/// <summary>
/// Check residuals of generalized eigenvalue problem.
/// </summary>
/// <param name="A">Symmetric matrix.</param>
/// <param name="B">Symmetric matrix.</param>
/// <param name="k">The number of converged eigenvalues.</param>
/// <param name="v">The eigenvalues array.</param>
/// <param name="X">The eigenvectors matrix.</param>
/// <param name="print">If true, print residuals.</param>
/// <returns>True, if all residuals are below threshold.</returns>
public static bool CheckResiduals(SparseMatrix A, SparseMatrix B, int k, Complex[] v, Matrix <Complex> X, bool print)
{
int N = A.RowCount;
// If more eigenvalues converged than were requested (real, non-symmetric case only).
k = Math.Min(k, X.ColumnCount);
if (print)
{
Console.WriteLine();
Console.WriteLine(" Lambda Residual");
}
var x = new Complex[N];
var y = new Complex[N];
bool ok = true;
for (int i = 0; i < k; i++)
{
var lambda = v[i];
X.Column(i, x);
CVector.Copy(x, y);
if (B != null)
{
// y = B*x
B.Multiply(x, y);
}
// y = A*x - lambda*B*x
A.Multiply(1.0, x, -lambda, y);
double r = CVector.Norm(y) / Complex.Abs(lambda);
if (r > ERROR_THRESHOLD)
{
ok = false;
}
if (print)
{
Console.WriteLine("{0,3}: {1,10:0.00000000} {2,10:0.00e+00}", i, lambda, r);
}
}
return(ok);
}
/// <summary>
/// Minimum degree ordering of A+A' (if A is symmetric) or A'A.
/// </summary>
/// <param name="order">0:natural, 1:Chol, 2:LU, 3:QR</param>
/// <param name="A">column-compressed matrix</param>
/// <returns>amd(A+A') if A is symmetric, or amd(A'A) otherwise, null on
/// error or for natural ordering</returns>
public static int[] AMD(int order, SparseMatrix A)
{
SparseMatrix C, A2, AT;
int[] Cp, Ci, P, W, nv, next, head, elen, degree, w,
hhead, ATp, ATi;
int d, dk, dext, lemax = 0, e, elenk, eln, i, j, k, k1,
k2, k3, jlast, ln, dense, nzmax, mindeg = 0, nvi, nvj, nvk, mark, wnvi,
cnz, nel = 0, p, p1, p2, p3, p4, pj, pk, pk1, pk2, pn, q, n, m;
bool ok;
int h;
// Construct matrix C
if (order <= 0 || order > 3)
{
return(null); // check
}
AT = (SparseMatrix)A.Transpose(false); // compute A'
if (AT == null)
{
return(null);
}
m = A.m; n = A.n;
dense = Math.Max(16, 10 * (int)Math.Sqrt(n)); // find dense threshold
dense = Math.Min(n - 2, dense);
if (order == 1 && n == m)
{
C = SparseMatrix.Add(A, AT, 0, 0); // C = A+A'
}
else if (order == 2)
{
ATp = AT.p; // drop dense columns from AT
ATi = AT.i;
for (p2 = 0, j = 0; j < m; j++)
{
p = ATp[j]; // column j of AT starts here
ATp[j] = p2; // new column j starts here
if (ATp[j + 1] - p > dense)
{
continue; // skip dense col j
}
for (; p < ATp[j + 1]; p++)
{
ATi[p2++] = ATi[p];
}
}
ATp[m] = p2; // finalize AT
A2 = (SparseMatrix)AT.Transpose(false); // A2 = AT'
C = A2 != null?SparseMatrix.Multiply(AT, A2) : null; // C=A'*A with no dense rows
}
else
{
C = SparseMatrix.Multiply(AT, A); // C=A'*A
}
if (C == null)
{
return(null);
}
C.Keep(KeepOffDiag, null); // drop diagonal entries
Cp = C.p;
cnz = Cp[n];
// add elbow room to C
if (!C.Resize(cnz + cnz / 5 + 2 * n))
{
return(null);
}
P = new int[n + 1]; // allocate result
W = new int[n + 1]; // get workspace
w = new int[n + 1];
degree = new int[n + 1];
elen = new int[n + 1]; // Initialized to 0's
// Initialize quotient graph
for (k = 0; k < n; k++)
{
W[k] = Cp[k + 1] - Cp[k];
}
W[n] = 0;
nzmax = C.nzmax;
Ci = C.i;
for (i = 0; i <= n; i++)
{
P[i] = -1;
w[i] = 1; // node i is alive
degree[i] = W[i]; // degree of node i
}
next = new int[n + 1];
hhead = new int[n + 1];
head = new int[n + 1];
nv = new int[n + 1];
Array.Copy(P, next, n + 1);
Array.Copy(P, head, n + 1); // degree list i is empty
Array.Copy(P, hhead, n + 1); // hash list i is empty
Array.Copy(w, nv, n + 1); // node i is just one node
mark = Clear(0, 0, w, n); // clear w
elen[n] = -2; // n is a dead element
Cp[n] = -1; // n is a root of assembly tree
w[n] = 0; // n is a dead element
// Initialize degree lists
for (i = 0; i < n; i++)
{
d = degree[i];
if (d == 0) // node i is empty
{
elen[i] = -2; // element i is dead
nel++;
Cp[i] = -1; // i is a root of assembly tree
w[i] = 0;
}
else if (d > dense) // node i is dense
{
nv[i] = 0; // absorb i into element n
elen[i] = -1; // node i is dead
nel++;
Cp[i] = FLIP(n);
nv[n]++;
}
else
{
if (head[d] != -1)
{
P[head[d]] = i;
}
next[i] = head[d]; // put node i in degree list d
head[d] = i;
}
}
while (nel < n) // while (selecting pivots) do
{
// Select node of minimum approximate degree
for (k = -1; mindeg < n && (k = head[mindeg]) == -1; mindeg++)
{
;
}
if (next[k] != -1)
{
P[next[k]] = -1;
}
head[mindeg] = next[k]; // remove k from degree list
elenk = elen[k]; // elenk = |Ek|
nvk = nv[k]; // # of nodes k represents
nel += nvk; // nv[k] nodes of A eliminated
// Garbage collection
if (elenk > 0 && cnz + mindeg >= nzmax)
{
for (j = 0; j < n; j++)
{
if ((p = Cp[j]) >= 0) // j is a live node or element
{
Cp[j] = Ci[p]; // save first entry of object
Ci[p] = FLIP(j); // first entry is now CS_FLIP(j)
}
}
for (q = 0, p = 0; p < cnz;) // scan all of memory
{
if ((j = FLIP(Ci[p++])) >= 0) // found object j
{
Ci[q] = Cp[j]; // restore first entry of object
Cp[j] = q++; // new pointer to object j
for (k3 = 0; k3 < W[j] - 1; k3++)
{
Ci[q++] = Ci[p++];
}
}
}
cnz = q; // Ci [cnz...nzmax-1] now free
}
// Construct new element
dk = 0;
nv[k] = -nvk; // flag k as in Lk
p = Cp[k];
pk1 = (elenk == 0) ? p : cnz; // do in place if elen[k] == 0
pk2 = pk1;
for (k1 = 1; k1 <= elenk + 1; k1++)
{
if (k1 > elenk)
{
e = k; // search the nodes in k
pj = p; // list of nodes starts at Ci[pj]*/
ln = W[k] - elenk; // length of list of nodes in k
}
else
{
e = Ci[p++]; // search the nodes in e
pj = Cp[e];
ln = W[e]; // length of list of nodes in e
}
for (k2 = 1; k2 <= ln; k2++)
{
i = Ci[pj++];
if ((nvi = nv[i]) <= 0)
{
continue; // node i dead, or seen
}
dk += nvi; // degree[Lk] += size of node i
nv[i] = -nvi; // negate nv[i] to denote i in Lk
Ci[pk2++] = i; // place i in Lk
if (next[i] != -1)
{
P[next[i]] = P[i];
}
if (P[i] != -1) // remove i from degree list
{
next[P[i]] = next[i];
}
else
{
head[degree[i]] = next[i];
}
}
if (e != k)
{
Cp[e] = FLIP(k); // absorb e into k
w[e] = 0; // e is now a dead element
}
}
if (elenk != 0)
{
cnz = pk2; // Ci [cnz...nzmax] is free
}
degree[k] = dk; // external degree of k - |Lk\i|
Cp[k] = pk1; // element k is in Ci[pk1..pk2-1]
W[k] = pk2 - pk1;
elen[k] = -2; // k is now an element
// Find set differences
mark = Clear(mark, lemax, w, n); // clear w if necessary
for (pk = pk1; pk < pk2; pk++) // scan 1: find |Le\Lk|
{
i = Ci[pk];
if ((eln = elen[i]) <= 0)
{
continue; // skip if elen[i] empty
}
nvi = -nv[i]; // nv [i] was negated
wnvi = mark - nvi;
for (p = Cp[i]; p <= Cp[i] + eln - 1; p++) // scan Ei
{
e = Ci[p];
if (w[e] >= mark)
{
w[e] -= nvi; // decrement |Le\Lk|
}
else if (w[e] != 0) // ensure e is a live element
{
w[e] = degree[e] + wnvi; // 1st time e seen in scan 1
}
}
}
// Degree update
for (pk = pk1; pk < pk2; pk++) // scan2: degree update
{
i = Ci[pk]; // consider node i in Lk
p1 = Cp[i];
p2 = p1 + elen[i] - 1;
pn = p1;
for (h = 0, d = 0, p = p1; p <= p2; p++) // scan Ei
{
e = Ci[p];
if (w[e] != 0) // e is an unabsorbed element
{
dext = w[e] - mark; // dext = |Le\Lk|
if (dext > 0)
{
d += dext; // sum up the set differences
Ci[pn++] = e; // keep e in Ei
h += e; // compute the hash of node i
}
else
{
Cp[e] = FLIP(k); // aggressive absorb. e.k
w[e] = 0; // e is a dead element
}
}
}
elen[i] = pn - p1 + 1; // elen[i] = |Ei|
p3 = pn;
p4 = p1 + W[i];
for (p = p2 + 1; p < p4; p++) // prune edges in Ai
{
j = Ci[p];
if ((nvj = nv[j]) <= 0)
{
continue; // node j dead or in Lk
}
d += nvj; // degree(i) += |j|
Ci[pn++] = j; // place j in node list of i
h += j; // compute hash for node i
}
if (d == 0) // check for mass elimination
{
Cp[i] = FLIP(k); // absorb i into k
nvi = -nv[i];
dk -= nvi; // |Lk| -= |i|
nvk += nvi; // |k| += nv[i]
nel += nvi;
nv[i] = 0;
elen[i] = -1; // node i is dead
}
else
{
degree[i] = Math.Min(degree[i], d); // update degree(i)
Ci[pn] = Ci[p3]; // move first node to end
Ci[p3] = Ci[p1]; // move 1st el. to end of Ei
Ci[p1] = k; // add k as 1st element in of Ei
W[i] = pn - p1 + 1; // new len of adj. list of node i
h = ((h < 0) ? (-h) : h) % n; // finalize hash of i
next[i] = hhead[h]; // place i in hash bucket
hhead[h] = i;
P[i] = h; // save hash of i in last[i]
}
} // scan2 is done
degree[k] = dk; // finalize |Lk|
lemax = Math.Max(lemax, dk);
mark = Clear(mark + lemax, lemax, w, n); // clear w
// Supernode detection
for (pk = pk1; pk < pk2; pk++)
{
i = Ci[pk];
if (nv[i] >= 0)
{
continue; // skip if i is dead
}
h = P[i]; // scan hash bucket of node i
i = hhead[h];
hhead[h] = -1; // hash bucket will be empty
for (; i != -1 && next[i] != -1; i = next[i], mark++)
{
ln = W[i];
eln = elen[i];
for (p = Cp[i] + 1; p <= Cp[i] + ln - 1; p++)
{
w[Ci[p]] = mark;
}
jlast = i;
for (j = next[i]; j != -1;) // compare i with all j
{
ok = (W[j] == ln) && (elen[j] == eln);
for (p = Cp[j] + 1; ok && p <= Cp[j] + ln - 1; p++)
{
if (w[Ci[p]] != mark)
{
ok = false; // compare i and j
}
}
if (ok) // i and j are identical
{
Cp[j] = FLIP(i); // absorb j into i
nv[i] += nv[j];
nv[j] = 0;
elen[j] = -1; // node j is dead
j = next[j]; // delete j from hash bucket
next[jlast] = j;
}
else
{
jlast = j; // j and i are different
j = next[j];
}
}
}
}
// Finalize new element
for (p = pk1, pk = pk1; pk < pk2; pk++) // finalize Lk
{
i = Ci[pk];
if ((nvi = -nv[i]) <= 0)
{
continue; // skip if i is dead
}
nv[i] = nvi; // restore nv[i]
d = degree[i] + dk - nvi; // compute external degree(i)
d = Math.Min(d, n - nel - nvi);
if (head[d] != -1)
{
P[head[d]] = i;
}
next[i] = head[d]; // put i back in degree list
P[i] = -1;
head[d] = i;
mindeg = Math.Min(mindeg, d); // find new minimum degree
degree[i] = d;
Ci[p++] = i; // place i in Lk
}
nv[k] = nvk; // # nodes absorbed into k
if ((W[k] = p - pk1) == 0) // length of adj list of element k
{
Cp[k] = -1; // k is a root of the tree
w[k] = 0; // k is now a dead element
}
if (elenk != 0)
{
cnz = p; // free unused space in Lk
}
}
// Postordering
for (i = 0; i < n; i++)
{
Cp[i] = FLIP(Cp[i]); // fix assembly tree
}
for (j = 0; j <= n; j++)
{
head[j] = -1;
}
for (j = n; j >= 0; j--) // place unordered nodes in lists
{
if (nv[j] > 0)
{
continue; // skip if j is an element
}
next[j] = head[Cp[j]]; // place j in list of its parent
head[Cp[j]] = j;
}
for (e = n; e >= 0; e--) // place elements in lists
{
if (nv[e] <= 0)
{
continue; // skip unless e is an element
}
if (Cp[e] != -1)
{
next[e] = head[Cp[e]]; // place e in list of its parent
head[Cp[e]] = e;
}
}
for (k = 0, i = 0; i <= n; i++) // postorder the assembly tree
{
if (Cp[i] == -1)
{
k = Common.TreeDepthFirstSearch(i, k, head, next, P, w);
}
}
return(P);
}