Пример #1
0
/*
 * 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 matrix.
 *
 * 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 applyHHTo2x1(tau, v, a1, A2, w1 *matrix.FloatMatrix, flags Flags) {

	tval := tau.GetAt(0, 0)
	if tval == 0.0 {
		return
	}

	// maybe with Scale(0.0), Axpy(w1, a1, 1.0)
	a1.CopyTo(w1)
	if flags&LEFT != 0 {
		// w1 = a1 + A2.T*v
		MVMult(w1, A2, v, 1.0, 1.0, TRANSA)
	} else {
		// w1 = a1 + A2*v
		MVMult(w1, A2, v, 1.0, 1.0, NOTRANS)
	}

	// w1 = tau*w1
	Scale(w1, tval)

	// a1 = a1 - w1
	a1.Minus(w1)

	// A2 = A2 - v*w1
	if flags&LEFT != 0 {
		MVRankUpdate(A2, v, w1, -1.0)
	} else {
		MVRankUpdate(A2, w1, v, -1.0)
	}
}
Пример #2
0
func blkUpperLDL(A, W *matrix.FloatMatrix, p *pPivots, nb int) (err error) {
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, A01, A02, A11, A12, A22 matrix.FloatMatrix
	var D1, wrk matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots

	err = nil
	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pBOTTOMRIGHT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pBOTTOM)

	for ATL.Rows() > 0 {
		repartition2x2to3x3(&ATL,
			&A00, &A01, &A02,
			nil, &A11, &A12,
			nil, nil, &A22, A, nb, pTOPLEFT)
		repartPivot2x1to3x1(&pT,
			&p0, &p1, &p2 /**/, p, nb, pTOP)

		// --------------------------------------------------------

		// A11 = LDL(A11)
		err = unblkUpperLDL(&A11, &p1)
		if err != nil {
			return
		}
		applyColPivots(&A01, &p1, 0, BACKWARD)
		applyRowPivots(&A12, &p1, 0, BACKWARD)
		scalePivots(&p1, ATL.Rows()-A11.Rows())

		A11.Diag(&D1)

		// A01 = A01*A11.-T
		SolveTrm(&A01, &A11, 1.0, UPPER|UNIT|RIGHT|TRANSA)
		// A01 = A01*D1.-1
		SolveDiag(&A01, &D1, RIGHT)

		// W = D1*U01.T = U01*D1
		W.SubMatrix(&wrk, 0, 0, A01.Rows(), nb)
		A01.CopyTo(&wrk)
		MultDiag(&wrk, &D1, RIGHT)

		// A00 = A00 - U01*D1*U01.T = A22 - U01*W.T
		UpdateTrm(&A00, &A01, &wrk, -1.0, 1.0, UPPER|TRANSB)

		// ---------------------------------------------------------

		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &A11, &A22, A, pTOPLEFT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pTOP)
	}
	return
}
Пример #3
0
/*
 * 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 unblkQRBlockReflector(T, A, tau *matrix.FloatMatrix) {
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a10, a11, A20, a21, A22 matrix.FloatMatrix
	var TTL, TTR, TBL, TBR matrix.FloatMatrix
	var T00, t01, T02, t11, t12, T22 matrix.FloatMatrix
	var tT, tB matrix.FloatMatrix
	var t0, tau1, t2 matrix.FloatMatrix

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partition2x2(
		&TTL, &TTR,
		&TBL, &TBR, T, 0, 0, pTOPLEFT)
	partition2x1(
		&tT,
		&tB, tau, 0, pTOP)

	for ABR.Rows() > 0 && ABR.Cols() > 0 {
		repartition2x2to3x3(&ATL,
			&A00, nil, nil,
			&a10, &a11, nil,
			&A20, &a21, &A22, A, 1, pBOTTOMRIGHT)
		repartition2x2to3x3(&TTL,
			&T00, &t01, &T02,
			nil, &t11, &t12,
			nil, nil, &T22, T, 1, pBOTTOMRIGHT)
		repartition2x1to3x1(&tT,
			&t0,
			&tau1,
			&t2, tau, 1, pBOTTOM)
		// --------------------------------------------------

		// t11 := tau
		tauval := tau1.GetAt(0, 0)
		if tauval != 0.0 {
			t11.SetAt(0, 0, tauval)

			// t01 := a10.T + &A20.T*a21
			a10.CopyTo(&t01)
			MVMult(&t01, &A20, &a21, -tauval, -tauval, TRANSA)
			// t01 := T00*t01
			MVMultTrm(&t01, &T00, UPPER)
			//t01.Scale(-tauval)
		}

		// --------------------------------------------------
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
		continue3x3to2x2(
			&TTL, &TTR,
			&TBL, &TBR, &T00, &t11, &T22, T, pBOTTOMRIGHT)
		continue3x1to2x1(
			&tT,
			&tB, &t0, &tau1, tau, pBOTTOM)
	}
}
Пример #4
0
/*
 * Unblocked QR decomposition with block reflector T.
 */
func unblockedQRT(A, T *matrix.FloatMatrix) {
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a10, a11, a12, A20, a21, A22 matrix.FloatMatrix
	var TTL, TTR, TBL, TBR matrix.FloatMatrix
	var T00, t01, T02, t11, t12, T22 matrix.FloatMatrix

	//As.SubMatrixOf(A, 0, 0, mlen, nb)
	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partition2x2(
		&TTL, &TTR,
		&TBL, &TBR, T, 0, 0, pTOPLEFT)

	for ABR.Rows() > 0 && ABR.Cols() > 0 {
		repartition2x2to3x3(&ATL,
			&A00, nil, nil,
			&a10, &a11, &a12,
			&A20, &a21, &A22, A, 1, pBOTTOMRIGHT)
		repartition2x2to3x3(&TTL,
			&T00, &t01, &T02,
			nil, &t11, &t12,
			nil, nil, &T22, T, 1, pBOTTOMRIGHT)

		// ------------------------------------------------------

		computeHouseholder(&a11, &a21, &t11, LEFT)

		// H*[a12 A22].T
		applyHouseholder(&t11, &a21, &a12, &A22, LEFT)

		// update T
		tauval := t11.GetAt(0, 0)
		if tauval != 0.0 {
			// t01 := -tauval*(a10.T + &A20.T*a21)
			a10.CopyTo(&t01)
			MVMult(&t01, &A20, &a21, -tauval, -tauval, TRANSA)
			// t01 := T00*t01
			MVMultTrm(&t01, &T00, UPPER)
		}

		// ------------------------------------------------------
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
		continue3x3to2x2(
			&TTL, &TTR,
			&TBL, &TBR, &T00, &t11, &T22, T, pBOTTOMRIGHT)
	}
}
Пример #5
0
func blkUpperLDLnoPiv(A, W *matrix.FloatMatrix, nb int) (err error) {
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, A01, A02, A11, A12, A22 matrix.FloatMatrix
	var D1, wrk matrix.FloatMatrix

	err = nil
	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pBOTTOMRIGHT)

	for ATL.Rows() > 0 {
		repartition2x2to3x3(&ATL,
			&A00, &A01, &A02,
			nil, &A11, &A12,
			nil, nil, &A22, A, nb, pTOPLEFT)

		// --------------------------------------------------------

		// A11 = LDL(A11)
		unblkUpperLDLnoPiv(&A11)
		A11.Diag(&D1)

		// A01 = A01*A11.-T
		SolveTrm(&A01, &A11, 1.0, UPPER|UNIT|RIGHT|TRANSA)
		// A01 = A01*D1.-1
		SolveDiag(&A01, &D1, RIGHT)

		// W = D1*U01.T = U01*D1
		W.SubMatrix(&wrk, 0, 0, A01.Rows(), nb)
		A01.CopyTo(&wrk)
		MultDiag(&wrk, &D1, RIGHT)

		// A00 = A00 - U01*D1*U01.T = A22 - U01*W.T
		UpdateTrm(&A00, &A01, &wrk, -1.0, 1.0, UPPER|TRANSB)

		// ---------------------------------------------------------

		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &A11, &A22, A, pTOPLEFT)
	}
	return
}
Пример #6
0
func findAndBuildPivot(AL, AR, WL, WR *matrix.FloatMatrix, k int) int {
	var dg, acol, wcol, wrow matrix.FloatMatrix

	// updated diagonal values on last column of workspace
	WR.SubMatrix(&dg, 0, WR.Cols()-1, AR.Rows(), 1)

	// find on-diagonal maximun value
	dmax := IAMax(&dg)
	//fmt.Printf("dmax=%d, val=%e\n", dmax, dg.GetAt(dmax, 0))

	// copy to first column of WR and update with factorized columns
	WR.SubMatrix(&wcol, 0, 0, WR.Rows(), 1)
	if dmax == 0 {
		AR.SubMatrix(&acol, 0, 0, AR.Rows(), 1)
		acol.CopyTo(&wcol)
	} else {
		AR.SubMatrix(&acol, dmax, 0, 1, dmax+1)
		acol.CopyTo(&wcol)
		if dmax < AR.Rows()-1 {
			var wrst matrix.FloatMatrix
			WR.SubMatrix(&wrst, dmax, 0, wcol.Rows()-dmax, 1)
			AR.SubMatrix(&acol, dmax, dmax, AR.Rows()-dmax, 1)
			acol.CopyTo(&wrst)
		}
	}
	if k > 0 {
		WL.SubMatrix(&wrow, dmax, 0, 1, WL.Cols())
		//fmt.Printf("update with wrow:%v\n", &wrow)
		//fmt.Printf("update wcol\n%v\n", &wcol)
		MVMult(&wcol, AL, &wrow, -1.0, 1.0, NOTRANS)
		//fmt.Printf("updated wcol:\n%v\n", &wcol)
	}
	if dmax > 0 {
		// pivot column in workspace
		t0 := WR.GetAt(0, 0)
		WR.SetAt(0, 0, WR.GetAt(dmax, 0))
		WR.SetAt(dmax, 0, t0)
		// pivot on diagonal
		t0 = dg.GetAt(0, 0)
		dg.SetAt(0, 0, dg.GetAt(dmax, 0))
		dg.SetAt(dmax, 0, t0)
	}
	return dmax
}
Пример #7
0
func unblkBoundedBKUpper(A, wrk *matrix.FloatMatrix, p *pPivots, ncol int) (error, int) {
	var err error
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a01, A02, a11, a12t, A22, a11inv matrix.FloatMatrix
	var w00, w01, w11 matrix.FloatMatrix
	var cwrk matrix.FloatMatrix
	var wx, Ax, wz matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots

	err = nil
	nc := 0
	if ncol > A.Cols() {
		ncol = A.Cols()
	}

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pBOTTOMRIGHT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pBOTTOM)

	// permanent working space for symmetric inverse of a11
	wrk.SubMatrix(&a11inv, wrk.Rows()-2, 0, 2, 2)
	a11inv.SetAt(0, 1, -1.0)
	a11inv.SetAt(1, 0, -1.0)

	for ATL.Cols() > 0 && nc < ncol {

		partition2x2(
			&w00, &w01,
			nil, &w11, wrk, nc, nc, pBOTTOMRIGHT)
		merge1x2(&wx, &w00, &w01)
		merge1x2(&Ax, &ATL, &ATR)

		//fmt.Printf("ATL:\n%v\n", &ATL)
		r, np := findAndBuildBKPivotUpper(&ATL, &ATR, &w00, &w01, nc)
		//fmt.Printf("[w00;w01]:\n%v\n", &wx)
		//fmt.Printf("after find: r=%d, np=%d, ncol=%d, nc=%d\n", r, np, ncol, nc)
		w00.SubMatrix(&wz, 0, w00.Cols()-2, w00.Rows(), 2)
		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 != -1 {
			// pivoting needed; np == 1, last row; np == 2; next to last rows
			nrow := ATL.Rows() - np
			applyBKPivotSym(&ATL, nrow, r, UPPER)
			// swap left hand rows to get correct updates
			swapRows(&ATR, nrow, r)
			swapRows(&w01, nrow, r)
			if np == 2 {
				/* pivot block on diagonal; -1,-1
				 * [r, r] | [r ,-1]
				 * ----------------  2-by-2 pivot, swapping [1,0] and [r,0]
				 * [r,-1] | [-1,-1]
				 */
				t0 := w00.GetAt(-2, -1)
				tr := w00.GetAt(r, -1)
				//fmt.Printf("nc=%d, t0=%e, tr=%e\n", nc, t0, tr)
				w00.SetAt(-2, -1, tr)
				w00.SetAt(r, -1, t0)
				// interchange diagonal entries on w11[:,1]
				t0 = w00.GetAt(-2, -2)
				tr = w00.GetAt(r, -2)
				w00.SetAt(-2, -2, tr)
				w00.SetAt(r, -2, t0)
				//fmt.Printf("wrk:\n%v\n", &wz)
			}
			//fmt.Printf("pivoted A:\n%v\n", &Ax)
			//fmt.Printf("pivoted wrk:\n%v\n", &wx)
		}

		// repartition according the pivot size
		repartition2x2to3x3(&ATL,
			&A00, &a01, &A02,
			nil, &a11, &a12t,
			nil, nil, &A22 /**/, A, np, pTOPLEFT)
		repartPivot2x1to3x1(&pT,
			&p0,
			&p1,
			&p2 /**/, p, np, pTOP)
		// ------------------------------------------------------------

		wlc := w00.Cols() - np
		//wlr := w00.Rows() - 1
		w00.SubMatrix(&cwrk, 0, wlc, a01.Rows(), np)
		if np == 1 {
			//fmt.Printf("wz:\n%v\n", &wz)
			//fmt.Printf("a11 <-- %e\n", w00.GetAt(a01.Rows(), wlc))

			//w00.SubMatrix(&cwrk, 0, wlc-np+1, a01.Rows(), np)
			a11.SetAt(0, 0, w00.GetAt(a01.Rows(), wlc))
			// a21 = a21/a11
			//fmt.Printf("np == 1: pre-update a01\n%v\n", &a01)
			cwrk.CopyTo(&a01)
			InvScale(&a01, a11.Float())
			//fmt.Printf("np == 1: cwrk\n%v\na21\n%v\n", &cwrk, &a21)
			// store pivot point relative to original matrix
			if r == -1 {
				p1.pivots[0] = ATL.Rows()
			} else {
				p1.pivots[0] = r + 1
			}
		} else if np == 2 {
			/*         d | b
			 * w00 == ------
			 *         . | a
			 */
			a := w00.GetAt(-1, -1)
			b := w00.GetAt(-2, -1)
			d := w00.GetAt(-2, -2)
			a11inv.SetAt(1, 1, d/b)
			a11inv.SetAt(0, 0, 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
			//fmt.Printf("a11inv:\n%v\n", &a11inv)

			// a01 = a01*a11.-1
			Mult(&a01, &cwrk, &a11inv, scale, 0.0, NOTRANS)
			a11.SetAt(1, 1, a)
			a11.SetAt(0, 1, b)
			a11.SetAt(0, 0, d)

			// store pivot point relative to original matrix
			p1.pivots[0] = -(r + 1)
			p1.pivots[1] = p1.pivots[0]
		}

		//fmt.Printf("end-of-loop: Ax r=%d, nc=%d. np=%d\n%v\n", r, nc, np, &Ax)
		//fmt.Printf("wx m(wblk)=%d:\n%v\n", m(&wx), &wx)

		// ------------------------------------------------------------
		nc += np
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pTOPLEFT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pTOP)

	}
	return err, nc
}
Пример #8
0
func findAndBuildBKPivotUpper(AL, AR, WL, WR *matrix.FloatMatrix, k int) (int, int) {
	var r, q int
	var rcol, qrow, src, wk, wkp1, wrow matrix.FloatMatrix

	lc := AL.Cols() - 1
	wc := WL.Cols() - 1
	lr := AL.Rows() - 1
	// Copy AR[:,lc] to WR[:,wc] and update with WL[0:]
	AL.SubMatrix(&src, 0, lc, AL.Rows(), 1)
	WL.SubMatrix(&wk, 0, wc, AL.Rows(), 1)
	src.CopyTo(&wk)
	if k > 0 {
		WR.SubMatrix(&wrow, lr, 0, 1, WR.Cols())
		//fmt.Printf("wrow: %v\n", &wrow)
		MVMult(&wk, AR, &wrow, -1.0, 1.0, NOTRANS)
		//fmt.Printf("wk after update:\n%v\n", &wk)
	}
	if AL.Rows() == 1 {
		return -1, 1
	}
	amax := math.Abs(WL.GetAt(lr, wc))

	// find max off-diagonal on first column.
	WL.SubMatrix(&rcol, 0, wc, lr, 1)
	//fmt.Printf("rcol:\n%v\n", &rcol)
	// r is row index and rmax is its absolute value
	r = IAMax(&rcol)
	rmax := math.Abs(rcol.GetAt(r, 0))
	//fmt.Printf("r=%d, amax=%e, rmax=%e\n", r, amax, rmax)
	if amax >= bkALPHA*rmax {
		// no pivoting, 1x1 diagonal
		return -1, 1
	}

	// Now we need to copy row r to WR[:,wc-1] and update it
	WL.SubMatrix(&wkp1, 0, wc-1, AL.Rows(), 1)
	if r > 0 {
		// above the diagonal part of AL
		AL.SubMatrix(&qrow, 0, r, r, 1)
		qrow.CopyTo(&wkp1)
	}
	//fmt.Printf("m(AR)=%d, r=%d, qrow: %v\n", AL.Rows(), r, &qrow)
	var wkr matrix.FloatMatrix
	AL.SubMatrix(&qrow, r, r, 1, AL.Rows()-r)
	wkp1.SubMatrix(&wkr, r, 0, AL.Rows()-r, 1)
	qrow.CopyTo(&wkr)
	//fmt.Printf("m(AR)=%d, r=%d, qrow: %v\n", AR.Rows(), r, &qrow)
	if k > 0 {
		// update wkp1
		WR.SubMatrix(&wrow, r, 0, 1, WR.Cols())
		//fmt.Printf("initial wpk1:\n%v\n", &wkp1)
		MVMult(&wkp1, AR, &wrow, -1.0, 1.0, NOTRANS)
	}
	//fmt.Printf("updated wpk1:\n%v\n", &wkp1)

	// set on-diagonal entry to zero to avoid hitting it.
	p1 := wkp1.GetAt(r, 0)
	wkp1.SetAt(r, 0, 0.0)
	// max off-diagonal on r'th column/row at index q
	q = IAMax(&wkp1)
	qmax := math.Abs(wkp1.GetAt(q, 0))
	wkp1.SetAt(r, 0, p1)
	//fmt.Printf("blk: r=%d, q=%d, amax=%e, rmax=%e, qmax=%e\n", r, q, amax, rmax, qmax)

	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.GetAt(r, wc-1)) >= bkALPHA*qmax {
		// 1x1 pivoting and interchange with k, r
		// pivot row in column WR[:,1] to W[:,0]
		//p1 := WL.GetAt(r, wc-1)
		WL.SubMatrix(&src, 0, wc-1, AL.Rows(), 1)
		WL.SubMatrix(&wkp1, 0, wc, AL.Rows(), 1)
		src.CopyTo(&wkp1)
		wkp1.SetAt(-1, 0, src.GetAt(r, 0))
		wkp1.SetAt(r, 0, src.GetAt(-1, 0))
		return r, 1
	} else {
		// 2x2 pivoting and interchange with k+1, r
		return r, 2
	}
	return -1, 1
}
Пример #9
0
/*
 * 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 *matrix.FloatMatrix, p *pPivots, ncol int) (error, int) {
	var err error
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a10t, a11, A20, a21, A22, a11inv matrix.FloatMatrix
	var w00, w10, w11 matrix.FloatMatrix
	var cwrk matrix.FloatMatrix
	//var s, d matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots

	err = nil
	nc := 0
	if ncol > A.Cols() {
		ncol = A.Cols()
	}

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pTOP)

	// permanent working space for symmetric inverse of a11
	wrk.SubMatrix(&a11inv, 0, wrk.Cols()-2, 2, 2)
	a11inv.SetAt(1, 0, -1.0)
	a11inv.SetAt(0, 1, -1.0)

	for ABR.Cols() > 0 && nc < ncol {

		partition2x2(
			&w00, nil,
			&w10, &w11, wrk, nc, nc, pTOPLEFT)

		//fmt.Printf("ABR:\n%v\n", &ABR)
		r, np := findAndBuildBKPivotLower(&ABL, &ABR, &w10, &w11, nc)
		//fmt.Printf("after find: r=%d, np=%d, ncol=%d, nc=%d\n", r, np, ncol, nc)
		if np > ncol-nc {
			// next pivot does not fit into ncol columns, restore last column,
			// return with number of factorized columns
			//fmt.Printf("np > ncol-nc: %d > %d\n", np, ncol-nc)
			return err, nc
			//goto undo
		}
		if r != 0 && r != np-1 {
			// pivoting needed; do swaping here
			applyBKPivotSym(&ABR, np-1, r, LOWER)
			// swap left hand rows to get correct updates
			swapRows(&ABL, np-1, r)
			swapRows(&w10, np-1, r)
			//ABL.SubMatrix(&s, np-1, 0, 1, ABL.Cols())
			//ABL.SubMatrix(&d, r,    0, 1, ABL.Cols())
			//Swap(&s, &d)
			//w10.SubMatrix(&s, np-1, 0, 1, w10.Cols())
			//w10.SubMatrix(&d, r,    0, 1, w10.Cols())
			//Swap(&s, &d)
			if np == 2 {
				/*
				 *          [0,0] | [r,0]
				 * a11 ==   -------------  2-by-2 pivot, swapping [1,0] and [r,0]
				 *          [r,0] | [r,r]
				 */
				t0 := w11.GetAt(1, 0)
				tr := w11.GetAt(r, 0)
				//fmt.Printf("nc=%d, t0=%e, tr=%e\n", nc, t0, tr)
				w11.SetAt(1, 0, tr)
				w11.SetAt(r, 0, t0)
				// interchange diagonal entries on w11[:,1]
				t0 = w11.GetAt(1, 1)
				tr = w11.GetAt(r, 1)
				w11.SetAt(1, 1, tr)
				w11.SetAt(r, 1, t0)
			}
			//fmt.Printf("pivoted A:\n%v\n", A)
			//fmt.Printf("pivoted wrk:\n%v\n", wrk)
		}

		// repartition according the pivot size
		repartition2x2to3x3(&ATL,
			&A00, nil, nil,
			&a10t, &a11, nil,
			&A20, &a21, &A22 /**/, A, np, pBOTTOMRIGHT)
		repartPivot2x1to3x1(&pT,
			&p0,
			&p1,
			&p2 /**/, p, np, pBOTTOM)
		// ------------------------------------------------------------

		if np == 1 {
			//
			w11.SubMatrix(&cwrk, np, 0, a21.Rows(), np)
			a11.SetAt(0, 0, w11.GetAt(0, 0))
			// a21 = a21/a11
			//fmt.Printf("np == 1: pre-update a21\n%v\n", &a21)
			cwrk.CopyTo(&a21)
			InvScale(&a21, a11.Float())
			//fmt.Printf("np == 1: cwrk\n%v\na21\n%v\n", &cwrk, &a21)
			// store pivot point relative to original matrix
			p1.pivots[0] = r + ATL.Rows() + 1
		} else if np == 2 {
			/*
			 * See comments for this block in unblkDecompBKLower().
			 */
			a := w11.GetAt(0, 0)
			b := w11.GetAt(1, 0)
			d := w11.GetAt(1, 1)
			a11inv.SetAt(0, 0, d/b)
			a11inv.SetAt(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

			w11.SubMatrix(&cwrk, np, 0, a21.Rows(), np)
			// a21 = a21*a11.-1
			Mult(&a21, &cwrk, &a11inv, scale, 0.0, NOTRANS)
			a11.SetAt(0, 0, a)
			a11.SetAt(1, 0, b)
			a11.SetAt(1, 1, d)

			// store pivot point relative to original matrix
			p1.pivots[0] = -(r + ATL.Rows() + 1)
			p1.pivots[1] = p1.pivots[0]
		}

		/*
		   if m(&ABR) < 5 {
		       var Ablk, wblk, w5 matrix.FloatMatrix
		       merge1x2(&Ablk, &ABL, &ABR)
		       merge1x2(&wblk, &w10, &w11)
		       wblk.SubMatrix(&w5, 0, 0, Ablk.Rows(), wblk.Cols())
		       fmt.Printf("blocked EOL: Ablk r=%d, nc=%d. np=%d\n%v\n", r, nc, np, &Ablk)
		       fmt.Printf("wblk m(wblk)=%d:\n%v\n", m(&w5), &w5)
		   }
		*/
		// ------------------------------------------------------------
		nc += np
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pBOTTOM)

	}
	// undo applied partial row pivots (AL, w00)
	//undo:
	return err, nc
}
Пример #10
0
/*
 * 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 *matrix.FloatMatrix, k int) (int, int) {
	var r, q int
	var rcol, qrow, src, wk, wkp1, wrow matrix.FloatMatrix

	// Copy AR column 0 to WR column 0 and update with WL[0:]
	AR.SubMatrix(&src, 0, 0, AR.Rows(), 1)
	WR.SubMatrix(&wk, 0, 0, AR.Rows(), 1)
	src.CopyTo(&wk)
	if k > 0 {
		WL.SubMatrix(&wrow, 0, 0, 1, WL.Cols())
		MVMult(&wk, AL, &wrow, -1.0, 1.0, NOTRANS)
		//fmt.Printf("wk after update:\n%v\n", &wk)
	}
	if AR.Rows() == 1 {
		return 0, 1
	}
	amax := math.Abs(WR.GetAt(0, 0))

	// find max off-diagonal on first column.
	WR.SubMatrix(&rcol, 1, 0, AR.Rows()-1, 1)
	//fmt.Printf("rcol:\n%v\n", &rcol)
	// r is row index and rmax is its absolute value
	r = IAMax(&rcol) + 1
	rmax := math.Abs(rcol.GetAt(r-1, 0))
	//fmt.Printf("r=%d, amax=%e, rmax=%e\n", r, amax, rmax)
	if amax >= bkALPHA*rmax {
		// no pivoting, 1x1 diagonal
		return 0, 1
	}
	// Now we need to copy row r to WR[:,1] and update it
	WR.SubMatrix(&wkp1, 0, 1, AR.Rows(), 1)
	AR.SubMatrix(&qrow, r, 0, 1, r+1)
	qrow.CopyTo(&wkp1)
	//fmt.Printf("m(AR)=%d, r=%d, qrow: %v\n", AR.Rows(), r, &qrow)
	if r < AR.Rows()-1 {
		var wkr matrix.FloatMatrix
		AR.SubMatrix(&qrow, r, r, AR.Rows()-r, 1)
		wkp1.SubMatrix(&wkr, r, 0, wkp1.Rows()-r, 1)
		qrow.CopyTo(&wkr)
		//fmt.Printf("m(AR)=%d, r=%d, qrow: %v\n", AR.Rows(), r, &qrow)
	}
	if k > 0 {
		// update wkp1
		WL.SubMatrix(&wrow, r, 0, 1, WL.Cols())
		//fmt.Printf("initial wpk1:\n%v\n", &wkp1)
		MVMult(&wkp1, AL, &wrow, -1.0, 1.0, NOTRANS)
		//fmt.Printf("updated wpk1:\n%v\n", &wkp1)
	}

	// set on-diagonal entry to zero to avoid finding it
	p1 := wkp1.GetAt(r, 0)
	wkp1.SetAt(r, 0, 0.0)
	// max off-diagonal on r'th column/row at index q
	q = IAMax(&wkp1)
	qmax := math.Abs(wkp1.GetAt(q, 0))
	// restore on-diagonal entry
	wkp1.SetAt(r, 0, p1)
	//arr := math.Abs(WR.GetAt(r, 1))
	//fmt.Printf("blk: r=%d, q=%d, amax=%e, rmax=%e, qmax=%e, Arr=%e\n", r, q, amax, rmax, qmax, arr)

	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.GetAt(r, 1)) >= bkALPHA*qmax {
		// 1x1 pivoting and interchange with k, r
		// pivot row in column WR[:,1] to W[:,0]
		//pr := WR.GetAt(r, 1)
		//_ = pr
		WR.SubMatrix(&src, 0, 1, AR.Rows(), 1)
		WR.SubMatrix(&wkp1, 0, 0, AR.Rows(), 1)
		src.CopyTo(&wkp1)
		wkp1.SetAt(0, 0, src.GetAt(r, 0))
		wkp1.SetAt(r, 0, src.GetAt(0, 0))
		return r, 1
	} else {
		// 2x2 pivoting and interchange with k+1, r
		return r, 2
	}
	return 0, 1
}
Пример #11
0
/*
 * Unblocked Bunch-Kauffman LDL factorization.
 *
 * Corresponds lapack.DSYTF2
 */
func unblkDecompBKUpper(A, wrk *matrix.FloatMatrix, p *pPivots) (error, int) {
	var err error
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a01, A02, a12t, a11, A22, a11inv matrix.FloatMatrix
	var cwrk matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots

	err = nil
	nc := 0

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pBOTTOMRIGHT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pBOTTOM)

	// permanent working space for symmetric inverse of a11
	wrk.SubMatrix(&a11inv, 0, wrk.Cols()-2, 2, 2)
	a11inv.SetAt(1, 0, -1.0)
	a11inv.SetAt(0, 1, -1.0)

	for ATL.Cols() > 0 {

		nr := ATL.Rows() - 1
		r, np := findBKPivot(&ATL, UPPER)
		if r != -1 /*&& r != np-1*/ {
			// pivoting needed; do swaping here
			//fmt.Printf("pre-pivot ATL [%d]:\n%v\n", ATL.Rows()-np, &ATL)
			applyBKPivotSym(&ATL, ATL.Rows()-np, r, UPPER)
			if np == 2 {
				/*
				 *         [r,r] | [r, nr]
				 * a11 ==  ---------------  2-by-2 pivot, swapping [nr-1,nr] and [r,nr]
				 *         [r,0] | [nr,nr]
				 */
				t := ATL.GetAt(nr-1, nr)
				ATL.SetAt(nr-1, nr, ATL.GetAt(r, nr))
				ATL.SetAt(r, nr, t)
			}
			//fmt.Printf("unblk: ATL after %d pivot [r=%d]:\n%v\n", np, r, &ATL)
		}

		// repartition according the pivot size
		repartition2x2to3x3(&ATL,
			&A00, &a01, &A02,
			nil, &a11, &a12t,
			nil, nil, &A22 /**/, A, np, pTOPLEFT)
		repartPivot2x1to3x1(&pT,
			&p0,
			&p1,
			&p2 /**/, p, np, pTOP)
		// ------------------------------------------------------------

		if np == 1 {
			// A00 = A00 - a01*a01.T/a11
			MVUpdateTrm(&A00, &a01, &a01, -1.0/a11.Float(), UPPER)
			// a01 = a01/a11
			InvScale(&a01, a11.Float())
			if r == -1 {
				p1.pivots[0] = ATL.Rows()
			} else {
				p1.pivots[0] = r + 1
			}
		} else if np == 2 {
			/*
			 * See comments on unblkDecompBKLower().
			 */
			a := a11.GetAt(0, 0)
			b := a11.GetAt(0, 1)
			d := a11.GetAt(1, 1)
			a11inv.SetAt(0, 0, d/b)
			a11inv.SetAt(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
			wrk.SubMatrix(&cwrk, 2, 0, a01.Rows(), a01.Cols())
			a01.CopyTo(&cwrk)
			//fmt.Printf("cwrk:\n%v\n", &cwrk)
			//fmt.Printf("a11inv:\n%v\n", &a11inv)
			// a01 = a01*a11.-1
			Mult(&a01, &cwrk, &a11inv, scale, 0.0, NOTRANS)
			// A00 = A00 - a01*a11.-1*a01.T = A00 - a01*cwrk.T
			UpdateTrm(&A00, &a01, &cwrk, -1.0, 1.0, UPPER|TRANSB)

			p1.pivots[0] = -(r + 1)
			p1.pivots[1] = p1.pivots[0]
		}

		// ------------------------------------------------------------
		nc += np
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pTOPLEFT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pTOP)

	}
	return err, nc
}
Пример #12
0
/*
 * Unblocked Bunch-Kauffman LDL factorization.
 *
 * Corresponds lapack.DSYTF2
 */
func unblkDecompBKLower(A, wrk *matrix.FloatMatrix, p *pPivots) (error, int) {
	var err error
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a10t, a11, A20, a21, A22, a11inv matrix.FloatMatrix
	var cwrk matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots

	err = nil
	nc := 0

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pTOP)

	// permanent working space for symmetric inverse of a11
	wrk.SubMatrix(&a11inv, 0, wrk.Cols()-2, 2, 2)
	a11inv.SetAt(1, 0, -1.0)
	a11inv.SetAt(0, 1, -1.0)

	for ABR.Cols() > 0 {

		r, np := findBKPivot(&ABR, LOWER)
		if r != 0 && r != np-1 {
			// pivoting needed; do swaping here
			applyBKPivotSym(&ABR, np-1, r, LOWER)
			if np == 2 {
				/*
				 *          [0,0] | [r,0]
				 * a11 ==   -------------  2-by-2 pivot, swapping [1,0] and [r,0]
				 *          [r,0] | [r,r]
				 */
				t := ABR.GetAt(1, 0)
				ABR.SetAt(1, 0, ABR.GetAt(r, 0))
				ABR.SetAt(r, 0, t)
			}
			//fmt.Printf("unblk: ABR after %d pivot [r=%d]:\n%v\n", np, r, &ABR)
		}

		// repartition according the pivot size
		repartition2x2to3x3(&ATL,
			&A00, nil, nil,
			&a10t, &a11, nil,
			&A20, &a21, &A22 /**/, A, np, pBOTTOMRIGHT)
		repartPivot2x1to3x1(&pT,
			&p0,
			&p1,
			&p2 /**/, p, np, pBOTTOM)
		// ------------------------------------------------------------

		if np == 1 {
			// A22 = A22 - a21*a21.T/a11
			MVUpdateTrm(&A22, &a21, &a21, -1.0/a11.Float(), LOWER)
			// a21 = a21/a11
			InvScale(&a21, a11.Float())
			// store pivot point relative to original matrix
			p1.pivots[0] = r + ATL.Rows() + 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.GetAt(0, 0)
			b := a11.GetAt(1, 0)
			d := a11.GetAt(1, 1)
			a11inv.SetAt(0, 0, d/b)
			a11inv.SetAt(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
			wrk.SubMatrix(&cwrk, 2, 0, a21.Rows(), a21.Cols())
			a21.CopyTo(&cwrk)
			// a21 = a21*a11.-1
			Mult(&a21, &cwrk, &a11inv, scale, 0.0, NOTRANS)
			// A22 = A22 - a21*a11.-1*a21.T = A22 - a21*cwrk.T
			UpdateTrm(&A22, &a21, &cwrk, -1.0, 1.0, LOWER|TRANSB)

			// store pivot point relative to original matrix
			p1.pivots[0] = -(r + ATL.Rows() + 1)
			p1.pivots[1] = p1.pivots[0]
		}

		/*
		   if m(&ABR) < 5 {
		       var Ablk matrix.FloatMatrix
		       merge1x2(&Ablk, &ABL, &ABR)
		       fmt.Printf("unblocked EOL: Ablk r=%d, nc=%d. np=%d\n%v\n", r, nc, np, &Ablk)
		   }
		*/
		// ------------------------------------------------------------
		nc += np
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pBOTTOM)

	}
	return err, nc
}
Пример #13
0
func unblkBoundedLowerLDL(A, W *matrix.FloatMatrix, p *pPivots, ncol int) (error, int) {
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a10, a11, A20, a21, A22, adiag, wcol matrix.FloatMatrix
	var w00, w10, w11 matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots
	var err error = nil

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pTOP)

	// copy current diagonal to last column of workspace
	W.SubMatrix(&wcol, 0, W.Cols()-1, A.Rows(), 1)
	A.Diag(&adiag)
	adiag.CopyTo(&wcol)
	//fmt.Printf("initial diagonal:\n%v\n", &wcol)

	nc := 0
	for ABR.Cols() > 0 && nc < ncol {

		partition2x2(
			&w00, nil,
			&w10, &w11, W, nc, nc, pTOPLEFT)

		dmax := findAndBuildPivot(&ABL, &ABR, &w10, &w11, nc)
		//fmt.Printf("dmax=%d\n", dmax)
		if dmax > 0 {
			// pivot diagonal in symmetric matrix; will swap a11 [0,0] and [imax,imax]
			applyPivotSym(&ABL, &ABR, dmax, LOWER)
			swapRows(&w10, 0, dmax)
			pB.pivots[0] = dmax + ATL.Rows() + 1
		} else {
			pB.pivots[0] = 0
		}

		//fmt.Printf("blk pivoted %d, A:\n%v\nW:\n%v\n", dmax, A, W)
		repartition2x2to3x3(&ATL,
			&A00, nil, nil,
			&a10, &a11, nil,
			&A20, &a21, &A22, A, 1, pBOTTOMRIGHT)
		repartPivot2x1to3x1(&pT,
			&p0, &p1, &p2 /**/, p, 1, pBOTTOM)

		// --------------------------------------------------------

		// Copy updated column from working space
		w11.SubMatrix(&wcol, 1, 0, a21.Rows(), 1)
		wcol.CopyTo(&a21)
		a11.SetAt(0, 0, w11.GetAt(0, 0))
		// l21 = a21/a11
		InvScale(&a21, a11.Float())
		// here: wcol == l21*d11 == a21
		if ncol-nc > 1 {
			// update diagonal in workspace if not last column of block
			w11.SubMatrix(&adiag, 1, w11.Cols()-1, a21.Rows(), 1)
			MVUpdateDiag(&adiag, &wcol, &wcol, -1.0/a11.Float())
		}
		//fmt.Printf("nc=%d, a11=%e\n", nc, a11.Float())
		//fmt.Printf("l21\n%v\n", &a21)
		//fmt.Printf("a21\n%v\n", &wcol)
		//fmt.Printf("diag\n%v\n", &adiag)
		//var Ablk, wblk matrix.FloatMatrix
		//merge1x2(&Ablk, &ABL, &ABR)
		//merge1x2(&wblk, &w10, &w11)
		//fmt.Printf("unblk Ablk:\n%v\n", &Ablk)
		//fmt.Printf("unblk wblk:\n%v\n", &wblk)

		// ---------------------------------------------------------

		nc++
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pBOTTOM)
	}
	return err, nc
}
Пример #14
0
/*
 * Blocked QR decomposition with compact WY transform. As implemented
 * in lapack.xGEQRF subroutine.
 */
func blockedQR(A, Tvec, Twork, W *matrix.FloatMatrix, nb int) {
	var ATL, ATR, ABL, ABR, AL, AR matrix.FloatMatrix
	var A00, A01, A02, A10, A11, A12, A20, A21, A22 matrix.FloatMatrix
	var TT, TB matrix.FloatMatrix
	var t0, tau, t2, Tdiag, WT, WB, W0, W1, W2 matrix.FloatMatrix
	//var Twork, W *matrix.FloatMatrix

	Tdiag.DiagOf(Twork)

	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partition2x1(
		&TT,
		&TB, Tvec, 0, pTOP)
	partition2x1(
		&WT,
		&WB, W, 0, pTOP)

	for ABR.Rows() > 0 && ABR.Cols() > 0 {
		repartition2x2to3x3(&ATL,
			&A00, &A01, &A02,
			&A10, &A11, &A12,
			&A20, &A21, &A22, A, nb, pBOTTOMRIGHT)
		repartition2x1to3x1(&TT,
			&t0,
			&tau,
			&t2, Tvec, nb, pBOTTOM)
		repartition2x1to3x1(&WT,
			&W0,
			&W1,
			&W2, W, nb, pBOTTOM)
		partition1x2(
			&AL, &AR, &ABR, nb, pLEFT)

		// current block size
		cb, rb := A11.Size()
		if rb < cb {
			cb = rb
		}

		// --------------------------------------------------------

		// decompose left side AL == /A11\
		//                           \A21/
		unblockedQRT(&AL, Twork)

		// copy scaling from T diagonal to tau-vector
		Tdiag.CopyTo(&tau)

		// update A'tail i.e. A12 and A22 with (I - Y*T*Y.T).T * A'tail
		// compute: C - Y*(C.T*Y*T).T
		updateWithQT(&A12, &A22, &A11, &A21, Twork, &W2, cb, true)

		// --------------------------------------------------------
		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &A11, &A22, A, pBOTTOMRIGHT)
		continue3x1to2x1(
			&TT,
			&TB, &t0, &tau, Tvec, pBOTTOM)
		continue3x1to2x1(
			&WT,
			&WB, &W0, &W1, W, pBOTTOM)
	}
}