// Solve secular function arising in symmetric eigenproblems. On exit 'Y' contains new
// eigenvalues and 'V' the rank-one update vector corresponding new eigenvalues.
// The matrix Qd holds for each eigenvalue then computed deltas as row vectors.
// On entry 'D' holds original eigenvalues and 'Z' is the rank-one update vector.
func TRDSecularSolveAll(y, v, Qd, d, z *cmat.FloatMatrix, rho float64, confs ...*gomas.Config) (err *gomas.Error) {
var delta cmat.FloatMatrix
var lmbda float64
var e, ei int
ei = 0
err = nil
if y.Len() != d.Len() || z.Len() != d.Len() || m(Qd) != n(Qd) || m(Qd) != d.Len() {
err = gomas.NewError(gomas.ESIZE, "TRDSecularSolveAll")
return
}
for i := 0; i < d.Len(); i++ {
delta.Row(Qd, i)
lmbda, e = trdsecRoot(d, z, &delta, i, rho)
if e < 0 && ei == 0 {
ei = -(i + 1)
}
y.SetAt(i, lmbda)
}
if ei == 0 {
trdsecUpdateVecDelta(v, Qd, d, rho)
} else {
err = gomas.NewError(gomas.ECONVERGE, "TRDSecularSolveAll", ei)
}
return
}
func sortEigenVec(D, U, V, C *cmat.FloatMatrix, updown int) {
var sD, m0, m1 cmat.FloatMatrix
N := D.Len()
for k := 0; k < N-1; k++ {
sD.SubVector(D, k, N-k)
pk := vecMinMax(&sD, -updown)
if pk != 0 {
t0 := D.GetAt(k)
D.SetAt(k, D.GetAt(pk+k))
D.SetAt(k+pk, t0)
if U != nil {
m0.Column(U, k)
m1.Column(U, k+pk)
blasd.Swap(&m1, &m0)
}
if V != nil {
m0.Row(V, k)
m1.Row(V, k+pk)
blasd.Swap(&m1, &m0)
}
if C != nil {
m0.Column(C, k)
m1.Column(C, k+pk)
blasd.Swap(&m1, &m0)
}
}
}
}
/*
* Compute
* B = B*diag(D).-1 flags & RIGHT == true
* B = diag(D).-1*B flags & LEFT == true
*
* If flags is LEFT (RIGHT) then element-wise divides columns (rows) of B with vector D.
*
* Arguments:
* B M-by-N matrix if flags&RIGHT == true or N-by-M matrix if flags&LEFT == true
*
* D N element column or row vector or N-by-N matrix
*
* flags Indicator bits, LEFT or RIGHT
*/
func SolveDiag(B, D *cmat.FloatMatrix, flags int, confs ...*gomas.Config) *gomas.Error {
var c, d0 cmat.FloatMatrix
var d *cmat.FloatMatrix
conf := gomas.CurrentConf(confs...)
d = D
if !D.IsVector() {
d0.Diag(D)
d = &d0
}
dn := d0.Len()
br, bc := B.Size()
switch flags & (gomas.LEFT | gomas.RIGHT) {
case gomas.LEFT:
if br != dn {
return gomas.NewError(gomas.ESIZE, "SolveDiag")
}
// scale rows;
for k := 0; k < dn; k++ {
c.Row(B, k)
blasd.InvScale(&c, d.GetAt(k), conf)
}
case gomas.RIGHT:
if bc != dn {
return gomas.NewError(gomas.ESIZE, "SolveDiag")
}
// scale columns
for k := 0; k < dn; k++ {
c.Column(B, k)
blasd.InvScale(&c, d.GetAt(k), conf)
}
}
return nil
}
// Compute eigenmatrix Q for updated eigenvalues in 'dl'.
func trdsecEigenBuild(Q, z, Q2 *cmat.FloatMatrix) {
var qi, delta cmat.FloatMatrix
for k := 0; k < z.Len(); k++ {
qi.Column(Q, k)
delta.Row(Q2, k)
trdsecEigenVecDelta(&qi, &delta, z)
}
}
func mNormInf(A *cmat.FloatMatrix) float64 {
var amax float64 = 0.0
var row cmat.FloatMatrix
arows, _ := A.Size()
for k := 0; k < arows; k++ {
row.Row(A, k)
rmax := blasd.ASum(&row)
if rmax > amax {
amax = rmax
}
}
return amax
}
/*
* Generate one of the orthogonal matrices Q or P.T determined by BDReduce() when
* reducing a real matrix A to bidiagonal form. Q and P.T are defined as products
* elementary reflectors H(i) or G(i) respectively.
*
* Orthogonal matrix Q is generated if flag WANTQ is set. And matrix P respectively
* if flag WANTP is set.
*/
func BDBuild(A, tau, W *cmat.FloatMatrix, K, flags int, confs ...*gomas.Config) *gomas.Error {
var Qh, Ph, tauh, d, s cmat.FloatMatrix
var err *gomas.Error = nil
if m(A) == 0 || n(A) == 0 {
return nil
}
if m(A) > n(A) || (m(A) == n(A) && flags&gomas.LOWER == 0) {
switch flags & (gomas.WANTQ | gomas.WANTP) {
case gomas.WANTQ:
tauh.SubMatrix(tau, 0, 0, n(A), 1)
err = QRBuild(A, &tauh, W, K, confs...)
case gomas.WANTP:
// Shift P matrix embedded in A down and fill first column and row
// to unit vector
for j := n(A) - 1; j > 0; j-- {
s.SubMatrix(A, j-1, j, 1, n(A)-j)
d.SubMatrix(A, j, j, 1, n(A)-j)
blasd.Copy(&d, &s)
A.Set(j, 0, 0.0)
}
// zero first row and set first entry to one
d.Row(A, 0)
blasd.Scale(&d, 0.0)
d.Set(0, 0, 1.0)
Ph.SubMatrix(A, 1, 1, n(A)-1, n(A)-1)
tauh.SubMatrix(tau, 0, 0, n(A)-1, 1)
if K > n(A)-1 {
K = n(A) - 1
}
err = LQBuild(&Ph, &tauh, W, K, confs...)
}
} else {
switch flags & (gomas.WANTQ | gomas.WANTP) {
case gomas.WANTQ:
// Shift Q matrix embedded in A right and fill first column and row
// to unit vector
for j := m(A) - 1; j > 0; j-- {
s.SubMatrix(A, j, j-1, m(A)-j, 1)
d.SubMatrix(A, j, j, m(A)-j, 1)
blasd.Copy(&d, &s)
A.Set(0, j, 0.0)
}
// zero first column and set first entry to one
d.Column(A, 0)
blasd.Scale(&d, 0.0)
d.Set(0, 0, 1.0)
Qh.SubMatrix(A, 1, 1, m(A)-1, m(A)-1)
tauh.SubMatrix(tau, 0, 0, m(A)-1, 1)
if K > m(A)-1 {
K = m(A) - 1
}
err = QRBuild(&Qh, &tauh, W, K, confs...)
case gomas.WANTP:
tauh.SubMatrix(tau, 0, 0, m(A), 1)
err = LQBuild(A, &tauh, W, K, confs...)
}
}
if err != nil {
err.Update("BDBuild")
}
return err
}
func svdSmall(S, U, V, A, W *cmat.FloatMatrix, bits int, conf *gomas.Config) (err *gomas.Error) {
var r cmat.FloatMatrix
var d0, d1, e0 float64
err = nil
K := m(A)
if n(A) < K {
K = n(A)
}
tau := cmat.NewMatrix(K, 1)
if m(A) >= n(A) {
if err = QRFactor(A, tau, W, conf); err != nil {
return
}
} else {
if err = LQFactor(A, tau, W, conf); err != nil {
return
}
}
if m(A) == 1 || n(A) == 1 {
// either tall M-by-1 or wide 1-by-N
S.SetAt(0, math.Abs(A.Get(0, 0)))
if bits&gomas.WANTU != 0 {
if n(A) == 1 {
if n(U) == n(A) {
// U is M-by-1
U.Copy(A)
QRBuild(U, tau, W, n(A), conf)
} else {
// U is M-by-M
eye := cmat.FloatDiagonalSource{1.0}
U.SetFrom(&eye, cmat.SYMM)
if err = QRMult(U, A, tau, W, gomas.RIGHT, conf); err != nil {
return
}
}
} else {
U.Set(0, 0, -1.0)
}
}
if bits&gomas.WANTV != 0 {
if m(A) == 1 {
if m(V) == m(A) {
// V is 1-by-N
V.Copy(A)
LQBuild(V, tau, W, m(A), conf)
} else {
// V is N-by-N
eye := cmat.FloatDiagonalSource{1.0}
V.SetFrom(&eye, cmat.SYMM)
if err = LQMult(V, A, tau, W, gomas.RIGHT, conf); err != nil {
return
}
}
} else {
// V is 1-by-1
V.Set(0, 0, -1.0)
}
}
return
}
// Use bdSvd2x2 functions
d0 = A.Get(0, 0)
d1 = A.Get(1, 1)
if m(A) >= n(A) {
e0 = A.Get(0, 1)
} else {
e0 = A.Get(1, 0)
}
if bits&(gomas.WANTU|gomas.WANTV) == 0 {
// no vectors
smin, smax := bdSvd2x2(d0, e0, d1)
S.SetAt(0, math.Abs(smax))
S.SetAt(1, math.Abs(smin))
return
}
// at least left or right eigenvector wanted
smin, smax, cosl, sinl, cosr, sinr := bdSvd2x2Vec(d0, e0, d1)
if bits&gomas.WANTU != 0 {
// left eigenvectors
if m(A) >= n(A) {
if n(U) == n(A) {
U.Copy(A)
if err = QRBuild(U, tau, W, n(A), conf); err != nil {
return
}
} else {
// U is M-by-M
eye := cmat.FloatDiagonalSource{1.0}
U.SetFrom(&eye, cmat.SYMM)
if err = QRMult(U, A, tau, W, gomas.RIGHT, conf); err != nil {
return
}
}
ApplyGivensRight(U, 0, 1, 0, m(A), cosl, sinl)
} else {
// U is 2-by-2
eye := cmat.FloatDiagonalSource{1.0}
U.SetFrom(&eye, cmat.SYMM)
ApplyGivensRight(U, 0, 1, 0, m(A), cosr, sinr)
}
}
if bits&gomas.WANTV != 0 {
if n(A) > m(A) {
if m(V) == m(A) {
V.Copy(A)
if err = LQBuild(V, tau, W, m(A), conf); err != nil {
return
}
} else {
eye := cmat.FloatDiagonalSource{1.0}
V.SetFrom(&eye, cmat.SYMM)
if err = LQMult(V, A, tau, W, gomas.RIGHT, conf); err != nil {
return
}
}
ApplyGivensLeft(V, 0, 1, 0, n(A), cosl, sinl)
} else {
// V is 2-by-2
eye := cmat.FloatDiagonalSource{1.0}
V.SetFrom(&eye, cmat.SYMM)
ApplyGivensLeft(V, 0, 1, 0, n(A), cosr, sinr)
}
if smax < 0.0 {
r.Row(V, 0)
blasd.Scale(&r, -1.0)
}
if smin < 0.0 {
r.Row(V, 1)
blasd.Scale(&r, -1.0)
}
}
S.SetAt(0, math.Abs(smax))
S.SetAt(1, math.Abs(smin))
return
}