n + 1 entries
const hhead = 7 * (n + 1) // last n + 1 entries
// use P as workspace for last
const last = P
// initialize quotient graph
let mark = _initializeQuotientGraph(n, cptr, W, len, head, last, next, hhead, nv, w, elen, degree)
// initialize degree lists
let nel = _initializeDegreeLists(n, cptr, W, degree, elen, w, dense, nv, head, last, next)
// minimum degree node
let mindeg = 0
// vars
let i, j, k, k1, k2, e, pj, ln, nvi, pk, eln, p1, p2, pn, h, d
// while (selecting pivots) do
while (nel < n) {
// select node of minimum approximate degree. amd() is now ready to start eliminating the graph. It first
// finds a node k of minimum degree and removes it from its degree list. The variable nel keeps track of thow
// many nodes have been eliminated.
for (k = -1; mindeg < n && (k = W[head + mindeg]) === -1; mindeg++);
if (W[next + k] !== -1) { last[W[next + k]] = -1 }
// remove k from degree list
W[head + mindeg] = W[next + k]
// elenk = |Ek|
const elenk = W[elen + k]
// # of nodes k represents
let nvk = W[nv + k]
// W[nv + k] nodes of A eliminated
nel += nvk
// Construct a new element. The new element Lk is constructed in place if |Ek| = 0. nv[i] is
// negated for all nodes i in Lk to flag them as members of this set. Each node i is removed from the
// degree lists. All elements e in Ek are absorved into element k.
let dk = 0
// flag k as in Lk
W[nv + k] = -nvk
let p = cptr[k]
// do in place if W[elen + k] === 0
const pk1 = (elenk === 0) ? p : cnz
let pk2 = pk1
for (k1 = 1; k1 <= elenk + 1; k1++) {
if (k1 > elenk) {
// search the nodes in k
e = k
// list of nodes starts at cindex[pj]
pj = p
// length of list of nodes in k
ln = W[len + k] - elenk
} else {
// search the nodes in e
e = cindex[p++]
pj = cptr[e]
// length of list of nodes in e
ln = W[len + e]
}
for (k2 = 1; k2 <= ln; k2++) {
i = cindex[pj++]
// check node i dead, or seen
if ((nvi = W[nv + i]) <= 0) { continue }
// W[degree + Lk] += size of node i
dk += nvi
// negate W[nv + i] to denote i in Lk
W[nv + i] = -nvi
// place i in Lk
cindex[pk2++] = i
if (W[next + i] !== -1) { last[W[next + i]] = last[i] }
// check we need to remove i from degree list
if (last[i] !== -1) { W[next + last[i]] = W[next + i] } else { W[head + W[degree + i]] = W[next + i] }
}
if (e !== k) {
// absorb e into k
cptr[e] = csFlip(k)
// e is now a dead element
W[w + e] = 0
}
}
// cindex[cnz...nzmax] is free
if (elenk !== 0) { cnz = pk2 }
// external degree of k - |Lk\i|
W[degree + k] = dk
// element k is in cindex[pk1..pk2-1]
cptr[k] = pk1
W[len + k] = pk2 - pk1
// k is now an element
W[elen + k] = -2
// Find set differences. The scan1 function now computes the set differences |Le \ Lk| for all elements e. At the start of the
// scan, no entry in the w array is greater than or equal to mark.
// clear w if necessary
mark = _wclear(mark, lemax, W, w, n)
// scan 1: find |Le\Lk|
for (pk = pk1; pk < pk2; pk++) {
i = cindex[pk]
// check if W[elen + i] empty, skip it
if ((eln = W[elen + i]) <= 0) { continue }
// W[nv + i] was negated
nvi = -W[nv + i]
const wnvi = mark - nvi
// scan Ei
for (p = cptr[i], p1 = cptr[i] + eln - 1; p <= p1; p++) {
e = cindex[p]
if (W[w + e] >= mark) {
// decrement |Le\Lk|
W[w + e] -= nvi
} else if (W[w + e] !== 0) {
// ensure e is a live element, 1st time e seen in scan 1
W[w + e] = W[degree + e] + wnvi
}
}
}
// degree update
// The second pass computes the approximate degree di, prunes the sets Ei and Ai, and computes a hash
// function h(i) for all nodes in Lk.
// scan2: degree update
for (pk = pk1; pk < pk2; pk++) {
// consider node i in Lk
i = cindex[pk]
p1 = cptr[i]
p2 = p1 + W[elen + i] - 1
pn = p1
// scan Ei
for (h = 0, d = 0, p = p1; p <= p2; p++) {
e = cindex[p]
// check e is an unabsorbed element
if (W[w + e] !== 0) {
// dext = |Le\Lk|
const dext = W[w + e] - mark
if (dext > 0) {
// sum up the set differences
d += dext
// keep e in Ei
cindex[pn++] = e
// compute the hash of node i
h += e
} else {
// aggressive absorb. e->k
cptr[e] = csFlip(k)
// e is a dead element
W[w + e] = 0
}
}
}
// W[elen + i] = |Ei|
W[elen + i] = pn - p1 + 1
const p3 = pn
const p4 = p1 + W[len + i]
// prune edges in Ai
for (p = p2 + 1; p < p4; p++) {
j = cindex[p]
// check node j dead or in Lk
const nvj = W[nv + j]
if (nvj <= 0) { continue }
// degree(i) += |j|
d += nvj
// place j in node list of i
cindex[pn++] = j
// compute hash for node i
h += j
}
// check for mass elimination
if (d === 0) {
// absorb i into k
cptr[i] = csFlip(k)
nvi = -W[nv + i]
// |Lk| -= |i|
dk -= nvi
// |k| += W[nv + i]
nvk += nvi
nel += nvi
W[nv + i] = 0
// node i is dead
W[elen + i] = -1
} else {
// update degree(i)
W[degree + i] = Math.min(W[degree + i], d)
// move first node to end
cindex[pn] = cindex[p3]
// move 1st el. to end of Ei
cindex[p3] = cindex[p1]
// add k as 1st element in of Ei
cindex[p1] = k
// new len of adj. list of node i
W[len + i] = pn - p1 + 1
// finalize hash of i
h = (h < 0 ? -h : h) % n
// place i in hash bucket
W[next + i] = W[hhead + h]
W[hhead + h] = i
// save hash of i in last[i]
last[i] = h
}
}
// finalize |Lk|
W[degree + k] = dk
lemax = Math.max(lemax, dk)
// clear w
mark = _wclear(mark + lemax, lemax, W, w, n)
// Supernode detection. Supernode detection relies on the hash function h(i) computed for each node i.
// If two nodes have identical adjacency lists, their hash functions wil be identical.
for (pk = pk1; pk < pk2; pk++) {
i = cindex[pk]
// check i is dead, skip it
if (W[nv + i] >= 0) { continue }
// scan hash bucket of node i
h = last[i]
i = W[hhead + h]
// hash bucket will be empty
W[hhead + h] = -1
for (; i !== -1 && W[next + i] !== -1; i = W[next + i], mark++) {
ln = W[len + i]
eln = W[elen + i]
for (p = cptr[i] + 1; p <= cptr[i] + ln - 1; p++) { W[w + cindex[p]] = mark }
let jlast = i
// compare i with all j
for (j = W[next + i]; j !== -1;) {
let ok = W[len + j] === ln && W[elen + j] === eln
for (p = cptr[j] + 1; ok && p <= cptr[j] + ln - 1; p++) {
// compare i and j
if (W[w + cindex[p]] !== mark) { ok = 0 }
}
// check i and j are identical
if (ok) {
// absorb j into i
cptr[j] = csFlip(i)
W[nv + i] += W[nv + j]
W[nv + j] = 0
// node j is dead
W[elen + j] = -1
// delete j from hash bucket
j = W[next + j]
W[next + jlast] = j
} else {
// j and i are different
jlast = j
j = W[next + j]
}
}
}
}
// Finalize new element. The elimination of node k is nearly complete. All nodes i in Lk are scanned one last time.
// Node i is removed from Lk if it is dead. The flagged status of nv[i] is cleared.
for (p = pk1, pk = pk1; pk < pk2; pk++) {
i = cindex[pk]
// check i is dead, skip it
if ((nvi = -W[nv + i]) <= 0) { continue }
// restore W[nv + i]
W[nv + i] = nvi
// compute external degree(i)
d = W[degree + i] + dk - nvi
d = Math.min(d, n - nel - nvi)
if (W[head + d] !== -1) { last[W[head + d]] = i }
// put i back in degree list
W[next + i] = W[head + d]
last[i] = -1
W[head + d] = i
// find new minimum degree
mindeg = Math.min(mindeg, d)
W[degree + i] = d
// place i in Lk
cindex[p++] = i
}
// # nodes absorbed into k
W[nv + k] = nvk
// length of adj list of element k
if ((W[len + k] = p - pk1) === 0) {
// k is a root of the tree
cptr[k] = -1
// k is now a dead element
W[w + k] = 0
}
if (elenk !== 0) {
// free unused space in Lk
cnz = p
}
}
// Postordering. The elimination is complete, but no permutation has been computed. All that is left
// of the graph is the assembly tree (ptr) and a set of dead nodes and elements (i is a dead node if
// nv[i] is zero and a dead element if nv[i] > 0). It is from this information only that the final permutation
// is computed. The tree is restored by unflipping all of ptr.
// fix assembly tree
for (i = 0; i < n; i++) { cptr[i] = csFlip(cptr[i]) }
for (j = 0; j <= n; j++) { W[head + j] = -1 }
// place unordered nodes in lists
for (j = n; j >= 0; j--) {
// skip if j is an element
if (W[nv + j] > 0) { continue }
// place j in list of its parent
W[next + j] = W[head + cptr[j]]
W[head + cptr[j]] = j
}
// place elements in lists
for (e = n; e >= 0; e--) {
// skip unless e is an element
if (W[nv + e] <= 0) { continue }
if (cptr[e] !== -1) {
// place e in list of its parent
W[next + e] = W[head + cptr[e]]
W[head + cptr[e]] = e
}
}
// postorder the assembly tree
for (k = 0, i = 0; i <= n; i++) {
if (cptr[i] === -1) { k = csTdfs(i, k, W, head, next, P, w) }
}
// remove last item in array
P.splice(P.length - 1, 1)
// return P
return P
}
/**
* Creates the matrix that will be used by the approximate minimum degree ordering algorithm. The function accepts the matrix M as input and returns a permutation
* vector P. The amd algorithm operates on a symmetrix matrix, so one of three symmetric matrices is formed.
*
* Order: 0
* A natural ordering P=null matrix is returned.
*
* Order: 1
* Matrix must be square. This is appropriate for a Cholesky or LU factorization.
* P = M + M'
*
* Order: 2
* Dense columns from M' are dropped, M recreated from M'. This is appropriatefor LU factorization of unsymmetric matrices.
* P = M' * M
*
* Order: 3
* This is best used for QR factorization or LU factorization is matrix M has no dense rows. A dense row is a row with more than 10*sqr(columns) entries.
* P = M' * M
*/
function _createTargetMatrix (order, a, m, n, dense) {
// compute A'
const at = transpose(a)
// check order = 1, matrix must be square
if (order === 1 && n === m) {
// C = A + A'
return add(a, at)
}
// check order = 2, drop dense columns from M'
if (order === 2) {
// transpose arrays
const tindex = at._index
const tptr = at._ptr
// new column index
let p2 = 0
// loop A' columns (rows)
for (let j = 0; j < m; j++) {
// column j of AT starts here
let p = tptr[j]
// new column j starts here
tptr[j] = p2
// skip dense col j
if (tptr[j + 1] - p > dense) { continue }
// map rows in column j of A
for (const p1 = tptr[j + 1]; p < p1; p++) { tindex[p2++] = tindex[p] }
}
// finalize AT
tptr[m] = p2
// recreate A from new transpose matrix
a = transpose(at)
// use A' * A
return multiply(at, a)
}
// use A' * A, square or rectangular matrix
return multiply(at, a)
}
/**
* Initialize quotient graph. There are four kind of nodes and elements that must be represented:
*
* - A live node is a node i (or a supernode) that has not been selected as a pivot nad has not been merged into another supernode.
* - A dead node i is one that has been removed from the graph, having been absorved into r = flip(ptr[i]).
* - A live element e is one that is in the graph, having been formed when node e was selected as the pivot.
* - A dead element e is one that has benn absorved into a subsequent element s = flip(ptr[e]).
*/
function _initializeQuotientGraph (n, cptr, W, len, head, last, next, hhead, nv, w, elen, degree) {
// Initialize quotient graph
for (let k = 0; k < n; k++) { W[len + k] = cptr[k + 1] - cptr[k] }
W[len + n] = 0
// initialize workspace
for (let i = 0; i <= n; i++) {
// degree list i is empty
W[head + i] = -1
last[i] = -1
W[next + i] = -1
// hash list i is empty
W[hhead + i] = -1
// node i is just one node
W[nv + i] = 1
// node i is alive
W[w + i] = 1
// Ek of node i is empty
W[elen + i] = 0
// degree of node i
W[degree + i] = W[len + i]
}
// clear w
const mark = _wclear(0, 0, W, w, n)
// n is a dead element
W[elen + n] = -2
// n is a root of assembly tree
cptr[n] = -1
// n is a dead element
W[w + n] = 0
// return mark
return mark
}
/**
* Initialize degree lists. Each node is placed in its degree lists. Nodes of zero degree are eliminated immediately. Nodes with
* degree >= dense are alsol eliminated and merged into a placeholder node n, a dead element. Thes nodes will appera last in the
* output permutation p.
*/
function _initializeDegreeLists (n, cptr, W, degree, elen, w, dense, nv, head, last, next) {
// result
let nel = 0
// loop columns
for (let i = 0; i < n; i++) {
// degree @ i
const d = W[degree + i]
// check node i is empty
if (d === 0) {
// element i is dead
W[elen + i] = -2
nel++
// i is a root of assembly tree
cptr[i] = -1
W[w + i] = 0
} else if (d > dense) {
// absorb i into element n
W[nv + i] = 0
// node i is dead
W[elen + i] = -1
nel++
cptr[i] = csFlip(n)
W[nv + n]++
} else {
const h = W[head + d]
if (h !== -1) { last[h] = i }
// put node i in degree list d
W[next + i] = W[head + d]
W[head + d] = i
}
}
return nel
}
function _wclear (mark, lemax, W, w, n) {
if (mark < 2 || (mark + lemax < 0)) {
for (let k = 0; k < n; k++) {
if (W[w + k] !== 0) { W[w + k] = 1 }
}
mark = 2
}
// at this point, W [0..n-1] < mark holds
return mark
}
function _diag (i, j) {
return i !== j
}
return csAmd
}
exports.name = 'csAmd'
exports.path = 'algebra.sparse'
exports.factory = factory