func EigenSym(D, A, W *cmat.FloatMatrix, bits int, confs ...*gomas.Config) (err *gomas.Error) {
var sD, sE, E, tau, Wred cmat.FloatMatrix
var vv *cmat.FloatMatrix
err = nil
vv = nil
conf := gomas.CurrentConf(confs...)
if m(A) != n(A) || D.Len() != m(A) {
err = gomas.NewError(gomas.ESIZE, "EigenSym")
return
}
if bits&gomas.WANTV != 0 && W.Len() < 3*n(A) {
err = gomas.NewError(gomas.EWORK, "EigenSym")
return
}
if bits&(gomas.LOWER|gomas.UPPER) == 0 {
bits = bits | gomas.LOWER
}
ioff := 1
if bits&gomas.LOWER != 0 {
ioff = -1
}
E.SetBuf(n(A)-1, 1, n(A)-1, W.Data())
tau.SetBuf(n(A), 1, n(A), W.Data()[n(A)-1:])
wrl := W.Len() - 2*n(A) - 1
Wred.SetBuf(wrl, 1, wrl, W.Data()[2*n(A)-1:])
// reduce to tridiagonal
if err = TRDReduce(A, &tau, &Wred, bits, conf); err != nil {
err.Update("EigenSym")
return
}
sD.Diag(A)
sE.Diag(A, ioff)
blasd.Copy(D, &sD)
blasd.Copy(&E, &sE)
if bits&gomas.WANTV != 0 {
if err = TRDBuild(A, &tau, &Wred, n(A), bits, conf); err != nil {
err.Update("EigenSym")
return
}
vv = A
}
// resize workspace
wrl = W.Len() - n(A) - 1
Wred.SetBuf(wrl, 1, wrl, W.Data()[n(A)-1:])
if err = TRDEigen(D, &E, vv, &Wred, bits, conf); err != nil {
err.Update("EigenSym")
return
}
return
}
/*
* Generates the real orthogonal matrix Q which is defined as the product of K elementary
* reflectors of order N embedded in matrix A as returned by TRDReduce().
*
* A On entry tridiagonal reduction as returned by TRDReduce().
* On exit the orthogonal matrix Q.
*
* tau Scalar coefficients of elementary reflectors.
*
* W Workspace
*
* K Number of reflectors , 0 < K < N
*
* flags LOWER or UPPER
*
* confs Optional blocking configuration
*
* If flags has UPPER set then
* Q = H(K)...H(1)H(0) where 0 < K < N-1
*
* If flags has LOWR set then
* Q = H(0)H(1)...H(K) where 0 < K < N-1
*/
func TRDBuild(A, tau, W *cmat.FloatMatrix, K, flags int, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
var Qh, tauh cmat.FloatMatrix
var s, d cmat.FloatMatrix
if K > m(A)-1 {
K = m(A) - 1
}
switch flags & (gomas.LOWER | gomas.UPPER) {
case gomas.LOWER:
// Shift Q matrix embedded in A right and fill first column
// unit column 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)
err = QRBuild(&Qh, &tauh, W, K, confs...)
case gomas.UPPER:
// Shift Q matrix embedded in A left and fill last column
// unit column vector
for j := 1; j < m(A); j++ {
s.SubMatrix(A, 0, j, j, 1)
d.SubMatrix(A, 0, j-1, j, 1)
blasd.Copy(&d, &s)
A.Set(-1, j-1, 0.0)
}
// zero last column and set last entry to one
d.Column(A, m(A)-1)
blasd.Scale(&d, 0.0)
d.Set(-1, 0, 1.0)
Qh.SubMatrix(A, 0, 0, m(A)-1, m(A)-1)
tauh.SubMatrix(tau, 0, 0, m(A)-1, 1)
err = QLBuild(&Qh, &tauh, W, K, confs...)
}
if err != nil {
err.Update("TRDBuild")
}
return err
}
func TestTrdMultUpper(t *testing.T) {
var dt, et, da, ea cmat.FloatMatrix
N := 843
nb := 48
conf := gomas.NewConf()
conf.LB = nb
A := cmat.NewMatrix(N, N)
tau := cmat.NewMatrix(N, 1)
src := cmat.NewFloatNormSource()
// create symmetric matrix
A.SetFrom(src, cmat.SYMM)
A0 := cmat.NewCopy(A)
W := lapackd.Workspace(lapackd.TRDReduceWork(A, conf))
lapackd.TRDReduce(A, tau, W, gomas.UPPER, conf)
// make tridiagonal matrix T
T0 := cmat.NewMatrix(N, N)
dt.Diag(T0)
da.Diag(A)
blasd.Copy(&dt, &da)
ea.Diag(A, 1)
et.Diag(T0, 1)
blasd.Copy(&et, &ea)
et.Diag(T0, -1)
blasd.Copy(&et, &ea)
T1 := cmat.NewCopy(T0)
// compute Q*T*Q.T (unblocked)
conf.LB = 0
lapackd.TRDMult(T0, A, tau, W, gomas.LEFT|gomas.UPPER, conf)
lapackd.TRDMult(T0, A, tau, W, gomas.RIGHT|gomas.TRANS|gomas.UPPER, conf)
blasd.Plus(T0, A0, 1.0, -1.0, gomas.NONE)
nrm := lapackd.NormP(T0, lapackd.NORM_ONE)
t.Logf("N=%d, unblk.||A - Q*T*Q.T||_1: %e\n", N, nrm)
// compute Q*T*Q.T (blocked)
conf.LB = nb
W = lapackd.Workspace(lapackd.TRDMultWork(A, gomas.LEFT|gomas.UPPER, conf))
lapackd.TRDMult(T1, A, tau, W, gomas.LEFT|gomas.UPPER, conf)
lapackd.TRDMult(T1, A, tau, W, gomas.RIGHT|gomas.TRANS|gomas.UPPER, conf)
blasd.Plus(T1, A0, 1.0, -1.0, gomas.NONE)
nrm = lapackd.NormP(T1, lapackd.NORM_ONE)
t.Logf("N=%d, blk.||A - Q*T*Q.T||_1: %e\n", N, nrm)
}
func TestBiredWide(t *testing.T) {
N := 811
M := 693
nb := 32
conf := gomas.NewConf()
conf.LB = 0
ediag := -1
zeromean := cmat.NewFloatNormSource()
A := cmat.NewMatrix(M, N)
A.SetFrom(zeromean)
A0 := cmat.NewCopy(A)
tauq := cmat.NewMatrix(N, 1)
taup := cmat.NewMatrix(N, 1)
W := lapackd.Workspace(M + N)
lapackd.BDReduce(A, tauq, taup, W, conf)
var D, E, Bd, Be cmat.FloatMatrix
D.Diag(A)
E.Diag(A, ediag)
B := cmat.NewMatrix(M, N)
Bd.Diag(B)
Be.Diag(B, ediag)
blasd.Copy(&Bd, &D)
blasd.Copy(&Be, &E)
Bt := cmat.NewMatrix(N, M)
blasd.Transpose(Bt, B)
conf.LB = nb
W0 := lapackd.Workspace(lapackd.BDMultWork(B, conf))
lapackd.BDMult(B, A, tauq, W0, gomas.MULTQ|gomas.LEFT, conf)
lapackd.BDMult(Bt, A, tauq, W0, gomas.MULTQ|gomas.RIGHT|gomas.TRANS, conf)
lapackd.BDMult(B, A, taup, W0, gomas.MULTP|gomas.RIGHT|gomas.TRANS, conf)
lapackd.BDMult(Bt, A, taup, W0, gomas.MULTP|gomas.LEFT, conf)
blasd.Plus(B, A0, 1.0, -1.0, gomas.NONE)
nrm := lapackd.NormP(B, lapackd.NORM_ONE)
t.Logf("M=%d, N=%d ||A - Q*B*P.T||_1 : %e\n", M, N, nrm)
blasd.Plus(Bt, A0, 1.0, -1.0, gomas.TRANSB)
nrm = lapackd.NormP(Bt, lapackd.NORM_ONE)
t.Logf("M=%d, N=%d ||A.T - P*B.T*Q.T||_1 : %e\n", M, N, nrm)
}
func main() {
flag.Parse()
M := N + N/10
conf := gomas.CurrentConf()
A := cmat.NewMatrix(M, N)
A0 := cmat.NewCopy(A)
tau := cmat.NewMatrix(N, 1)
W := lapackd.Workspace(lapackd.QRFactorWork(A, conf))
zeromean := cmat.NewFloatNormSource()
A.SetFrom(zeromean)
cumtime := 0.0
mintime := 0.0
maxtime := 0.0
for i := 0; i < count; i++ {
flushCache()
t1 := time.Now()
// ----------------------------------------------
lapackd.QRFactor(A, tau, W, conf)
// ----------------------------------------------
t2 := time.Now()
tm := t2.Sub(t1)
if mintime == 0.0 || tm.Seconds() < mintime {
mintime = tm.Seconds()
}
if maxtime == 0.0 || tm.Seconds() > maxtime {
maxtime = tm.Seconds()
}
cumtime += tm.Seconds()
if verbose {
fmt.Printf("%3d %12.4f msec, %9.4f gflops\n",
i, 1e+3*tm.Seconds(), gflops(M, N, tm.Seconds()))
}
blasd.Copy(A, A0)
}
cumtime /= float64(count)
minflops := gflops(M, N, maxtime)
avgflops := gflops(M, N, cumtime)
maxflops := gflops(M, N, mintime)
fmt.Printf("%5d %5d %3d %9.4f %9.4f %9.4f Gflops\n", M, N, conf.LB, minflops, avgflops, maxflops)
}
/*
* Update vector with compact WY Householder block
* (I - Y*T*Y.T)*v = v - Y*T*Y.T*v
*
* LEFT:
* 1 | 0 * v0 = v0 = v0
* 0 | Q v1 Q*v1 = v1 - Y*T*Y.T*v1
*
* 1 | 0 * v0 = v0 = v0
* 0 | Q.T v1 Q.T*v1 = v1 - Y*T.T*Y.T*v1
*
* RIGHT:
* v0 | v1 * 1 | 0 = v0 | v1*Q = v0 | v1 - v1*Y*T*Y.T
* 0 | Q
*
* v0 | v1 * 1 | 0 = v0 | v1*Q.T = v0 | v1 - v1*Y*T.T*Y.T
* 0 | Q.T
*/
func updateVecLeftWY2(v, Y1, Y2, T, w *cmat.FloatMatrix, bits int) {
var v1, v2 cmat.FloatMatrix
var w0 cmat.FloatMatrix
v1.SubMatrix(v, 1, 0, n(Y1), 1)
v2.SubMatrix(v, n(Y1)+1, 0, m(Y2), 1)
w0.SubMatrix(w, 0, 0, m(Y1), 1)
// w0 := Y1.T*v1 + Y2.T*v2
blasd.Copy(&w0, &v1)
blasd.MVMultTrm(&w0, Y1, 1.0, gomas.LOWER|gomas.UNIT|gomas.TRANS)
blasd.MVMult(&w0, Y2, &v2, 1.0, 1.0, gomas.TRANS)
// w0 := op(T)*w0
blasd.MVMultTrm(&w0, T, 1.0, bits|gomas.UPPER)
// v2 := v2 - Y2*w0
blasd.MVMult(&v2, Y2, &w0, -1.0, 1.0, gomas.NONE)
// v1 := v1 - Y1*w0
blasd.MVMultTrm(&w0, Y1, 1.0, gomas.LOWER|gomas.UNIT)
blasd.Axpy(&v1, &w0, -1.0)
}
func unblkBoundedBKUpper(A, wrk *cmat.FloatMatrix, p *Pivots, ncol int, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12, A22, a11inv cmat.FloatMatrix
var w00, w01, w11 cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
if ncol > n(A) {
ncol = n(A)
}
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, m(wrk)-2, 0, 2, 2)
a11inv.Set(0, 1, -1.0)
a11inv.Set(1, 0, -1.0)
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
partitionPivot2x1(
&pT,
&pB, *p, 0, util.PBOTTOM)
for n(&ATL) > 0 && nc < ncol {
util.Partition2x2(
&w00, &w01,
nil, &w11, wrk, nc, nc, util.PBOTTOMRIGHT)
r, np := findAndBuildBKPivotUpper(&ATL, &ATR, &w00, &w01, nc)
if np > ncol-nc {
// next pivot does not fit into ncol columns,
// return with number of factorized columns
return err, nc
}
cwrk.SubMatrix(&w00, 0, n(&w00)-np, m(&ATL), np)
if r != -1 {
// pivoting needed; do swaping here
k := m(&ATL) - np
applyBKPivotSymUpper(&ATL, k, r)
// swap right hand rows to get correct updates
swapRows(&ATR, k, r)
swapRows(&w01, k, r)
if np == 2 && r != k {
/* for 2x2 blocks we need diagonal pivots.
* [r, r] | [ r,-1]
* a11 == ---------------- 2-by-2 pivot, swapping [1,0] and [r,0]
* [-1,r] | [-1,-1]
*/
t0 := w00.Get(k, -1)
tr := w00.Get(r, -1)
w00.Set(k, -1, tr)
w00.Set(r, -1, t0)
t0 = w00.Get(k, -2)
tr = w00.Get(r, -2)
w00.Set(k, -2, tr)
w00.Set(r, -2, t0)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22 /**/, A, np, util.PTOPLEFT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, *p, np, util.PTOP)
// ------------------------------------------------------------
wlc := n(&w00) - np
cwrk.SubMatrix(&w00, 0, wlc, m(&a01), n(&a01))
if np == 1 {
//
a11.Set(0, 0, w00.Get(m(&a01), wlc))
// a21 = a21/a11
blasd.Copy(&a01, &cwrk)
blasd.InvScale(&a01, a11.Get(0, 0))
// store pivot point relative to original matrix
if r == -1 {
p1[0] = m(&ATL)
} else {
p1[0] = r + 1
}
} else if np == 2 {
/* a | b d/b | -1
* w00 == ------ == a11 --> a11.-1 == -------- * scale
* . | d -1 | a/b
*/
a := w00.Get(m(&ATL)-2, -2)
b := w00.Get(m(&ATL)-2, -1)
d := w00.Get(m(&ATL)-1, -1)
a11inv.Set(0, 0, d/b)
a11inv.Set(1, 1, a/b)
// denominator: (a/b)*(d/b)-1.0 == (a*d - b^2)/b^2
scale := 1.0 / ((a/b)*(d/b) - 1.0)
scale /= b
// a01 = a01*a11.-1
blasd.Mult(&a01, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
a11.Set(0, 0, a)
a11.Set(0, 1, b)
a11.Set(1, 1, d)
// store pivot point relative to original matrix
p1[0] = -(r + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, *p, util.PTOP)
}
return err, nc
}
/*
* Find diagonal pivot and build incrementaly updated block.
*
* d x r2 x x x c1 | x x kp1 k | w w
* d r2 x x x c1 | x x kp1 k | w w
* r2 r2 r2 r2 c1 | x x kp1 k | w w
* d x x c1 | x x kp1 k | w w
* d x c1 | x x kp1 k | w w
* d c1 | x x kp1 k | w w
* c1 | x x kp1 k | w w
* -------------------------- -------------
* (AL) (AR) (WL) (WR)
*
* Matrix AL contains the unfactored part of the matrix and AR the already
* factored columns. Matrix WR is updated values of factored part ie.
* w(i) = l(i)d(i). Matrix WL will have updated values for next column.
* Column WL(k) contains updated AL(c1) and WL(kp1) possible pivot row AL(r2).
*
* On exit, for 1x1 diagonal the rightmost column of WL (k) holds the updated
* value of AL(c1). If pivoting this required the WL(k) holds the actual pivoted
* column/row.
*
* For 2x2 diagonal blocks WL(k) holds the updated AL(c1) and WL(kp1) holds
* actual values of pivot column/row AL(r2), without the diagonal pivots.
*/
func findAndBuildBKPivotUpper(AL, AR, WL, WR *cmat.FloatMatrix, k int) (int, int) {
var r, q int
var rcol, qrow, src, wk, wkp1, wrow cmat.FloatMatrix
lc := n(AL) - 1
wc := n(WL) - 1
lr := m(AL) - 1
// Copy AL[:,lc] to WL[:,wc] and update with WR[0:]
src.SubMatrix(AL, 0, lc, m(AL), 1)
wk.SubMatrix(WL, 0, wc, m(AL), 1)
blasd.Copy(&wk, &src)
if k > 0 {
wrow.SubMatrix(WR, lr, 0, 1, n(WR))
blasd.MVMult(&wk, AR, &wrow, -1.0, 1.0, gomas.NONE)
}
if m(AL) == 1 {
return -1, 1
}
// amax is on-diagonal element of current column
amax := math.Abs(WL.Get(lr, wc))
// find max off-diagonal on first column.
rcol.SubMatrix(WL, 0, wc, lr, 1)
// r is row index and rmax is its absolute value
r = blasd.IAmax(&rcol)
rmax := math.Abs(rcol.Get(r, 0))
if amax >= bkALPHA*rmax {
// no pivoting, 1x1 diagonal
return -1, 1
}
// Now we need to copy row r to WL[:,wc-1] and update it
wkp1.SubMatrix(WL, 0, wc-1, m(AL), 1)
if r > 0 {
// above the diagonal part of AL
qrow.SubMatrix(AL, 0, r, r, 1)
blasd.Copy(&wkp1, &qrow)
}
var wkr cmat.FloatMatrix
qrow.SubMatrix(AL, r, r, 1, m(AL)-r)
wkr.SubMatrix(&wkp1, r, 0, m(AL)-r, 1)
blasd.Copy(&wkr, &qrow)
if k > 0 {
// update wkp1
wrow.SubMatrix(WR, r, 0, 1, n(WR))
blasd.MVMult(&wkp1, AR, &wrow, -1.0, 1.0, gomas.NONE)
}
// set on-diagonal entry to zero to avoid hitting it.
p1 := wkp1.Get(r, 0)
wkp1.Set(r, 0, 0.0)
// max off-diagonal on r'th column/row at index q
q = blasd.IAmax(&wkp1)
qmax := math.Abs(wkp1.Get(q, 0))
wkp1.Set(r, 0, p1)
if amax >= bkALPHA*rmax*(rmax/qmax) {
// no pivoting, 1x1 diagonal
return -1, 1
}
// if q == r then qmax is not off-diagonal, qmax == WR[r,1] and
// we get 1x1 pivot as following is always true
if math.Abs(WL.Get(r, wc-1)) >= bkALPHA*qmax {
// 1x1 pivoting and interchange with k, r
// pivot row in column WL[:,-2] to WL[:,-1]
src.SubMatrix(WL, 0, wc-1, m(AL), 1)
wkp1.SubMatrix(WL, 0, wc, m(AL), 1)
blasd.Copy(&wkp1, &src)
wkp1.Set(-1, 0, src.Get(r, 0))
wkp1.Set(r, 0, src.Get(-1, 0))
return r, 1
} else {
// 2x2 pivoting and interchange with k+1, r
return r, 2
}
return -1, 1
}
/*
* Unblocked Bunch-Kauffman LDL factorization.
*
* Corresponds lapack.DSYTF2
*/
func unblkDecompBKUpper(A, wrk *cmat.FloatMatrix, p Pivots, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12, A22, a11inv cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
partitionPivot2x1(
&pT,
&pB, p, 0, util.PBOTTOM)
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, 0, n(wrk)-2, 2, 2)
a11inv.Set(1, 0, -1.0)
a11inv.Set(0, 1, -1.0)
for n(&ATL) > 0 {
nr := m(&ATL) - 1
r, np := findBKPivotUpper(&ATL)
if r != -1 {
cwrk.SubMatrix(&ATL, 0, n(&ATL)-np, m(&ATL), np)
// pivoting needed; do swaping here
applyBKPivotSymUpper(&ATL, m(&ATL)-np, r)
if np == 2 {
/* [r, r] | [r, nr]
* a11 == ---------------- 2-by-2 pivot, swapping [nr-1,nr] and [r,nr]
* [nr,r] | [nr,nr] (nr is the current diagonal entry)
*/
t := ATL.Get(nr-1, nr)
ATL.Set(nr-1, nr, ATL.Get(r, nr))
ATL.Set(r, nr, t)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22 /**/, A, np, util.PTOPLEFT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, p, np, util.PTOP)
// ------------------------------------------------------------
if np == 1 {
// A00 = A00 - a01*a01.T/a11
blasd.MVUpdateTrm(&A00, &a01, &a01, -1.0/a11.Get(0, 0), gomas.UPPER)
// a01 = a01/a11
blasd.InvScale(&a01, a11.Get(0, 0))
// store pivot point relative to original matrix
if r == -1 {
p1[0] = m(&ATL)
} else {
p1[0] = r + 1
}
} else if np == 2 {
/* see comments on unblkDecompBKLower() */
a := a11.Get(0, 0)
b := a11.Get(0, 1)
d := a11.Get(1, 1)
a11inv.Set(0, 0, d/b)
a11inv.Set(1, 1, a/b)
// denominator: (a/b)*(d/b)-1.0 == (a*d - b^2)/b^2
scale := 1.0 / ((a/b)*(d/b) - 1.0)
scale /= b
// cwrk = a21
cwrk.SubMatrix(wrk, 2, 0, m(&a01), n(&a01))
blasd.Copy(&cwrk, &a01)
// a01 = a01*a11.-1
blasd.Mult(&a01, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
// A00 = A00 - a01*a11.-1*a01.T = A00 - a01*cwrk.T
blasd.UpdateTrm(&A00, &a01, &cwrk, -1.0, 1.0, gomas.UPPER|gomas.TRANSB, conf)
// store pivot point relative to original matrix
p1[0] = -(r + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, p, util.PTOP)
}
return err, nc
}
// test: M < N, U=[m,m] and V=[m,n] or V=[n,n] (square)
func testWide(M, N int, square bool, t *testing.T) {
var A, A0, W, S, U, Uu, V, Vv *cmat.FloatMatrix
var sD cmat.FloatMatrix
var s string
wsize := M * N
if wsize < 100 {
wsize = 100
}
S = cmat.NewMatrix(M, 1)
A = cmat.NewMatrix(M, N)
U = cmat.NewMatrix(M, M)
Uu = cmat.NewMatrix(M, M)
if square {
V = cmat.NewMatrix(N, N)
Vv = cmat.NewMatrix(N, N)
} else {
V = cmat.NewMatrix(M, N)
Vv = cmat.NewMatrix(M, M)
}
src := cmat.NewFloatNormSource()
A.SetFrom(src)
A0 = cmat.NewCopy(A)
W = cmat.NewMatrix(wsize, 1)
if err := lapackd.SVD(S, U, V, A, W, gomas.WANTU|gomas.WANTV); err != nil {
t.Errorf("SVD error: %v\n", err)
return
}
// ||I - U.T*U||
sD.Diag(Uu)
blasd.Mult(Uu, U, U, 1.0, 0.0, gomas.TRANSA)
blasd.Add(&sD, -1.0)
nrm0 := lapackd.NormP(Uu, lapackd.NORM_ONE)
// ||I - V*V.T||
sD.Diag(Vv)
blasd.Mult(Vv, V, V, 1.0, 0.0, gomas.TRANSB)
blasd.Add(&sD, -1.0)
nrm1 := lapackd.NormP(Vv, lapackd.NORM_ONE)
if square {
// right vectors are N-by-N
Sg := cmat.NewMatrix(M, N)
A1 := cmat.NewMatrix(M, N)
sD.Diag(Sg)
blasd.Copy(&sD, S)
blasd.Mult(A1, Sg, V, 1.0, 0.0, gomas.NONE)
blasd.Mult(A0, U, A1, -1.0, 1.0, gomas.NONE)
s = "U=[m,m], V=[n,n]"
} else {
// right vectors are M-by-N
lapackd.MultDiag(V, S, gomas.LEFT)
blasd.Mult(A0, U, V, -1.0, 1.0, gomas.NONE)
s = "U=[m,m], V=[m,n]"
}
nrm2 := lapackd.NormP(A0, lapackd.NORM_ONE)
if N < 10 {
t.Logf("A - U*S*V.T:\n%v\n", A0)
}
t.Logf("M=%d, N=%d, %s ||A - U*S*V.T||_1 :%e\n", M, N, s, nrm2)
t.Logf(" ||I - U.T*U||_1 : %e\n", nrm0)
t.Logf(" ||I - V*V.T||_1 : %e\n", nrm1)
}
func svdWide(S, U, V, A, W *cmat.FloatMatrix, bits int, conf *gomas.Config) (err *gomas.Error) {
var uu, vv *cmat.FloatMatrix
var tauq, taup, Wred, sD, sE, L, Vm cmat.FloatMatrix
if (bits & (gomas.WANTU | gomas.WANTV)) != 0 {
if W.Len() < 4*n(A) {
err = gomas.NewError(gomas.ESIZE, "SVD")
return
}
}
tauq.SetBuf(m(A)-1, 1, m(A)-1, W.Data())
taup.SetBuf(m(A), 1, m(A), W.Data()[tauq.Len():])
wrl := W.Len() - 2*m(A) - 1
Wred.SetBuf(wrl, 1, wrl, W.Data()[2*m(A)-1:])
if svdCrossover(n(A), m(A)) {
goto do_n_much_bigger
}
// reduce to bidiagonal form
if err = BDReduce(A, &tauq, &taup, &Wred, conf); err != nil {
return
}
sD.Diag(A)
sE.Diag(A, -1)
blasd.Copy(S, &sD)
// leftt vectors
if bits&gomas.WANTU != 0 {
L.SubMatrix(A, 0, 0, m(A), m(A))
U.Copy(&L)
cmat.TriL(U, 0)
if err = BDBuild(U, &tauq, &Wred, m(U), gomas.WANTQ|gomas.LOWER, conf); err != nil {
return
}
uu = U
}
// right vectors
if bits&gomas.WANTV != 0 {
if m(V) == m(A) {
// V is M-by-N; copy and make upper triangular
V.Copy(A)
//cmat.TriU(V, 0)
if err = BDBuild(V, &taup, &Wred, m(V), gomas.WANTP, conf); err != nil {
return
}
} else {
// V is N-by-N
eye := cmat.FloatDiagonalSource{1.0}
V.SetFrom(&eye, cmat.SYMM)
err = BDMult(V, A, &taup, &Wred, gomas.MULTP|gomas.LEFT|gomas.TRANS, conf)
if err != nil {
return
}
}
vv = V
}
err = BDSvd(S, &sE, uu, vv, W, bits|gomas.LOWER)
return
do_n_much_bigger:
// here N >> M, use LQ factor first
if err = LQFactor(A, &taup, &Wred, conf); err != nil {
return
}
if bits&gomas.WANTV != 0 {
if m(V) == m(A) {
V.Copy(A)
if err = LQBuild(V, &taup, &Wred, m(A), conf); err != nil {
return
}
} else {
// V is N-by-N
eye := cmat.FloatDiagonalSource{1.0}
V.SetFrom(&eye, cmat.SYMM)
if err = LQMult(V, A, &taup, &Wred, gomas.RIGHT, conf); err != nil {
return
}
}
}
L.SubMatrix(A, 0, 0, m(A), m(A))
cmat.TriL(&L, 0)
// resize tauq/taup for UPPER bidiagonal reduction
tauq.SetBuf(m(A), 1, m(A), W.Data())
taup.SetBuf(m(A)-1, 1, m(A)-1, W.Data()[tauq.Len():])
// bidiagonal reduce
if err = BDReduce(&L, &tauq, &taup, &Wred, conf); err != nil {
return
}
if bits&gomas.WANTV != 0 {
Vm.SubMatrix(V, 0, 0, m(A), n(A))
err = BDMult(&Vm, &L, &taup, &Wred, gomas.MULTP|gomas.LEFT|gomas.TRANS, conf)
if err != nil {
return
}
vv = V
}
if bits&gomas.WANTU != 0 {
U.Copy(&L)
if err = BDBuild(U, &tauq, &Wred, m(U), gomas.WANTQ, conf); err != nil {
return
}
uu = U
}
sD.Diag(A)
sE.Diag(A, 1)
blasd.Copy(S, &sD)
err = BDSvd(S, &sE, uu, vv, W, bits|gomas.UPPER, conf)
return
}
/*
* Find diagonal pivot and build incrementaly updated block.
*
* (AL) (AR) (WL) (WR)
* -------------------------- ---------- k'th row in W
* x x | c1 w w | k kp1
* x x | c1 d w w | k kp1
* x x | c1 x d w w | k kp1
* x x | c1 x x d w w | k kp1
* x x | c1 r2 r2 r2 r2 w w | k kp1
* x x | c1 x x x r2 d w w | k kp1
* x x | c1 x x x r2 x d w w | k kp1
*
* Matrix AR contains the unfactored part of the matrix and AL the already
* factored columns. Matrix WL is updated values of factored part ie.
* w(i) = l(i)d(i). Matrix WR will have updated values for next column.
* Column WR(k) contains updated AR(c1) and WR(kp1) possible pivot row AR(r2).
*/
func findAndBuildBKPivotLower(AL, AR, WL, WR *cmat.FloatMatrix, k int) (int, int) {
var r, q int
var rcol, qrow, src, wk, wkp1, wrow cmat.FloatMatrix
// Copy AR column 0 to WR column 0 and update with WL[0:]
src.SubMatrix(AR, 0, 0, m(AR), 1)
wk.SubMatrix(WR, 0, 0, m(AR), 1)
wk.Copy(&src)
if k > 0 {
wrow.SubMatrix(WL, 0, 0, 1, n(WL))
blasd.MVMult(&wk, AL, &wrow, -1.0, 1.0, gomas.NONE)
}
if m(AR) == 1 {
return 0, 1
}
amax := math.Abs(WR.Get(0, 0))
// find max off-diagonal on first column.
rcol.SubMatrix(WR, 1, 0, m(AR)-1, 1)
// r is row index and rmax is its absolute value
r = blasd.IAmax(&rcol) + 1
rmax := math.Abs(rcol.Get(r-1, 0))
if amax >= bkALPHA*rmax {
// no pivoting, 1x1 diagonal
return 0, 1
}
// Now we need to copy row r to WR[:,1] and update it
wkp1.SubMatrix(WR, 0, 1, m(AR), 1)
qrow.SubMatrix(AR, r, 0, 1, r+1)
blasd.Copy(&wkp1, &qrow)
if r < m(AR)-1 {
var wkr cmat.FloatMatrix
qrow.SubMatrix(AR, r, r, m(AR)-r, 1)
wkr.SubMatrix(&wkp1, r, 0, m(&wkp1)-r, 1)
blasd.Copy(&wkr, &qrow)
}
if k > 0 {
// update wkp1
wrow.SubMatrix(WL, r, 0, 1, n(WL))
blasd.MVMult(&wkp1, AL, &wrow, -1.0, 1.0, gomas.NONE)
}
// set on-diagonal entry to zero to avoid finding it
p1 := wkp1.Get(r, 0)
wkp1.Set(r, 0, 0.0)
// max off-diagonal on r'th column/row at index q
q = blasd.IAmax(&wkp1)
qmax := math.Abs(wkp1.Get(q, 0))
// restore on-diagonal entry
wkp1.Set(r, 0, p1)
if amax >= bkALPHA*rmax*(rmax/qmax) {
// no pivoting, 1x1 diagonal
return 0, 1
}
// if q == r then qmax is not off-diagonal, qmax == WR[r,1] and
// we get 1x1 pivot as following is always true
if math.Abs(WR.Get(r, 1)) >= bkALPHA*qmax {
// 1x1 pivoting and interchange with k, r
// pivot row in column WR[:,1] to W[:,0]
src.SubMatrix(WR, 0, 1, m(AR), 1)
wkp1.SubMatrix(WR, 0, 0, m(AR), 1)
blasd.Copy(&wkp1, &src)
wkp1.Set(0, 0, src.Get(r, 0))
wkp1.Set(r, 0, src.Get(0, 0))
return r, 1
} else {
// 2x2 pivoting and interchange with k+1, r
return r, 2
}
return 0, 1
}
/*
* Unblocked Bunch-Kauffman LDL factorization.
*
* Corresponds lapack.DSYTF2
*/
func unblkDecompBKLower(A, wrk *cmat.FloatMatrix, p Pivots, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10t, a11, A20, a21, A22, a11inv cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
partitionPivot2x1(
&pT,
&pB, p, 0, util.PTOP)
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, 0, n(wrk)-2, 2, 2)
a11inv.Set(1, 0, -1.0)
a11inv.Set(0, 1, -1.0)
for n(&ABR) > 0 {
r, np := findBKPivotLower(&ABR)
if r != 0 && r != np-1 {
// pivoting needed; do swaping here
applyBKPivotSymLower(&ABR, np-1, r)
if np == 2 {
/* [0,0] | [r,0]
* a11 == ------------- 2-by-2 pivot, swapping [1,0] and [r,0]
* [r,0] | [r,r]
*/
t := ABR.Get(1, 0)
ABR.Set(1, 0, ABR.Get(r, 0))
ABR.Set(r, 0, t)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10t, &a11, nil,
&A20, &a21, &A22 /**/, A, np, util.PBOTTOMRIGHT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, p, np, util.PBOTTOM)
// ------------------------------------------------------------
if np == 1 {
// A22 = A22 - a21*a21.T/a11
blasd.MVUpdateTrm(&A22, &a21, &a21, -1.0/a11.Get(0, 0), gomas.LOWER)
// a21 = a21/a11
blasd.InvScale(&a21, a11.Get(0, 0))
// store pivot point relative to original matrix
p1[0] = r + m(&ATL) + 1
} else if np == 2 {
/* from Bunch-Kaufmann 1977:
* (E2 C.T) = ( I2 0 )( E 0 )( I[n-2] E.-1*C.T )
* (C B ) ( C*E.-1 I[n-2] )( 0 A[n-2] )( 0 I2 )
*
* A[n-2] = B - C*E.-1*C.T
*
* E.-1 is inverse of a symmetric matrix, cannot use
* triangular solve. We calculate inverse of 2x2 matrix.
* Following is inspired by lapack.SYTF2
*
* a | b 1 d | -b b d/b | -1
* inv ----- = ------ * ------ = ----------- * --------
* b | d (ad-b^2) -b | a (a*d - b^2) -1 | a/b
*
*/
a := a11.Get(0, 0)
b := a11.Get(1, 0)
d := a11.Get(1, 1)
a11inv.Set(0, 0, d/b)
a11inv.Set(1, 1, a/b)
// denominator: (a/b)*(d/b)-1.0 == (a*d - b^2)/b^2
scale := 1.0 / ((a/b)*(d/b) - 1.0)
scale /= b
// cwrk = a21
cwrk.SubMatrix(wrk, 2, 0, m(&a21), n(&a21))
blasd.Copy(&cwrk, &a21)
// a21 = a21*a11.-1
blasd.Mult(&a21, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
// A22 = A22 - a21*a11.-1*a21.T = A22 - a21*cwrk.T
blasd.UpdateTrm(&A22, &a21, &cwrk, -1.0, 1.0, gomas.LOWER|gomas.TRANSB, conf)
// store pivot point relative to original matrix
p1[0] = -(r + m(&ATL) + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, p, util.PBOTTOM)
}
return err, nc
}
/*
* 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
}
/*
* Unblocked, bounded Bunch-Kauffman LDL factorization for at most ncol columns.
* At most ncol columns are factorized and trailing matrix updates are restricted
* to ncol columns. Also original columns are accumulated to working matrix, which
* is used by calling blocked algorithm to update the trailing matrix with BLAS3
* update.
*
* Corresponds lapack.DLASYF
*/
func unblkBoundedBKLower(A, wrk *cmat.FloatMatrix, p *Pivots, ncol int, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10t, a11, A20, a21, A22, a11inv cmat.FloatMatrix
var w00, w10, w11 cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
if ncol > n(A) {
ncol = n(A)
}
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, 0, n(wrk)-2, 2, 2)
a11inv.Set(1, 0, -1.0)
a11inv.Set(0, 1, -1.0)
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
partitionPivot2x1(
&pT,
&pB, *p, 0, util.PTOP)
for n(&ABR) > 0 && nc < ncol {
util.Partition2x2(
&w00, nil,
&w10, &w11, wrk, nc, nc, util.PTOPLEFT)
r, np := findAndBuildBKPivotLower(&ABL, &ABR, &w10, &w11, nc)
if np > ncol-nc {
// next pivot does not fit into ncol columns, restore last column,
// return with number of factorized columns
return err, nc
}
if r != 0 && r != np-1 {
// pivoting needed; do swaping here
applyBKPivotSymLower(&ABR, np-1, r)
// swap left hand rows to get correct updates
swapRows(&ABL, np-1, r)
swapRows(&w10, np-1, r)
if np == 2 {
/*
* [0,0] | [r,0]
* a11 == ------------- 2-by-2 pivot, swapping [1,0] and [r,0]
* [r,0] | [r,r]
*/
t0 := w11.Get(1, 0)
tr := w11.Get(r, 0)
w11.Set(1, 0, tr)
w11.Set(r, 0, t0)
// interchange diagonal entries on w11[:,1]
t0 = w11.Get(1, 1)
tr = w11.Get(r, 1)
w11.Set(1, 1, tr)
w11.Set(r, 1, t0)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10t, &a11, nil,
&A20, &a21, &A22 /**/, A, np, util.PBOTTOMRIGHT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, *p, np, util.PBOTTOM)
// ------------------------------------------------------------
if np == 1 {
//
cwrk.SubMatrix(&w11, np, 0, m(&a21), np)
a11.Set(0, 0, w11.Get(0, 0))
// a21 = a21/a11
blasd.Copy(&a21, &cwrk)
blasd.InvScale(&a21, a11.Get(0, 0))
// store pivot point relative to original matrix
p1[0] = r + m(&ATL) + 1
} else if np == 2 {
/*
* See comments for this block in unblkDecompBKLower().
*/
a := w11.Get(0, 0)
b := w11.Get(1, 0)
d := w11.Get(1, 1)
a11inv.Set(0, 0, d/b)
a11inv.Set(1, 1, a/b)
// denominator: (a/b)*(d/b)-1.0 == (a*d - b^2)/b^2
scale := 1.0 / ((a/b)*(d/b) - 1.0)
scale /= b
cwrk.SubMatrix(&w11, np, 0, m(&a21), np)
// a21 = a21*a11.-1
blasd.Mult(&a21, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
a11.Set(0, 0, a)
a11.Set(1, 0, b)
a11.Set(1, 1, d)
// store pivot point relative to original matrix
p1[0] = -(r + m(&ATL) + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, *p, util.PBOTTOM)
}
return err, nc
}
/*
* Blocked version of Hessenberg reduction algorithm as presented in (1). This
* version uses compact-WY transformation.
*
* Some notes:
*
* Elementary reflectors stored in [A11; A21].T are not on diagonal of A11. Update of
* a block aligned with A11; A21 is as follow
*
* 1. Update from left Q(k)*C:
* c0 0 c0
* (I - Y*T*Y.T).T*C = C - Y*(C.T*Y)*T.T = C1 - Y1 * (C1.T.Y1+C2.T*Y2)*T.T = C1-Y1*W
* C2 Y2 C2-Y2*W
*
* where W = (C1.T*Y1+C2.T*Y2)*T.T and first row of C is not affected by update
*
* 2. Update from right C*Q(k):
* 0
* C - C*Y*T*Y.T = c0;C1;C2 - c0;C1;C2 * Y1 *T*(0;Y1;Y2) = c0; C1-W*Y1; C2-W*Y2
* Y2
* where W = (C1*Y1 + C2*Y2)*T and first column of C is not affected
*
*/
func blkHessGQvdG(A, Tvec, W *cmat.FloatMatrix, nb int, conf *gomas.Config) *gomas.Error {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, A11, A12, A21, A22, A2 cmat.FloatMatrix
var tT, tB, td cmat.FloatMatrix
var t0, t1, t2, T cmat.FloatMatrix
var V, VT, VB /*V0, V1, V2,*/, Y1, Y2, W0 cmat.FloatMatrix
//fmt.Printf("blkHessGQvdG...\n")
T.SubMatrix(W, 0, 0, conf.LB, conf.LB)
V.SubMatrix(W, conf.LB, 0, m(A), conf.LB)
td.Diag(&T)
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x1(
&tT,
&tB, Tvec, 0, util.PTOP)
for m(&ABR) > nb+1 && n(&ABR) > nb {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
nil, &A11, &A12,
nil, &A21, &A22, A, nb, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tT,
&t0,
&t1,
&t2, Tvec, nb, util.PBOTTOM)
util.Partition2x1(
&VT,
&VB, &V, m(&ATL), util.PTOP)
// ------------------------------------------------------
unblkBuildHessGQvdG(&ABR, &T, &VB, nil)
blasd.Copy(&t1, &td)
// m(Y) == m(ABR)-1, n(Y) == n(A11)
Y1.SubMatrix(&ABR, 1, 0, n(&A11), n(&A11))
Y2.SubMatrix(&ABR, 1+n(&A11), 0, m(&A21)-1, n(&A11))
// [A01; A02] == ATR := ATR*(I - Y*T*Y.T)
updateHessRightWY(&ATR, &Y1, &Y2, &T, &VT, conf)
// A2 = [A12; A22].T
util.Merge2x1(&A2, &A12, &A22)
// A2 := A2 - VB*T*A21.T
be := A21.Get(0, -1)
A21.Set(0, -1, 1.0)
blasd.MultTrm(&VB, &T, 1.0, gomas.UPPER|gomas.RIGHT)
blasd.Mult(&A2, &VB, &A21, -1.0, 1.0, gomas.TRANSB, conf)
A21.Set(0, -1, be)
// A2 := (I - Y*T*Y.T).T * A2
W0.SubMatrix(&V, 0, 0, n(&A2), n(&Y2))
updateHessLeftWY(&A2, &Y1, &Y2, &T, &W0, conf)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &A11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tT,
&tB, &t0, &t1, Tvec, util.PBOTTOM)
}
if m(&ABR) > 1 {
// do the rest with unblocked
util.Merge2x1(&A2, &ATR, &ABR)
W0.SetBuf(m(A), 1, m(A), W.Data())
unblkHessGQvdG(&A2, &tB, &W0, m(&ATR))
}
return nil
}
// Compute SVD when m(A) >= n(A)
func svdTall(S, U, V, A, W *cmat.FloatMatrix, bits int, conf *gomas.Config) (err *gomas.Error) {
var uu, vv *cmat.FloatMatrix
var tauq, taup, Wred, sD, sE, R, Un cmat.FloatMatrix
if (bits & (gomas.WANTU | gomas.WANTV)) != 0 {
if W.Len() < 4*n(A) {
err = gomas.NewError(gomas.ESIZE, "SVD")
return
}
}
tauq.SetBuf(n(A), 1, n(A), W.Data())
taup.SetBuf(n(A)-1, 1, n(A)-1, W.Data()[tauq.Len():])
wrl := W.Len() - 2*n(A) - 1
Wred.SetBuf(wrl, 1, wrl, W.Data()[2*n(A)-1:])
if svdCrossover(m(A), n(A)) {
goto do_m_much_bigger
}
// reduce to bidiagonal form
if err = BDReduce(A, &tauq, &taup, &Wred, conf); err != nil {
return
}
sD.Diag(A)
sE.Diag(A, 1)
blasd.Copy(S, &sD)
// left vectors
if bits&gomas.WANTU != 0 {
if n(U) == n(A) {
// U is M-by-N; copy and make lower triangular
U.Copy(A)
cmat.TriL(U, 0)
if err = BDBuild(U, &tauq, &Wred, n(U), gomas.WANTQ, conf); err != nil {
return
}
} else {
// U is M-by-M
eye := cmat.FloatDiagonalSource{1.0}
U.SetFrom(&eye, cmat.SYMM)
if err = BDMult(U, A, &tauq, &Wred, gomas.MULTQ|gomas.RIGHT, conf); err != nil {
return
}
}
uu = U
}
// right vectors
if bits&gomas.WANTV != 0 {
R.SubMatrix(A, 0, 0, n(A), n(A))
V.Copy(&R)
cmat.TriU(V, 0)
if err = BDBuild(V, &taup, &Wred, m(V), gomas.WANTP, conf); err != nil {
return
}
vv = V
}
err = BDSvd(S, &sE, uu, vv, W, bits|gomas.UPPER)
return
do_m_much_bigger:
// M >> N here; first use QR factorization
if err = QRFactor(A, &tauq, &Wred, conf); err != nil {
return
}
if bits&gomas.WANTU != 0 {
if n(U) == n(A) {
U.Copy(A)
if err = QRBuild(U, &tauq, &Wred, n(U), 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, &tauq, &Wred, gomas.LEFT, conf); err != nil {
return
}
}
}
R.SubMatrix(A, 0, 0, n(A), n(A))
cmat.TriU(&R, 0)
// bidiagonal reduce
if err = BDReduce(&R, &tauq, &taup, &Wred, conf); err != nil {
return
}
if bits&gomas.WANTU != 0 {
Un.SubMatrix(U, 0, 0, m(A), n(A))
if err = BDMult(&Un, &R, &tauq, &Wred, gomas.MULTQ|gomas.RIGHT, conf); err != nil {
return
}
uu = U
}
if bits&gomas.WANTV != 0 {
V.Copy(&R)
if err = BDBuild(V, &taup, &Wred, m(V), gomas.WANTP, conf); err != nil {
return
}
vv = V
}
sD.Diag(A)
sE.Diag(A, 1)
blasd.Copy(S, &sD)
err = BDSvd(S, &sE, uu, vv, W, bits|gomas.UPPER, conf)
return
}
