func TestBidiagReduceUnblocked(t *testing.T) {
N := 217
M := 269
conf := gomas.NewConf()
conf.LB = 0
zeromean := cmat.NewFloatNormSource()
A := cmat.NewMatrix(M, N)
A.SetFrom(zeromean)
tauq := cmat.NewMatrix(N, 1)
taup := cmat.NewMatrix(N, 1)
At := cmat.NewMatrix(N, M)
blasd.Transpose(At, A)
tauqt := cmat.NewMatrix(N, 1)
taupt := cmat.NewMatrix(N, 1)
W := lapackd.Workspace(M + N)
lapackd.BDReduce(A, tauq, taup, W, conf)
lapackd.BDReduce(At, tauqt, taupt, W, conf)
// BiRed(A) == BiRed(A.T).T
blasd.Plus(At, A, 1.0, -1.0, gomas.TRANSB)
blasd.Axpy(tauqt, taup, -1.0)
blasd.Axpy(taupt, tauq, -1.0)
nrm := lapackd.NormP(At, lapackd.NORM_ONE)
t.Logf("M=%d, N=%d || BiRed(A) - BiRed(A.T).T||_1 : %e\n", M, N, nrm)
nrm = lapackd.NormP(taupt, lapackd.NORM_ONE)
t.Logf(" || BiRed(A).tauq - BiRed(A.T).taup||_1 : %e\n", nrm)
nrm = lapackd.NormP(tauqt, lapackd.NORM_ONE)
t.Logf(" || BiRed(A).taup - BiRed(A.T).tauq||_1 : %e\n", nrm)
}
func TestTriRedUpper(t *testing.T) {
N := 843
nb := 48
conf := gomas.NewConf()
conf.LB = 0
A := cmat.NewMatrix(N, N)
tau := cmat.NewMatrix(N, 1)
src := cmat.NewFloatNormSource()
A.SetFrom(src, cmat.UPPER)
A1 := cmat.NewCopy(A)
tau1 := cmat.NewCopy(tau)
_ = A1
W := lapackd.Workspace(N)
W1 := lapackd.Workspace(N * nb)
lapackd.TRDReduce(A, tau, W, gomas.UPPER, conf)
conf.LB = nb
lapackd.TRDReduce(A1, tau1, W1, gomas.UPPER, conf)
blasd.Plus(A, A1, -1.0, 1.0, gomas.NONE)
nrm := lapackd.NormP(A, lapackd.NORM_ONE)
t.Logf("N=%d, ||unblk.Trired(A) - blk.Trired(A)||_1: %e\n", N, nrm)
blasd.Axpy(tau, tau1, -1.0)
nrm = blasd.Nrm2(tau)
t.Logf(" ||unblk.Trired(tau) - blk.Trired(tau)||_1: %e\n", nrm)
}
/*
* Applies a real elementary reflector H to a real m by n matrix A,
* from either the left or the right. H is represented in the form
*
* H = I - tau * ( 1 ) * ( 1 v.T )
* ( v )
*
* where tau is a real scalar and v is a real vector.
*
* If tau = 0, then H is taken to be the unit cmat.
*
* A is /a1\ a1 := a1 - w1
* \A2/ A2 := A2 - v*w1
* w1 := tau*(a1 + A2.T*v) if side == LEFT
* := tau*(a1 + A2*v) if side == RIGHT
*
* Intermediate work space w1 required as parameter, no allocation.
*/
func applyHouseholder2x1(tau, v, a1, A2, w1 *cmat.FloatMatrix, flags int) *gomas.Error {
var err *gomas.Error = nil
tval := tau.Get(0, 0)
if tval == 0.0 {
return err
}
// shape oblivious vector copy.
blasd.Axpby(w1, a1, 1.0, 0.0)
if flags&gomas.LEFT != 0 {
// w1 = a1 + A2.T*v
err = blasd.MVMult(w1, A2, v, 1.0, 1.0, gomas.TRANSA)
} else {
// w1 = a1 + A2*v
err = blasd.MVMult(w1, A2, v, 1.0, 1.0, gomas.NONE)
}
// w1 = tau*w1
blasd.Scale(w1, tval)
// a1 = a1 - w1
blasd.Axpy(a1, w1, -1.0)
// A2 = A2 - v*w1
if flags&gomas.LEFT != 0 {
err = blasd.MVUpdate(A2, v, w1, -1.0)
} else {
err = blasd.MVUpdate(A2, w1, v, -1.0)
}
return err
}
/*
* Reduce upper triangular matrix to tridiagonal.
*
* Elementary reflectors Q = H(n-1)...H(2)H(1) are stored on upper
* triangular part of A. Reflector H(n-1) saved at column A(n) and
* scalar multiplier to tau[n-1]. If parameter `tail` is true then
* this function is used to reduce tail part of partially reduced
* matrix and tau-vector partitioning is starting from last position.
*/
func unblkReduceTridiagUpper(A, tauq, W *cmat.FloatMatrix, tail bool) {
var ATL, ABR cmat.FloatMatrix
var A00, a01, a11, A22 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var y21 cmat.FloatMatrix
var v0 float64
toff := 1
if tail {
toff = 0
}
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x1(
&tqT,
&tqB, tauq, toff, util.PBOTTOM)
for n(&ATL) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, nil,
nil, &a11, nil,
nil, nil, &A22, A, 1, util.PTOPLEFT)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, 1, util.PTOP)
// set temp vectors for this round
y21.SetBuf(n(&A00), 1, n(&A00), W.Data())
// ------------------------------------------------------
// Compute householder to zero super-diagonal entries
computeHouseholderRev(&a01, &tauq1)
tauqv := tauq1.Get(0, 0)
// set superdiagonal to unit
v0 = a01.Get(-1, 0)
a01.Set(-1, 0, 1.0)
// y21 := A22*a12t
blasd.MVMultSym(&y21, &A00, &a01, tauqv, 0.0, gomas.UPPER)
// beta := tauq*a12t*y21
beta := tauqv * blasd.Dot(&a01, &y21)
// y21 := y21 - 0.5*beta*a125
blasd.Axpy(&y21, &a01, -0.5*beta)
// A22 := A22 - a12t*y21.T - y21*a12.T
blasd.MVUpdate2Sym(&A00, &a01, &y21, -1.0, gomas.UPPER)
// restore superdiagonal value
a01.Set(-1, 0, v0)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PTOP)
}
}
func test_bdsvd(N, flags, kind int, verbose bool, t *testing.T) {
var At, sD, sE, tmp cmat.FloatMatrix
uplo := "upper"
offdiag := 1
if flags&gomas.LOWER != 0 {
offdiag = -1
uplo = "lower"
}
A0 := cmat.NewMatrix(N, N)
desc := setDiagonals(A0, offdiag, kind)
At.SubMatrix(A0, 0, 0, N, N)
sD.Diag(A0, 0)
sE.Diag(A0, offdiag)
D := cmat.NewCopy(&sD)
E := cmat.NewCopy(&sE)
// unit singular vectors
U := cmat.NewMatrix(N, N)
sD.Diag(U, 0)
sD.Add(1.0)
V := cmat.NewMatrix(N, N)
sD.Diag(V, 0)
sD.Add(1.0)
W := cmat.NewMatrix(4*N, 1)
C := cmat.NewMatrix(N, N)
lapackd.BDSvd(D, E, U, V, W, flags|gomas.WANTU|gomas.WANTV)
blasd.Mult(C, U, U, 1.0, 0.0, gomas.TRANSA)
sD.Diag(C)
sD.Add(-1.0)
nrmu := lapackd.NormP(C, lapackd.NORM_ONE)
blasd.Mult(C, V, V, 1.0, 0.0, gomas.TRANSB)
sD.Add(-1.0)
nrmv := lapackd.NormP(C, lapackd.NORM_ONE)
blasd.Mult(C, U, A0, 1.0, 0.0, gomas.TRANSA)
blasd.Mult(&At, C, V, 1.0, 0.0, gomas.TRANSB)
if verbose && N < 10 {
t.Logf("D:\n%v\n", asRow(&tmp, D))
t.Logf("U:\n%v\n", U)
t.Logf("V:\n%v\n", V)
t.Logf("U.T*A*V\n%v\n", &At)
}
sD.Diag(&At)
blasd.Axpy(&sD, D, -1.0)
nrma := lapackd.NormP(&At, lapackd.NORM_ONE)
t.Logf("N=%d [%s,%s] ||U.T*A*V - bdsvd(A)||_1: %e\n", N, uplo, desc, nrma)
t.Logf(" ||I - U.T*U||_1: %e\n", nrmu)
t.Logf(" ||I - V.T*V||_1: %e\n", nrmv)
}
/*
* Tridiagonal reduction of LOWER triangular symmetric matrix, zero elements below 1st
* subdiagonal:
*
* A = (1 - tau*u*u.t)*A*(1 - tau*u*u.T)
* = (I - tau*( 0 0 )) (a11 a12) (I - tau*( 0 0 ))
* ( ( 0 u*u.t)) (a21 A22) ( ( 0 u*u.t))
*
* a11, a12, a21 not affected
*
* from LEFT:
* A22 = A22 - tau*u*u.T*A22
* from RIGHT:
* A22 = A22 - tau*A22*u.u.T
*
* LEFT and RIGHT:
* A22 = A22 - tau*u*u.T*A22 - tau*(A22 - tau*u*u.T*A22)*u*u.T
* = A22 - tau*u*u.T*A22 - tau*A22*u*u.T + tau*tau*u*u.T*A22*u*u.T
* [x = tau*A22*u (vector)] (SYMV)
* A22 = A22 - u*x.T - x*u.T + tau*u*u.T*x*u.T
* [beta = tau*u.T*x (scalar)] (DOT)
* = A22 - u*x.T - x*u.T + beta*u*u.T
* = A22 - u*(x - 0.5*beta*u).T - (x - 0.5*beta*u)*u.T
* [w = x - 0.5*beta*u] (AXPY)
* = A22 - u*w.T - w*u.T (SYR2)
*
* Result of reduction for N = 5:
* ( d . . . . )
* ( e d . . . )
* ( v1 e d . . )
* ( v1 v2 e d . )
* ( v1 v2 v3 e d )
*/
func unblkReduceTridiagLower(A, tauq, W *cmat.FloatMatrix) {
var ATL, ABR cmat.FloatMatrix
var A00, a11, a21, A22 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var y21 cmat.FloatMatrix
var v0 float64
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x1(
&tqT,
&tqB, tauq, 0, util.PTOP)
for m(&ABR) > 0 && n(&ABR) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
nil, &a11, nil,
nil, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, 1, util.PBOTTOM)
// set temp vectors for this round
y21.SetBuf(n(&A22), 1, n(&A22), W.Data())
// ------------------------------------------------------
// Compute householder to zero subdiagonal entries
computeHouseholderVec(&a21, &tauq1)
tauqv := tauq1.Get(0, 0)
// set subdiagonal to unit
v0 = a21.Get(0, 0)
a21.Set(0, 0, 1.0)
// y21 := tauq*A22*a21
blasd.MVMultSym(&y21, &A22, &a21, tauqv, 0.0, gomas.LOWER)
// beta := tauq*a21.T*y21
beta := tauqv * blasd.Dot(&a21, &y21)
// y21 := y21 - 0.5*beta*a21
blasd.Axpy(&y21, &a21, -0.5*beta)
// A22 := A22 - a21*y21.T - y21*a21.T
blasd.MVUpdate2Sym(&A22, &a21, &y21, -1.0, gomas.LOWER)
// restore subdiagonal
a21.Set(0, 0, v0)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PBOTTOM)
}
}
func TestReduceBidiagBlkWide(t *testing.T) {
N := 911
M := 823
nb := 48
conf := gomas.NewConf()
conf.LB = 0
zeromean := cmat.NewFloatNormSource()
A := cmat.NewMatrix(M, N)
A.SetFrom(zeromean)
tauq := cmat.NewMatrix(N, 1)
taup := cmat.NewMatrix(N, 1)
A1 := cmat.NewCopy(A)
tauq1 := cmat.NewMatrix(N, 1)
taup1 := cmat.NewMatrix(N, 1)
W := lapackd.Workspace(M + N)
lapackd.BDReduce(A, tauq, taup, W, conf)
conf.LB = nb
W1 := lapackd.Workspace(lapackd.BDReduceWork(A1, conf))
lapackd.BDReduce(A1, tauq1, taup1, W1, conf)
// BiRed(A) == BiRed(A.T).T
blasd.Plus(A1, A, 1.0, -1.0, gomas.NONE)
blasd.Axpy(tauq1, tauq, -1.0)
blasd.Axpy(taup1, taup, -1.0)
nrm := lapackd.NormP(A1, lapackd.NORM_ONE)
t.Logf("M=%d, N=%d || BiRed(A) - blk.BiRed(A)||_1 : %e\n", M, N, nrm)
nrm = lapackd.NormP(taup1, lapackd.NORM_ONE)
t.Logf(" || BiRed(A).tauq - blk.BiRed(A).taup||_1 : %e\n", nrm)
nrm = lapackd.NormP(tauq1, lapackd.NORM_ONE)
t.Logf(" || BiRed(A).taup - blk.BiRed(A).tauq||_1 : %e\n", nrm)
}
func TestBidiagReduceBlockedTall(t *testing.T) {
N := 711
M := 883
nb := 64
conf := gomas.NewConf()
conf.LB = 0
zeromean := cmat.NewFloatNormSource()
A := cmat.NewMatrix(M, N)
A.SetFrom(zeromean)
tauq := cmat.NewMatrix(N, 1)
taup := cmat.NewMatrix(N, 1)
A1 := cmat.NewCopy(A)
tauq1 := cmat.NewMatrix(N, 1)
taup1 := cmat.NewMatrix(N, 1)
W := lapackd.Workspace(M + N)
W1 := lapackd.Workspace(nb * (M + N + 1))
lapackd.BDReduce(A, tauq, taup, W, conf)
conf.LB = nb
lapackd.BDReduce(A1, tauq1, taup1, W1, conf)
// unblk.BiRed(A) == blk.BiRed(A)
blasd.Plus(A1, A, 1.0, -1.0, gomas.NONE)
blasd.Axpy(tauq1, tauq, -1.0)
blasd.Axpy(taup1, taup, -1.0)
nrm := lapackd.NormP(A1, lapackd.NORM_ONE)
t.Logf("M=%d, N=%d || unblk.BiRed(A) - blk.BiRed(A)||_1 : %e\n", M, N, nrm)
nrm = lapackd.NormP(taup1, lapackd.NORM_ONE)
t.Logf(" || unblk.BiRed(A).tauq - blk.BiRed(A).taup||_1 : %e\n", nrm)
nrm = lapackd.NormP(tauq1, lapackd.NORM_ONE)
t.Logf(" || unblk.BiRed(A).taup - blk.BiRed(A).tauq||_1 : %e\n", nrm)
}
func test_trdevd(N, flags, kind int, verbose bool, t *testing.T) {
var At, sD, sE, tmp cmat.FloatMatrix
A0 := cmat.NewMatrix(N, N)
desc := setTrdDiagonals(A0, kind)
At.SubMatrix(A0, 0, 0, N, N)
sD.Diag(A0, 0)
sE.Diag(A0, 1)
D := cmat.NewCopy(&sD)
E := cmat.NewCopy(&sE)
V := cmat.NewMatrix(N, N)
sD.Diag(V, 0)
sD.Add(1.0)
W := cmat.NewMatrix(4*N, 1)
C := cmat.NewMatrix(N, N)
if verbose && N < 10 {
t.Logf("A0:\n%v\n", A0.ToString("%6.3f"))
t.Logf("V.pre:\n%v\n", V.ToString("%6.3f"))
}
lapackd.TRDEigen(D, E, V, W, flags|gomas.WANTV)
for k := 0; k < N-1; k++ {
if E.GetAt(k) != 0.0 {
t.Logf("E[%d] != 0.0 (%e)\n", k, E.GetAt(k))
}
}
blasd.Mult(C, V, V, 1.0, 0.0, gomas.TRANSB)
sD.Diag(C)
sD.Add(-1.0)
nrmv := lapackd.NormP(C, lapackd.NORM_ONE)
blasd.Mult(C, V, A0, 1.0, 0.0, gomas.TRANSA)
blasd.Mult(&At, C, V, 1.0, 0.0, gomas.NONE)
if verbose && N < 10 {
t.Logf("D:\n%v\n", asRow(&tmp, D).ToString("%6.3f"))
t.Logf("V:\n%v\n", V.ToString("%6.3f"))
t.Logf("V.T*A*V\n%v\n", At.ToString("%6.3f"))
}
sD.Diag(&At)
blasd.Axpy(&sD, D, -1.0)
nrma := lapackd.NormP(&At, lapackd.NORM_ONE)
t.Logf("N=%d [%s] ||V.T*A*V - eigen(A)||_1: %e\n", N, desc, nrma)
t.Logf(" ||I - V.T*V||_1: %e\n", nrmv)
}
// test: unblk.QLFactor == blk.QLFactor
func TestQLFactor(t *testing.T) {
var t0 cmat.FloatMatrix
M := 911
N := 835
nb := 32
conf := gomas.NewConf()
A := cmat.NewMatrix(M, N)
src := cmat.NewFloatNormSource()
A.SetFrom(src)
tau := cmat.NewMatrix(M, 1)
W := cmat.NewMatrix(M+N, 1)
A1 := cmat.NewCopy(A)
tau1 := cmat.NewCopy(tau)
conf.LB = 0
lapackd.QLFactor(A, tau, W, conf)
conf.LB = nb
W1 := lapackd.Workspace(lapackd.QLFactorWork(A1, conf))
lapackd.QLFactor(A1, tau1, W1, conf)
if N < 10 {
t.Logf("unblkQL(A):\n%v\n", A)
t0.SetBuf(1, tau.Len(), 1, tau.Data())
t.Logf("unblkQL.tau:\n%v\n", &t0)
t.Logf("blkQL(A):\n%v\n", A1)
t0.SetBuf(1, tau1.Len(), 1, tau1.Data())
t.Logf("blkQL.tau:\n%v\n", &t0)
}
blasd.Plus(A1, A, 1.0, -1.0, gomas.NONE)
nrm := lapackd.NormP(A1, lapackd.NORM_ONE)
t.Logf("M=%d, N=%d ||blkQL(A) - unblkQL(A)||_1: %e\n", M, N, nrm)
blasd.Axpy(tau1, tau, -1.0)
nrm = blasd.Nrm2(tau1)
t.Logf(" ||blkQL.tau - unblkQL.tau||_1: %e\n", nrm)
}
/*
* 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)
}
/* From LAPACK/dlarf.f
*
* Applies a real elementary reflector H to a real m by n matrix A,
* from either the left or the right. H is represented in the form
*
* H = I - tau * ( 1 ) * ( 1 v.T )
* ( v )
*
* where tau is a real scalar and v is a real vector.
*
* If tau = 0, then H is taken to be the unit cmat.
*
* A is /a1\ a1 := a1 - w1
* \A2/ A2 := A2 - v*w1
* w1 := tau*(a1 + A2.T*v) if side == LEFT
* := tau*(a1 + A2*v) if side == RIGHT
*
* Allocates/frees intermediate work space matrix w1.
*/
func applyHouseholder(tau, v, a1, A2 *cmat.FloatMatrix, flags int) {
tval := tau.Get(0, 0)
if tval == 0.0 {
return
}
w1 := cmat.NewCopy(a1)
if flags&gomas.LEFT != 0 {
// w1 = a1 + A2.T*v
blasd.MVMult(w1, A2, v, 1.0, 1.0, gomas.TRANSA)
} else {
// w1 = a1 + A2*v
blasd.MVMult(w1, A2, v, 1.0, 1.0, gomas.NONE)
}
// w1 = tau*w1
blasd.Scale(w1, tval)
// a1 = a1 - w1
blasd.Axpy(a1, w1, -1.0)
// A2 = A2 - v*w1
blasd.MVUpdate(A2, v, w1, -1.0)
}
/*
* This is adaptation of TRIRED_LAZY_UNB algorithm from (1).
*/
func unblkBuildTridiagUpper(A, tauq, Y, W *cmat.FloatMatrix) {
var ATL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12, A22 cmat.FloatMatrix
var YTL, YBR cmat.FloatMatrix
var Y00, y01, Y02, y11, y12, Y22 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var w12 cmat.FloatMatrix
var v0 float64
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x2(
&YTL, nil,
nil, &YBR, Y, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x1(
&tqT,
&tqB, tauq, 0, util.PBOTTOM)
k := 0
for k < n(Y) {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22, A, 1, util.PTOPLEFT)
util.Repartition2x2to3x3(&YTL,
&Y00, &y01, &Y02,
nil, &y11, &y12,
nil, nil, &Y22, Y, 1, util.PTOPLEFT)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, 1, util.PTOP)
// set temp vectors for this round
w12.SubMatrix(Y, -1, 0, 1, n(&Y02))
// ------------------------------------------------------
if n(&Y02) > 0 {
aa := blasd.Dot(&a12, &y12)
aa += blasd.Dot(&y12, &a12)
a11.Set(0, 0, a11.Get(0, 0)-aa)
// a01 := a01 - A02*y12
blasd.MVMult(&a01, &A02, &y12, -1.0, 1.0, gomas.NONE)
// a01 := a01 - Y02*a12
blasd.MVMult(&a01, &Y02, &a12, -1.0, 1.0, gomas.NONE)
// restore superdiagonal value
a12.Set(0, 0, v0)
}
// Compute householder to zero subdiagonal entries
computeHouseholderRev(&a01, &tauq1)
tauqv := tauq1.Get(0, 0)
// set sub&iagonal to unit
v0 = a01.Get(-1, 0)
a01.Set(-1, 0, 1.0)
// y01 := tauq*A00*a01
blasd.MVMultSym(&y01, &A00, &a01, tauqv, 0.0, gomas.UPPER)
// w12 := A02.T*a01
blasd.MVMult(&w12, &A02, &a01, 1.0, 0.0, gomas.TRANS)
// y01 := y01 - Y02*(A02.T*a01)
blasd.MVMult(&y01, &Y02, &w12, -tauqv, 1.0, gomas.NONE)
// w12 := Y02.T*a01
blasd.MVMult(&w12, &Y02, &a01, 1.0, 0.0, gomas.TRANS)
// y01 := y01 - A02*(Y02.T*a01)
blasd.MVMult(&y01, &A02, &w12, -tauqv, 1.0, gomas.NONE)
// beta := tauq*a01.T*y01
beta := tauqv * blasd.Dot(&a01, &y01)
// y01 := y01 - 0.5*beta*a01
blasd.Axpy(&y01, &a01, -0.5*beta)
// ------------------------------------------------------
k += 1
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
util.Continue3x3to2x2(
&YTL, nil,
nil, &YBR, &Y00, &y11, &Y22, A, util.PTOPLEFT)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PTOP)
}
// restore superdiagonal value
A.Set(m(&ATL)-1, n(&ATL), v0)
}
/*
* This is adaptation of TRIRED_LAZY_UNB algorithm from (1).
*/
func unblkBuildTridiagLower(A, tauq, Y, W *cmat.FloatMatrix) {
var ATL, ABR cmat.FloatMatrix
var A00, a10, a11, A20, a21, A22 cmat.FloatMatrix
var YTL, YBR cmat.FloatMatrix
var Y00, y10, y11, Y20, y21, Y22 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var w12 cmat.FloatMatrix
var v0 float64
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x2(
&YTL, nil,
nil, &YBR, Y, 0, 0, util.PTOPLEFT)
util.Partition2x1(
&tqT,
&tqB, tauq, 0, util.PTOP)
k := 0
for k < n(Y) {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, nil,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x2to3x3(&YTL,
&Y00, nil, nil,
&y10, &y11, nil,
&Y20, &y21, &Y22, Y, 1, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, 1, util.PBOTTOM)
// set temp vectors for this round
//w12.SetBuf(y10.Len(), 1, y10.Len(), W.Data())
w12.SubMatrix(Y, 0, 0, 1, n(&Y00))
// ------------------------------------------------------
if n(&Y00) > 0 {
aa := blasd.Dot(&a10, &y10)
aa += blasd.Dot(&y10, &a10)
a11.Set(0, 0, a11.Get(0, 0)-aa)
// a21 := a21 - A20*y10
blasd.MVMult(&a21, &A20, &y10, -1.0, 1.0, gomas.NONE)
// a21 := a21 - Y20*a10
blasd.MVMult(&a21, &Y20, &a10, -1.0, 1.0, gomas.NONE)
// restore subdiagonal value
a10.Set(0, -1, v0)
}
// Compute householder to zero subdiagonal entries
computeHouseholderVec(&a21, &tauq1)
tauqv := tauq1.Get(0, 0)
// set subdiagonal to unit
v0 = a21.Get(0, 0)
a21.Set(0, 0, 1.0)
// y21 := tauq*A22*a21
blasd.MVMultSym(&y21, &A22, &a21, tauqv, 0.0, gomas.LOWER)
// w12 := A20.T*a21
blasd.MVMult(&w12, &A20, &a21, 1.0, 0.0, gomas.TRANS)
// y21 := y21 - Y20*(A20.T*a21)
blasd.MVMult(&y21, &Y20, &w12, -tauqv, 1.0, gomas.NONE)
// w12 := Y20.T*a21
blasd.MVMult(&w12, &Y20, &a21, 1.0, 0.0, gomas.TRANS)
// y21 := y21 - A20*(Y20.T*a21)
blasd.MVMult(&y21, &A20, &w12, -tauqv, 1.0, gomas.NONE)
// beta := tauq*a21.T*y21
beta := tauqv * blasd.Dot(&a21, &y21)
// y21 := y21 - 0.5*beta*a21
blasd.Axpy(&y21, &a21, -0.5*beta)
// ------------------------------------------------------
k += 1
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x3to2x2(
&YTL, nil,
nil, &YBR, &Y00, &y11, &Y22, A, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PBOTTOM)
}
// restore subdiagonal value
A.Set(m(&ATL), n(&ATL)-1, v0)
}
/*
* Compute unblocked bidiagonal reduction for A when M >= N
*
* Diagonal and first super/sub diagonal are overwritten with the
* upper/lower bidiagonal matrix B.
*
* This computing (1-tauq*v*v.T)*A*(1-taup*u.u.T) from left to right.
*/
func unblkReduceBidiagLeft(A, tauq, taup, W *cmat.FloatMatrix) {
var ATL, ABR cmat.FloatMatrix
var A00, a11, a12t, a21, A22 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var tpT, tpB, tp0, taup1, tp2 cmat.FloatMatrix
var y21, z21 cmat.FloatMatrix
var v0 float64
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x1(
&tqT,
&tqB, tauq, 0, util.PTOP)
util.Partition2x1(
&tpT,
&tpB, taup, 0, util.PTOP)
for m(&ABR) > 0 && n(&ABR) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
nil, &a11, &a12t,
nil, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, 1, util.PBOTTOM)
util.Repartition2x1to3x1(&tpT,
&tp0,
&taup1,
&tp2, taup, 1, util.PBOTTOM)
// set temp vectors for this round
y21.SetBuf(n(&a12t), 1, n(&a12t), W.Data())
z21.SetBuf(m(&a21), 1, m(&a21), W.Data()[y21.Len():])
// ------------------------------------------------------
// Compute householder to zero subdiagonal entries
computeHouseholder(&a11, &a21, &tauq1)
// y21 := a12 + A22.T*a21
blasd.Axpby(&y21, &a12t, 1.0, 0.0)
blasd.MVMult(&y21, &A22, &a21, 1.0, 1.0, gomas.TRANSA)
// a12t := a12t - tauq*y21
tauqv := tauq1.Get(0, 0)
blasd.Axpy(&a12t, &y21, -tauqv)
// Compute householder to zero elements above 1st superdiagonal
computeHouseholderVec(&a12t, &taup1)
v0 = a12t.Get(0, 0)
a12t.Set(0, 0, 1.0)
taupv := taup1.Get(0, 0)
// [v == a12t, u == a21]
beta := blasd.Dot(&y21, &a12t)
// z21 := tauq*beta*u
blasd.Axpby(&z21, &a21, tauqv*beta, 0.0)
// z21 := A22*v - z21
blasd.MVMult(&z21, &A22, &a12t, 1.0, -1.0, gomas.NONE)
// A22 := A22 - tauq*u*y21
blasd.MVUpdate(&A22, &a21, &y21, -tauqv)
// A22 := A22 - taup*z21*v
blasd.MVUpdate(&A22, &z21, &a12t, -taupv)
a12t.Set(0, 0, v0)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PBOTTOM)
util.Continue3x1to2x1(
&tpT,
&tpB, &tp0, &taup1, taup, util.PBOTTOM)
}
}
/*
* This is adaptation of BIRED_LAZY_UNB algorithm from (1).
*
* Z matrix accumulates updates of row transformations i.e. first
* Householder that zeros off diagonal entries on row. Vector z21
* is updates for current round, Z20 are already accumulated updates.
* Vector z21 updates a12 before next transformation.
*
* Y matrix accumulates updates on column tranformations ie Householder
* that zeros elements below sub-diagonal. Vector y21 is updates for current
* round, Y20 are already accumulated updates. Vector y21 updates
* a21 befor next transformation.
*
* Z, Y matrices upper trigonal part is not needed, temporary vector
* w00 that has maximum length of n(Y) is placed on the last column of
* Z matrix on each iteration.
*/
func unblkBuildBidiagRight(A, tauq, taup, Y, Z *cmat.FloatMatrix) {
var ATL, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a10, a11, a12t, A20, a21, A22 cmat.FloatMatrix
var YTL, YBR, ZTL, ZBR cmat.FloatMatrix
var Y00, y10, Y20, y11, y21, Y22 cmat.FloatMatrix
var Z00, z10, Z20, z11, z21, Z22 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var tpT, tpB, tp0, taup1, tp2 cmat.FloatMatrix
var w00 cmat.FloatMatrix
var v0 float64
// Y is workspace for building updates for first Householder.
// And Z is space for build updates for second Householder
// Y is n(A)-2,nb and Z is m(A)-1,nb
util.Partition2x2(
&ATL, nil,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x2(
&YTL, nil,
nil, &YBR, Y, 0, 0, util.PTOPLEFT)
util.Partition2x2(
&ZTL, nil,
nil, &ZBR, Z, 0, 0, util.PTOPLEFT)
util.Partition2x1(
&tqT,
&tqB, tauq, 0, util.PTOP)
util.Partition2x1(
&tpT,
&tpB, taup, 0, util.PTOP)
k := 0
for k < n(Y) {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
&a10, &a11, &a12t,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x2to3x3(&YTL,
&Y00, nil, nil,
&y10, &y11, nil,
&Y20, &y21, &Y22, Y, 1, util.PBOTTOMRIGHT)
util.Repartition2x2to3x3(&ZTL,
&Z00, nil, nil,
&z10, &z11, nil,
&Z20, &z21, &Z22, Z, 1, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, 1, util.PBOTTOM)
util.Repartition2x1to3x1(&tpT,
&tp0,
&taup1,
&tp2, taup, 1, util.PBOTTOM)
// set temp vectors for this round,
w00.SubMatrix(Z, 0, n(Z)-1, m(&A02), 1)
// ------------------------------------------------------
// u10 == a10, U20 == A20, u21 == a21,
// v10 == a01, V20 == A02, v21 == a12t
if n(&Y20) > 0 {
// a11 := a11 - u10t*z10 - y10*v10
aa := blasd.Dot(&a10, &z10)
aa += blasd.Dot(&y10, &a01)
a11.Set(0, 0, a11.Get(0, 0)-aa)
// a12t := a12t - V20*z10 - Z20*u10
blasd.MVMult(&a12t, &A02, &y10, -1.0, 1.0, gomas.TRANS)
blasd.MVMult(&a12t, &Z20, &a10, -1.0, 1.0, gomas.NONE)
// a21 := a21 - Y20*v10 - U20*z10
blasd.MVMult(&a21, &Y20, &a01, -1.0, 1.0, gomas.NONE)
blasd.MVMult(&a21, &A20, &z10, -1.0, 1.0, gomas.NONE)
// here restore bidiagonal entry
a10.Set(0, -1, v0)
}
// Compute householder to zero superdiagonal entries
computeHouseholder(&a11, &a12t, &taup1)
taupv := taup1.Get(0, 0)
// y21 := a21 + A22*v21 - Y20*U20.T*v21 - V20*Z20.T*v21
blasd.Axpby(&y21, &a21, 1.0, 0.0)
blasd.MVMult(&y21, &A22, &a12t, 1.0, 1.0, gomas.NONE)
// w00 := U20.T*v21 [= A02*a12t]
blasd.MVMult(&w00, &A02, &a12t, 1.0, 0.0, gomas.NONE)
// y21 := y21 - U20*w00 [U20 == A20]
blasd.MVMult(&y21, &Y20, &w00, -1.0, 1.0, gomas.NONE)
// w00 := Z20.T*v21
blasd.MVMult(&w00, &Z20, &a12t, 1.0, 0.0, gomas.TRANS)
// y21 := y21 - V20*w00 [V20 == A02.T]
blasd.MVMult(&y21, &A20, &w00, -1.0, 1.0, gomas.NONE)
// a21 := a21 - taup*y21
blasd.Scale(&y21, taupv)
blasd.Axpy(&a21, &y21, -1.0)
// Compute householder to zero elements below 1st subdiagonal
computeHouseholderVec(&a21, &tauq1)
v0 = a21.Get(0, 0)
a21.Set(0, 0, 1.0)
tauqv := tauq1.Get(0, 0)
// z21 := tauq*(A22*y - V20*Y20.T*u - Z20*U20.T*u - beta*v)
// [v == a12t, u == a21]
beta := blasd.Dot(&y21, &a21)
// z21 := beta*v
blasd.Axpby(&z21, &a12t, beta, 0.0)
// w00 = Y20.T*u
blasd.MVMult(&w00, &Y20, &a21, 1.0, 0.0, gomas.TRANS)
// z21 = z21 + V20*w00 == A02.T*w00
blasd.MVMult(&z21, &A02, &w00, 1.0, 1.0, gomas.TRANS)
// w00 := U20.T*u (U20.T == A20.T)
blasd.MVMult(&w00, &A20, &a21, 1.0, 0.0, gomas.TRANS)
// z21 := z21 + Z20*w00
blasd.MVMult(&z21, &Z20, &w00, 1.0, 1.0, gomas.NONE)
// z21 := -tauq*z21 + tauq*A22*v
blasd.MVMult(&z21, &A22, &a21, tauqv, -tauqv, gomas.TRANS)
// ------------------------------------------------------
k += 1
util.Continue3x3to2x2(
&ATL, nil,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x3to2x2(
&YTL, nil,
nil, &YBR, &Y00, &y11, &Y22, Y, util.PBOTTOMRIGHT)
util.Continue3x3to2x2(
&ZTL, nil,
nil, &ZBR, &Z00, &z11, &Z22, Z, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PBOTTOM)
util.Continue3x1to2x1(
&tpT,
&tpB, &tp0, &taup1, taup, util.PBOTTOM)
}
// restore
ABL.Set(0, -1, v0)
}