/*
* 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
}
func TestPartition2D(t *testing.T) {
var ATL, ATR, ABL, ABR, As cmat.FloatMatrix
var A00, a01, A02, a10, a11, a12, A20, a21, A22 cmat.FloatMatrix
csource := cmat.NewFloatConstSource(1.0)
A := cmat.NewMatrix(6, 6)
As.SubMatrix(A, 1, 1, 4, 4)
As.SetFrom(csource)
Partition2x2(&ATL, &ATR, &ABL, &ABR, &As, 0, 0, PTOPLEFT)
t.Logf("ATL:\n%v\n", &ATL)
t.Logf("n(ATL)=%d, n(As)=%d\n", n(&ATL), n(&As))
k := 0
for n(&ATL) < n(&As) && k < n(&As) {
Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
&a10, &a11, &a12,
&A20, &a21, &A22, &As, 1, PBOTTOMRIGHT)
t.Logf("n(A00)=%d, n(a01)=%d, n(A02)=%d\n", n(&A00), n(&a01), n(&A02))
t.Logf("n(a10)=%d, n(a11)=%d, n(a12)=%d\n", n(&a10), n(&a11), n(&a12))
t.Logf("n(A20)=%d, n(a21)=%d, n(A22)=%d\n", n(&A20), n(&a21), n(&A22))
//t.Logf("n(a12)=%d [%d], n(a11)=%d\n", n(&a12), a12.Len(), a11.Len())
a11.Set(0, 0, a11.Get(0, 0)+1.0)
addConst(&a21, -2.0)
Continue3x3to2x2(&ATL, &ATR, &ABL, &ABR, &A00, &a11, &A22, &As, PBOTTOMRIGHT)
t.Logf("n(ATL)=%d, n(As)=%d\n", n(&ATL), n(&As))
k += 1
}
t.Logf("A:\n%v\n", A)
}
// unblocked LU decomposition w/o pivots, FLAME LU nopivots variant 5
func unblockedLUnoPiv(A *cmat.FloatMatrix, conf *gomas.Config) *gomas.Error {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a10, a11, a12, A20, a21, A22 cmat.FloatMatrix
var err *gomas.Error = nil
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
for m(&ATL) < m(A) {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
&a10, &a11, &a12,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
// a21 = a21/a11
blasd.InvScale(&a21, a11.Get(0, 0))
// A22 = A22 - a21*a12
blasd.MVUpdate(&A22, &a21, &a12, -1.0)
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
}
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 blkReduceTridiagUpper(A, tauq, Y, W *cmat.FloatMatrix, lb int, conf *gomas.Config) {
var ATL, ABR cmat.FloatMatrix
var A00, A01, A11, A22 cmat.FloatMatrix
var YT, YB, Y0, Y1, Y2 cmat.FloatMatrix
var tqT, tqB, tq0, tauq1, tq2 cmat.FloatMatrix
var v0 float64
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x1(
&YT,
&YB, Y, 0, util.PBOTTOM)
util.Partition2x1(
&tqT,
&tqB, tauq, 1, util.PBOTTOM)
for m(&ATL)-lb > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, &A01, nil,
nil, &A11, nil,
nil, nil, &A22, A, lb, util.PTOPLEFT)
util.Repartition2x1to3x1(&YT,
&Y0,
&Y1,
&Y2, Y, lb, util.PTOP)
util.Repartition2x1to3x1(&tqT,
&tq0,
&tauq1,
&tq2, tauq, lb, util.PTOP)
// ------------------------------------------------------
unblkBuildTridiagUpper(&ATL, &tauq1, &YT, W)
// set subdiagonal entry to unit
v0 = A01.Get(-1, 0)
A01.Set(-1, 0, 1.0)
// A22 := A22 - A01*Y0.T - Y0*A01.T
blasd.Update2Sym(&A00, &A01, &Y0, -1.0, 1.0, gomas.UPPER, conf)
// restore subdiagonal entry
A01.Set(-1, 0, v0)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &A11, &A22, A, util.PTOPLEFT)
util.Continue3x1to2x1(
&YT,
&YB, &Y0, &Y1, Y, util.PTOP)
util.Continue3x1to2x1(
&tqT,
&tqB, &tq0, &tauq1, tauq, util.PTOP)
}
if m(&ATL) > 0 {
unblkReduceTridiagUpper(&ATL, &tqT, W, true)
}
}
/*
* like LAPACK/dlafrt.f
*
* Build block reflector T from HH reflector stored in TriLU(A) and coefficients
* in tau.
*
* Q = I - Y*T*Y.T; Householder H = I - tau*v*v.T
*
* T = | T z | z = -tau*T*Y.T*v
* | 0 c | c = tau
*
* Q = H(1)H(2)...H(k) building forward here.
*/
func unblkQLBlockReflector(T, A, tau *cmat.FloatMatrix) {
var ATL, ABR cmat.FloatMatrix
var A00, a01, a11, A02, a12, A22 cmat.FloatMatrix
var TTL, TBR cmat.FloatMatrix
var T00, t11, t21, T22 cmat.FloatMatrix
var tT, tB cmat.FloatMatrix
var t0, tau1, t2 cmat.FloatMatrix
util.Partition2x2(
&ATL, nil,
nil, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x2(
&TTL, nil,
nil, &TBR, T, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x1(
&tT,
&tB, tau, 0, util.PBOTTOM)
for m(&ATL) > 0 && n(&ATL) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22, A, 1, util.PTOPLEFT)
util.Repartition2x2to3x3(&TTL,
&T00, nil, nil,
nil, &t11, nil,
nil, &t21, &T22, T, 1, util.PTOPLEFT)
util.Repartition2x1to3x1(&tT,
&t0,
&tau1,
&t2, tau, 1, util.PTOP)
// --------------------------------------------------
// t11 := tau
tauval := tau1.Get(0, 0)
if tauval != 0.0 {
t11.Set(0, 0, tauval)
// t21 := -tauval*(a12.T + &A02.T*a12)
blasd.Axpby(&t21, &a12, 1.0, 0.0)
blasd.MVMult(&t21, &A02, &a01, -tauval, -tauval, gomas.TRANSA)
// t21 := T22*t01
blasd.MVMultTrm(&t21, &T22, 1.0, gomas.LOWER)
}
// --------------------------------------------------
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
util.Continue3x3to2x2(
&TTL, nil,
nil, &TBR, &T00, &t11, &T22, T, util.PTOPLEFT)
util.Continue3x1to2x1(
&tT,
&tB, &t0, &tau1, tau, util.PTOP)
}
}
/*
* like LAPACK/dlafrt.f
*
* Build block reflector T from HH reflector stored in TriLU(A) and coefficients
* in tau.
*
* Q = I - Y*T*Y.T; Householder H = I - tau*v*v.T
*
* T = | T 0 | z = -tau*T*Y.T*v
* | z c | c = tau
*
* Q = H(1)H(2)...H(k) building forward here.
*/
func unblkBlockReflectorRQ(T, A, tau *cmat.FloatMatrix) {
var ATL, ABR cmat.FloatMatrix
var A00, a10, A20, a11, a21, A22 cmat.FloatMatrix
var TTL, TBR cmat.FloatMatrix
var T00, t11, t21, T22 cmat.FloatMatrix
var tT, tB cmat.FloatMatrix
var t0, tau1, t2 cmat.FloatMatrix
util.Partition2x2(
&ATL, nil,
nil, &ABR /**/, A, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x2(
&TTL, nil,
nil, &TBR /**/, T, 0, 0, util.PBOTTOMRIGHT)
util.Partition2x1(
&tT,
&tB /**/, tau, 0, util.PBOTTOM)
for m(&ATL) > 0 && n(&ATL) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, nil,
&A20, &a21, &A22 /**/, A, 1, util.PTOPLEFT)
util.Repartition2x2to3x3(&TTL,
&T00, nil, nil,
nil, &t11, nil,
nil, &t21, &T22 /**/, T, 1, util.PTOPLEFT)
util.Repartition2x1to3x1(&tT,
&t0,
&tau1,
&t2 /**/, tau, 1, util.PTOP)
// --------------------------------------------------
// t11 := tau
tauval := tau1.Get(0, 0)
if tauval != 0.0 {
t11.Set(0, 0, tauval)
// t21 := -tauval*(a21 + A20*a10)
blasd.Axpby(&t21, &a21, 1.0, 0.0)
blasd.MVMult(&t21, &A20, &a10, -tauval, -tauval, gomas.NONE)
// t21 := T22*t21
blasd.MVMultTrm(&t21, &T22, 1.0, gomas.LOWER)
}
// --------------------------------------------------
util.Continue3x3to2x2(
&ATL, nil,
nil, &ABR /**/, &A00, &a11, &A22, A, util.PTOPLEFT)
util.Continue3x3to2x2(
&TTL, nil,
nil, &TBR /**/, &T00, &t11, &T22, T, util.PTOPLEFT)
util.Continue3x1to2x1(
&tT,
&tB /**/, &t0, &tau1, tau, util.PTOP)
}
}
/*
* like LAPACK/dlafrt.f
*
* Build block reflector T from HH reflector stored in TriLU(A) and coefficients
* in tau.
*
* Q = I - Y*T*Y.T; Householder H = I - tau*v*v.T
*
* T = | T z | z = -tau*T*Y.T*v
* | 0 c | c = tau
*
* Q = H(1)H(2)...H(k) building forward here.
*/
func unblkBlockReflectorLQ(T, A, tau *cmat.FloatMatrix) {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12, A22 cmat.FloatMatrix
var TTL, TTR, TBL, TBR cmat.FloatMatrix
var T00, t01, T02, t11, t12, T22 cmat.FloatMatrix
var tT, tB cmat.FloatMatrix
var t0, tau1, t2 cmat.FloatMatrix
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x2(
&TTL, &TTR,
&TBL, &TBR, T, 0, 0, util.PTOPLEFT)
util.Partition2x1(
&tT,
&tB, tau, 0, util.PTOP)
for m(&ABR) > 0 && n(&ABR) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x2to3x3(&TTL,
&T00, &t01, &T02,
nil, &t11, &t12,
nil, nil, &T22, T, 1, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tT,
&t0,
&tau1,
&t2, tau, 1, util.PBOTTOM)
// --------------------------------------------------
// t11 := tau
tauval := tau1.Get(0, 0)
if tauval != 0.0 {
t11.Set(0, 0, tauval)
// t01 := -tauval*(a01 + A02*a12)
blasd.Axpby(&t01, &a01, 1.0, 0.0)
blasd.MVMult(&t01, &A02, &a12, -tauval, -tauval, gomas.NONE)
// t01 := T00*t01
blasd.MVMultTrm(&t01, &T00, 1.0, gomas.UPPER)
}
// --------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x3to2x2(
&TTL, &TTR,
&TBL, &TBR, &T00, &t11, &T22, T, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tT,
&tB, &t0, &tau1, tau, util.PBOTTOM)
}
}
/*
* 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)
}
}
/*
* Unblocked QR decomposition with block reflector T.
*/
func unblockedQRT(A, T, W *cmat.FloatMatrix) *gomas.Error {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10, a11, a12, A20, a21, A22 cmat.FloatMatrix
var TTL, TTR, TBL, TBR cmat.FloatMatrix
var T00, t01, T02, t11, t12, T22, w12 cmat.FloatMatrix
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition2x2(
&TTL, &TTR,
&TBL, &TBR, T, 0, 0, util.PTOPLEFT)
for m(&ABR) > 0 && n(&ABR) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, &a12,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x2to3x3(&TTL,
&T00, &t01, &T02,
nil, &t11, &t12,
nil, nil, &T22, T, 1, util.PBOTTOMRIGHT)
// ------------------------------------------------------
computeHouseholder(&a11, &a21, &t11)
// H*[a12 A22].T
w12.SubMatrix(W, 0, 0, a12.Len(), 1)
applyHouseholder2x1(&t11, &a21, &a12, &A22, &w12, gomas.LEFT)
// update T
tauval := t11.Get(0, 0)
if tauval != 0.0 {
// t01 := -tauval*(a10.T + &A20.T*a21)
//a10.CopyTo(&t01)
blasd.Axpby(&t01, &a10, 1.0, 0.0)
blasd.MVMult(&t01, &A20, &a21, -tauval, -tauval, gomas.TRANSA)
// t01 := T00*t01
blasd.MVMultTrm(&t01, &T00, 1.0, gomas.UPPER)
}
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x3to2x2(
&TTL, &TTR,
&TBL, &TBR, &T00, &t11, &T22, T, util.PBOTTOMRIGHT)
}
return err
}
func Amax(X *cmat.FloatMatrix, confs ...*gomas.Config) float64 {
switch X.Len() {
case 0:
return 0.0
case 1:
return X.Get(0, 0)
}
ix := IAmax(X, confs...)
if ix == -1 {
return 0.0
}
return X.Get(ix, 0)
}
func TestPartition1H(t *testing.T) {
var AL, AR, A0, a1, A2 cmat.FloatMatrix
A := cmat.NewMatrix(1, 6)
Partition1x2(&AL, &AR, A, 0, PLEFT)
t.Logf("n(AL)=%d, n(AR)=%d\n", n(&AL), n(&AR))
for n(&AL) < n(A) {
addConst(&AR, 1.0)
t.Logf("n(AR)=%d; %v\n", n(&AR), &AR)
Repartition1x2to1x3(&AL, &A0, &a1, &A2, A, 1, PRIGHT)
t.Logf("n(A0)=%d, n(A2)=%d, a1=%.1f\n", n(&A0), n(&A2), a1.Get(0, 0))
Continue1x3to1x2(&AL, &AR, &A0, &a1, A, PRIGHT)
}
t.Logf("A:%v\n", A)
}
/*
* Unblocked code for generating M by N matrix Q with orthogonal columns which
* are defined as the last N columns of the product of K first elementary
* reflectors.
*
* Parameter nk is last nk elementary reflectors that are not used in computing
* the matrix Q. Parameter mk length of the first unused elementary reflectors
* First nk columns are zeroed and subdiagonal mk-nk is set to unit.
*
* Compatible with lapack.DORG2L subroutine.
*/
func unblkBuildQL(A, Tvec, W *cmat.FloatMatrix, mk, nk int, mayClear bool) {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, a10, a11, a21, A22 cmat.FloatMatrix
var tT, tB cmat.FloatMatrix
var t0, tau1, t2, w12, D cmat.FloatMatrix
// (mk, nk) = (rows, columns) of upper left partition
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, mk, nk, util.PTOPLEFT)
util.Partition2x1(
&tT,
&tB, Tvec, nk, util.PTOP)
// zero the left side
if nk > 0 && mayClear {
blasd.Scale(&ABL, 0.0)
blasd.Scale(&ATL, 0.0)
D.Diag(&ATL, nk-mk)
blasd.Add(&D, 1.0)
}
for m(&ABR) > 0 && n(&ABR) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, nil,
&a10, &a11, nil,
nil, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition2x1to3x1(&tT,
&t0,
&tau1,
&t2, Tvec, 1, util.PBOTTOM)
// ------------------------------------------------------
w12.SubMatrix(W, 0, 0, a10.Len(), 1)
applyHouseholder2x1(&tau1, &a01, &a10, &A00, &w12, gomas.LEFT)
blasd.Scale(&a01, -tau1.Get(0, 0))
a11.Set(0, 0, 1.0-tau1.Get(0, 0))
// zero bottom elements
blasd.Scale(&a21, 0.0)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue3x1to2x1(
&tT,
&tB, &t0, &tau1, Tvec, util.PBOTTOM)
}
}
/*
* Unblocked code for generating M by N matrix Q with orthogonal columns which
* are defined as the first N columns of the product of K first elementary
* reflectors.
*
* Parameters nk = n(A)-K, mk = m(A)-K define the initial partitioning of
* matrix A.
*
* Q = H(k)H(k-1)...H(1) , 0 < k <= M, where H(i) = I - tau*v*v.T
*
* Computation is ordered as H(k)*H(k-1)...*H(1)*I ie. from bottom to top.
*
* If k < M rows k+1:M are cleared and diagonal entries [k+1:M,k+1:M] are
* set to unit. Then the matrix Q is generated by right multiplying elements below
* of i'th elementary reflector H(i).
*
* Compatible to lapack.xORG2L subroutine.
*/
func unblkBuildLQ(A, Tvec, W *cmat.FloatMatrix, mk, nk int, mayClear bool) {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10, a11, a12, a21, A22 cmat.FloatMatrix
var tT, tB cmat.FloatMatrix
var t0, tau1, t2, w12, D cmat.FloatMatrix
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, mk, nk, util.PBOTTOMRIGHT)
util.Partition2x1(
&tT,
&tB, Tvec, mk, util.PBOTTOM)
// zero the bottom part
if mk > 0 && mayClear {
blasd.Scale(&ABL, 0.0)
blasd.Scale(&ABR, 0.0)
D.Diag(&ABR)
blasd.Add(&D, 1.0)
}
for m(&ATL) > 0 && n(&ATL) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, &a12,
nil, &a21, &A22, A, 1, util.PTOPLEFT)
util.Repartition2x1to3x1(&tT,
&t0,
&tau1,
&t2, Tvec, 1, util.PTOP)
// ------------------------------------------------------
w12.SubMatrix(W, 0, 0, a21.Len(), 1)
applyHouseholder2x1(&tau1, &a12, &a21, &A22, &w12, gomas.RIGHT)
blasd.Scale(&a12, -tau1.Get(0, 0))
a11.Set(0, 0, 1.0-tau1.Get(0, 0))
// zero
blasd.Scale(&a10, 0.0)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
util.Continue3x1to2x1(
&tT,
&tB, &t0, &tau1, Tvec, util.PTOP)
}
}
func findBKPivotUpper(A *cmat.FloatMatrix) (int, int) {
var r, q int
var rcol, qrow cmat.FloatMatrix
if m(A) == 1 {
return 0, 1
}
lastcol := m(A) - 1
amax := math.Abs(A.Get(lastcol, lastcol))
// column above [A.Rows()-1, A.Rows()-1]
rcol.SubMatrix(A, 0, lastcol, lastcol, 1)
r = blasd.IAmax(&rcol)
// max off-diagonal on first column at index r
rmax := math.Abs(A.Get(r, lastcol))
if amax >= bkALPHA*rmax {
// no pivoting, 1x1 diagonal
return -1, 1
}
// max off-diagonal on r'th row at index q
// a) rest of the r'th row above diagonal
qmax := 0.0
if r > 0 {
qrow.SubMatrix(A, 0, r, r, 1)
q = blasd.IAmax(&qrow)
qmax = math.Abs(A.Get(q, r /*+1*/))
}
// b) elements right of diagonal
qrow.SubMatrix(A, r, r+1, 1, lastcol-r)
q = blasd.IAmax(&qrow)
qmax2 := math.Abs(qrow.Get(0, q))
if qmax2 > qmax {
qmax = qmax2
}
if amax >= bkALPHA*rmax*(rmax/qmax) {
// no pivoting, 1x1 diagonal
return -1, 1
}
if math.Abs(A.Get(r, r)) >= bkALPHA*qmax {
// 1x1 pivoting and interchange with k, r
return r, 1
} else {
// 2x2 pivoting and interchange with k+1, r
return r, 2
}
return 0, 1
}
/*
* Apply diagonal pivot (row and column swapped) to symmetric matrix blocks.
*
* LOWER triangular; moving from top-left to bottom-right
*
* -----------------------
* | d
* | x P1 x x x P2 -- current row/col 'srcix'
* | x S2 d x x x
* | x S2 x d x x
* | x S2 x x d x
* | x P2 D2 D2 D2 P3 -- swap with row/col 'dstix'
* | x S3 x x x D3 d
* | x S3 x x x D3 x d
* (AR)
*/
func applyBKPivotSymLower(AR *cmat.FloatMatrix, srcix, dstix int) {
var s, d cmat.FloatMatrix
// S2 -- D2
s.SubMatrix(AR, srcix+1, srcix, dstix-srcix-1, 1)
d.SubMatrix(AR, dstix, srcix+1, 1, dstix-srcix-1)
blasd.Swap(&s, &d)
// S3 -- D3
s.SubMatrix(AR, dstix+1, srcix, m(AR)-dstix-1, 1)
d.SubMatrix(AR, dstix+1, dstix, m(AR)-dstix-1, 1)
blasd.Swap(&s, &d)
// swap P1 and P3
p1 := AR.Get(srcix, srcix)
p3 := AR.Get(dstix, dstix)
AR.Set(srcix, srcix, p3)
AR.Set(dstix, dstix, p1)
return
}
/*
* Apply diagonal pivot (row and column swapped) to symmetric matrix blocks.
*
* UPPER triangular; moving from bottom-right to top-left
*
* d x D3 x x x S3 x |
* d D3 x x x S3 x |
* P3 D2 D2 D2 P2 x | -- dstinx
* d x x S2 x |
* d x S2 x |
* d S2 x |
* P1 x | -- srcinx
* d |
* ----------------------
* (ABR)
*/
func applyBKPivotSymUpper(AR *cmat.FloatMatrix, srcix, dstix int) {
var s, d cmat.FloatMatrix
// AL is ATL, AR is ATR; P1 is AL[srcix, srcix];
// S2 -- D2
s.SubMatrix(AR, dstix+1, srcix, srcix-dstix-1, 1)
d.SubMatrix(AR, dstix, dstix+1, 1, srcix-dstix-1)
blasd.Swap(&s, &d)
// S3 -- D3
s.SubMatrix(AR, 0, srcix, dstix, 1)
d.SubMatrix(AR, 0, dstix, dstix, 1)
blasd.Swap(&s, &d)
// swap P1 and P3
p1 := AR.Get(srcix, srcix)
p3 := AR.Get(dstix, dstix)
AR.Set(srcix, srcix, p3)
AR.Set(dstix, dstix, p1)
return
}
func TestViews(t *testing.T) {
var As cmat.FloatMatrix
N := 7
A := cmat.NewMatrix(N, N)
zeromean := cmat.NewFloatNormSource()
A.SetFrom(zeromean)
dlast := A.Get(-1, -1)
t.Logf("A[-1,-1] = %10.3e\n", dlast)
for i := 0; i < N; i++ {
As.SubMatrix(A, i, i)
As.Set(0, 0, As.Get(0, 0)+float64(i+1))
if N < 10 {
t.Logf("A[%d,%d] = %10.3e\n", i, i, As.Get(0, 0))
}
}
ok := float64(N)+dlast == A.Get(-1, -1)
nC := int(A.Get(-1, -1) - dlast)
t.Logf("add 1.0 %d times [%10.3e to %10.3e]: %v\n", nC, dlast, A.Get(-1, -1), ok)
}
/*
* α = (1 + sqrt(17))/8
* λ = |a(r,1)| = max{|a(2,1)|, . . . , |a(m,1)|}
* if λ > 0
* if |a(1,1)| ≥ αλ
* use a11 as 1-by-1 pivot
* else
* σ = |a(p,r)| = max{|a(1,r)|,..., |a(r−1,r)|, |a(r+1,r)|,..., |a(m,r)|}
* if |a(1,1) |σ ≥ αλ^2
* use a(1,1) as 1-by-1 pivot
* else if |a(r,r)| ≥ ασ
* use a(r,r) as 1-by-1 pivot
* else
* a11 | ar1
* use -------- as 2-by-2 pivot
* ar1 | arr
* end
* end
* end
*
* -----------------------
* | d
* | x P1 x x x P2 -- current row/col 'srcix'
* | x S2 d x x x
* | x S2 x d x x
* | x S2 x x d x
* | x P2 D2 D2 D2 P3 -- swap with row/col 'dstix'
* | x S3 x x x D3 d
* | x S3 x x x D3 x d
* (AR)
*/
func findBKPivotLower(A *cmat.FloatMatrix) (int, int) {
var r, q int
var rcol, qrow cmat.FloatMatrix
if m(A) == 1 {
return 0, 1
}
amax := math.Abs(A.Get(0, 0))
// column below diagonal at [0, 0]
rcol.SubMatrix(A, 1, 0, m(A)-1, 1)
r = blasd.IAmax(&rcol) + 1
// max off-diagonal on first column at index r
rmax := math.Abs(A.Get(r, 0))
if amax >= bkALPHA*rmax {
// no pivoting, 1x1 diagonal
return 0, 1
}
// max off-diagonal on r'th row at index q
qrow.SubMatrix(A, r, 0, 1, r /*+1*/)
q = blasd.IAmax(&qrow)
qmax := math.Abs(A.Get(r, q /*+1*/))
if r < m(A)-1 {
// rest of the r'th row after diagonal
qrow.SubMatrix(A, r+1, r, m(A)-r-1, 1)
q = blasd.IAmax(&qrow)
qmax2 := math.Abs(qrow.Get(q, 0))
if qmax2 > qmax {
qmax = qmax2
}
}
if amax >= bkALPHA*rmax*(rmax/qmax) {
// no pivoting, 1x1 diagonal
return 0, 1
}
if math.Abs(A.Get(r, r)) >= bkALPHA*qmax {
// 1x1 pivoting and interchange with k, r
return r, 1
} else {
// 2x2 pivoting and interchange with k+1, r
return r, 2
}
return 0, 1
}
/*
* Compute A[i:i+1,j:j+nc] = G(c,s)*A[i:i+1,j:j+nc]
*
* Applies Givens rotation to nc columns on rows r1,r2 of A starting from col.
*/
func ApplyGivensLeft(A *cmat.FloatMatrix, r1, r2, col, ncol int, cos, sin float64) {
if col >= n(A) {
return
}
if r1 == r2 {
// one row
for k := col; k < col+ncol; k++ {
v0 := A.Get(r1, k)
A.Set(r1, k, cos*v0)
}
return
}
for k := col; k < col+ncol; k++ {
v0 := A.Get(r1, k)
v1 := A.Get(r2, k)
y0 := cos*v0 + sin*v1
y1 := cos*v1 - sin*v0
A.Set(r1, k, y0)
A.Set(r2, k, y1)
}
}
/*
* Compute A[i:i+nr,c1;c2] = A[i:i+nr,c1;c2]*G(c,s)
*
* Applies Givens rotation to nr rows of columns c1, c2 of A starting at row.
*
*/
func ApplyGivensRight(A *cmat.FloatMatrix, c1, c2, row, nrow int, cos, sin float64) {
if row >= m(A) {
return
}
if c1 == c2 {
// one column
for k := row; k < row+nrow; k++ {
v0 := A.Get(k, c1)
A.Set(k, c1, cos*v0)
}
return
}
for k := row; k < row+nrow; k++ {
v0 := A.Get(k, c1)
v1 := A.Get(k, c2)
y0 := cos*v0 + sin*v1
y1 := cos*v1 - sin*v0
A.Set(k, c1, y0)
A.Set(k, c2, y1)
}
}
/*
* Apply diagonal pivot (row and column swapped) to symmetric matrix blocks.
* AR[0,0] is on diagonal and AL is block to the left of diagonal and AR the
* triangular diagonal block. Need to swap row and column.
*
* LOWER triangular; moving from top-left to bottom-right
*
* d
* x d
* x x d |
* --------------------------
* S1 S1 S1 | P1 x x x P2 -- current row
* x x x | S2 d x x x
* x x x | S2 x d x x
* x x x | S2 x x d x
* D1 D1 D1 | P2 D2 D2 D2 P3 -- swap with row 'index'
* x x x | S3 x x x D3 d
* x x x | S3 x x x D3 x d
* (ABL) (ABR)
*
* UPPER triangular; moving from bottom-right to top-left
*
* (ATL) (ATR)
* d x x D3 x x x | S3 x x
* d x D3 x x x | S3 x x
* d D3 x x x | S3 x x
* P3 D2 D2 D2| P2 D1 D1
* d x x | S2 x x
* d x | S2 x x
* d | S2 x x
* -----------------------------
* | P1 S1 S1
* | d x
* | d
* (ABR)
*/
func applyPivotSym(AL, AR *cmat.FloatMatrix, index int, flags int) {
var s, d cmat.FloatMatrix
lr, lc := AL.Size()
rr, rc := AR.Size()
if flags&gomas.LOWER != 0 {
// AL is [ABL]; AR is [ABR]; P1 is AR[0,0], P2 is AR[index, 0]
// S1 -- D1
s.SubMatrix(AL, 0, 0, 1, lc)
d.SubMatrix(AL, index, 0, 1, lc)
blasd.Swap(&s, &d)
// S2 -- D2
s.SubMatrix(AR, 1, 0, index-1, 1)
d.SubMatrix(AR, index, 1, 1, index-1)
blasd.Swap(&s, &d)
// S3 -- D3
s.SubMatrix(AR, index+1, 0, rr-index-1, 1)
d.SubMatrix(AR, index+1, index, rr-index-1, 1)
blasd.Swap(&s, &d)
// swap P1 and P3
p1 := AR.Get(0, 0)
p3 := AR.Get(index, index)
AR.Set(0, 0, p3)
AR.Set(index, index, p1)
return
}
if flags&gomas.UPPER != 0 {
// AL is merged from [ATL, ATR], AR is [ABR]; P1 is AR[0, 0]; P2 is AL[index, -1]
colno := lc - rc
// S1 -- D1; S1 is on the first row of AR
s.SubMatrix(AR, 0, 1, 1, rc-1)
d.SubMatrix(AL, index, colno+1, 1, rc-1)
blasd.Swap(&s, &d)
// S2 -- D2
s.SubMatrix(AL, index+1, colno, lr-index-2, 1)
d.SubMatrix(AL, index, index+1, 1, colno-index-1)
blasd.Swap(&s, &d)
// S3 -- D3
s.SubMatrix(AL, 0, index, index, 1)
d.SubMatrix(AL, 0, colno, index, 1)
blasd.Swap(&s, &d)
//fmt.Printf("3, AR=%v\n", AR)
// swap P1 and P3
p1 := AR.Get(0, 0)
p3 := AL.Get(index, index)
AR.Set(0, 0, p3)
AL.Set(index, index, p1)
return
}
}
func MVMultTrm(X, A *cmat.FloatMatrix, alpha float64, bits int, confs ...*gomas.Config) *gomas.Error {
ar, ac := A.Size()
xr, xc := X.Size()
if ar*ac == 0 {
return nil
}
if xr != 1 && xc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVMultTrm")
}
nx := X.Len()
if ac != nx || ar != ac {
return gomas.NewError(gomas.ESIZE, "MVMultTrm")
}
if ar == 1 {
vscal(X, alpha*A.Get(0, 0), nx)
} else {
trmv(X, A, alpha, bits, nx)
}
return nil
}
/*
* Apply diagonal pivot (row and column swapped) to symmetric matrix blocks.
* AR[0,0] is on diagonal and AL is block to the left of diagonal and AR the
* triangular diagonal block. Need to swap row and column.
*
* LOWER triangular; moving from top-left to bottom-right
*
* d
* x d |
* --------------------------
* x x | d
* S1 S1| S1 P1 x x x P2 -- current row/col 'srcix'
* x x | x S2 d x x x
* x x | x S2 x d x x
* x x | x S2 x x d x
* D1 D1| D1 P2 D2 D2 D2 P3 -- swap with row/col 'dstix'
* x x | x S3 x x x D3 d
* x x | x S3 x x x D3 x d
* (ABL) (ABR)
*
* UPPER triangular; moving from bottom-right to top-left
*
* (ATL) (ATR)
* d x x D3 x x x S3 x | x
* d x D3 x x x S3 x | x
* d D3 x x x S3 x | x
* P3 D2 D2 D2 P2 D1| D1 -- dstinx
* d x x S2 x | x
* d x S2 x | x
* d S2 x | x
* P1 S1| S1 -- srcinx
* d | x
* -----------------------------
* | d
* (ABR)
*/
func applyPivotSym2(AL, AR *cmat.FloatMatrix, srcix, dstix int, flags int) {
var s, d cmat.FloatMatrix
_, lc := AL.Size()
rr, rc := AR.Size()
if flags&gomas.LOWER != 0 {
// AL is [ABL]; AR is [ABR]; P1 is AR[0,0], P2 is AR[index, 0]
// S1 -- D1
AL.SubMatrix(&s, srcix, 0, 1, lc)
AL.SubMatrix(&d, dstix, 0, 1, lc)
blasd.Swap(&s, &d)
if srcix > 0 {
AR.SubMatrix(&s, srcix, 0, 1, srcix)
AR.SubMatrix(&d, dstix, 0, 1, srcix)
blasd.Swap(&s, &d)
}
// S2 -- D2
AR.SubMatrix(&s, srcix+1, srcix, dstix-srcix-1, 1)
AR.SubMatrix(&d, dstix, srcix+1, 1, dstix-srcix-1)
blasd.Swap(&s, &d)
// S3 -- D3
AR.SubMatrix(&s, dstix+1, srcix, rr-dstix-1, 1)
AR.SubMatrix(&d, dstix+1, dstix, rr-dstix-1, 1)
blasd.Swap(&s, &d)
// swap P1 and P3
p1 := AR.Get(srcix, srcix)
p3 := AR.Get(dstix, dstix)
AR.Set(srcix, srcix, p3)
AR.Set(dstix, dstix, p1)
return
}
if flags&gomas.UPPER != 0 {
// AL is ATL, AR is ATR; P1 is AL[srcix, srcix];
// S1 -- D1;
AR.SubMatrix(&s, srcix, 0, 1, rc)
AR.SubMatrix(&d, dstix, 0, 1, rc)
blasd.Swap(&s, &d)
if srcix < lc-1 {
// not the corner element
AL.SubMatrix(&s, srcix, srcix+1, 1, srcix)
AL.SubMatrix(&d, dstix, srcix+1, 1, srcix)
blasd.Swap(&s, &d)
}
// S2 -- D2
AL.SubMatrix(&s, dstix+1, srcix, srcix-dstix-1, 1)
AL.SubMatrix(&d, dstix, dstix+1, 1, srcix-dstix-1)
blasd.Swap(&s, &d)
// S3 -- D3
AL.SubMatrix(&s, 0, srcix, dstix, 1)
AL.SubMatrix(&d, 0, dstix, dstix, 1)
blasd.Swap(&s, &d)
//fmt.Printf("3, AR=%v\n", AR)
// swap P1 and P3
p1 := AR.Get(0, 0)
p3 := AL.Get(dstix, dstix)
AR.Set(srcix, srcix, p3)
AL.Set(dstix, dstix, p1)
return
}
}
func trdsecEigenBuildInplace(Q, z *cmat.FloatMatrix) {
var QTL, QBR, Q00, q11, q12, q21, Q22, qi cmat.FloatMatrix
var zk0, zk1, dk0, dk1 float64
util.Partition2x2(
&QTL, nil,
nil, &QBR /**/, Q, 0, 0, util.PTOPLEFT)
for m(&QBR) > 0 {
util.Repartition2x2to3x3(&QTL,
&Q00, nil, nil,
nil, &q11, &q12,
nil, &q21, &Q22 /**/, Q, 1, util.PBOTTOMRIGHT)
//---------------------------------------------------------------
k := m(&Q00)
zk0 = z.GetAt(k)
dk0 = q11.Get(0, 0)
q11.Set(0, 0, zk0/dk0)
for i := 0; i < q12.Len(); i++ {
zk1 = z.GetAt(k + i + 1)
dk0 = q12.GetAt(i)
dk1 = q21.GetAt(i)
q12.SetAt(i, zk0/dk1)
q21.SetAt(i, zk1/dk0)
}
//---------------------------------------------------------------
util.Continue3x3to2x2(
&QTL, nil,
nil, &QBR /**/, &Q00, &q11, &Q22 /**/, Q, util.PBOTTOMRIGHT)
}
// scale column eigenvectors
for k := 0; k < z.Len(); k++ {
qi.Column(Q, k)
t := blasd.Nrm2(&qi)
blasd.InvScale(&qi, t)
}
}
/*
* Apply elementary Householder reflector v to matrix A2.
*
* H = I - tau*v*v.t;
*
* RIGHT: A = A*H = A - tau*A*v*v.T = A - tau*w1*v.T
* LEFT: A = H*A = A - tau*v*v.T*A = A - tau*v*A.T*v = A - tau*v*w1
*/
func applyHouseholder1x1(tau, v, A2, w1 *cmat.FloatMatrix, flags int) *gomas.Error {
var err *gomas.Error = nil
tval := tau.Get(0, 0)
if tval == 0.0 {
return nil
}
if flags&gomas.LEFT != 0 {
// w1 = A2.T*v
err = blasd.MVMult(w1, A2, v, 1.0, 0.0, gomas.TRANSA)
if err == nil {
// A2 = A2 - tau*v*w1; m(A2) == len(v) && n(A2) == len(w1)
err = blasd.MVUpdate(A2, v, w1, -tval)
}
} else {
// w1 = A2*v
err = blasd.MVMult(w1, A2, v, 1.0, 0.0, gomas.NONE)
if err == nil {
// A2 = A2 - tau*w1*v; m(A2) == len(w1) && n(A2) == len(v)
err = blasd.MVUpdate(A2, w1, v, -tval)
}
}
return err
}
func unblockedLowerCHOL(A *cmat.FloatMatrix, flags int, nr int) (err *gomas.Error) {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10, a11, A20, a21, A22 cmat.FloatMatrix
err = nil
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
for m(&ATL) < m(A) {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, nil,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
// a11 = sqrt(a11)
aval := a11.Get(0, 0)
if aval < 0.0 {
if err == nil {
err = gomas.NewError(gomas.ENEGATIVE, "DecomposeCHOL", m(&ATL)+nr)
}
} else {
a11.Set(0, 0, math.Sqrt(aval))
}
// a21 = a21/a11
blasd.InvScale(&a21, a11.Get(0, 0))
// A22 = A22 - a21*a21' (SYR)
blasd.MVUpdateSym(&A22, &a21, -1.0, gomas.LOWER)
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
}
return
}
/* 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)
}
/* From LAPACK/dlarfg.f
*
* Generates a real elementary reflector H of order n, such
* that
*
* H * ( alpha ) = ( beta ), H**T * H = I.
* ( x ) ( 0 )
*
* where alpha and beta are scalars, and x is an (n-1)-element real
* vector. 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 (n-1)-element
* vector.
*
* If the elements of x are all zero, then tau = 0 and H is taken to be
* the unit cmat.
*
* Otherwise 1 <= tau <= 2.
*/
func computeHouseholder(a11, x, tau *cmat.FloatMatrix) {
// norm_x2 = ||x||_2
norm_x2 := blasd.Nrm2(x)
if norm_x2 == 0.0 {
tau.Set(0, 0, 0.0)
return
}
alpha := a11.Get(0, 0)
sign := 1.0
if math.Signbit(alpha) {
sign = -1.0
}
// beta = -(alpha / |alpha|) * ||alpha x||
// = -sign(alpha) * sqrt(alpha**2, norm_x2**2)
beta := -sign * sqrtX2Y2(alpha, norm_x2)
// x = x /(a11 - beta)
blasd.InvScale(x, alpha-beta)
tau.Set(0, 0, (beta-alpha)/beta)
a11.Set(0, 0, beta)
}
/*
* Computes upper Hessenberg reduction of N-by-N matrix A using unblocked
* algorithm as described in (1).
*
* Hessengerg reduction: A = Q.T*B*Q, Q unitary, B upper Hessenberg
* Q = H(0)*H(1)*...*H(k) where H(k) is k'th Householder reflector.
*
* Compatible with lapack.DGEHD2.
*/
func unblkHessGQvdG(A, Tvec, W *cmat.FloatMatrix, row int) {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a11, a21, A22 cmat.FloatMatrix
var AL, AR, A0, a1, A2 cmat.FloatMatrix
var tT, tB cmat.FloatMatrix
var t0, tau1, t2, w12, v1 cmat.FloatMatrix
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, row, 0, util.PTOPLEFT)
util.Partition1x2(
&AL, &AR, A, 0, util.PLEFT)
util.Partition2x1(
&tT,
&tB, Tvec, 0, util.PTOP)
v1.SubMatrix(W, 0, 0, m(A), 1)
for m(&ABR) > 1 && n(&ABR) > 0 {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
nil, &a11, nil,
nil, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
util.Repartition1x2to1x3(&AL,
&A0, &a1, &A2, A, 1, util.PRIGHT)
util.Repartition2x1to3x1(&tT,
&t0,
&tau1,
&t2, Tvec, 1, util.PBOTTOM)
// ------------------------------------------------------
// a21 = [beta; H(k)].T
computeHouseholderVec(&a21, &tau1)
tauval := tau1.Get(0, 0)
beta := a21.Get(0, 0)
a21.Set(0, 0, 1.0)
// v1 := A2*a21
blasd.MVMult(&v1, &A2, &a21, 1.0, 0.0, gomas.NONE)
// A2 := A2 - tau*v1*a21 (A2 := A2*H(k))
blasd.MVUpdate(&A2, &v1, &a21, -tauval)
w12.SubMatrix(W, 0, 0, n(&A22), 1)
// w12 := a21.T*A22 = A22.T*a21
blasd.MVMult(&w12, &A22, &a21, 1.0, 0.0, gomas.TRANS)
// A22 := A22 - tau*a21*w12 (A22 := H(k)*A22)
blasd.MVUpdate(&A22, &a21, &w12, -tauval)
a21.Set(0, 0, beta)
// ------------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue1x3to1x2(
&AL, &AR, &A0, &a1, A, util.PRIGHT)
util.Continue3x1to2x1(
&tT,
&tB, &t0, &tau1, Tvec, util.PBOTTOM)
}
}