Exemple #1
0
func newSimilarityCircuit(uVal, uGrad, vVal, vGrad []float64) *similarityCircuit {
	s := similarityCircuit{
		UVal:  uVal,
		UGrad: uGrad,
		VVal:  vVal,
		VGrad: vGrad,
	}
	u := blas64.Vector{Inc: 1, Data: uVal}
	v := blas64.Vector{Inc: 1, Data: vVal}
	s.UV = blas64.Dot(len(uVal), u, v)
	s.Unorm = blas64.Nrm2(len(uVal), u)
	s.Vnorm = blas64.Nrm2(len(vVal), v)
	s.TopVal = s.UV / (s.Unorm * s.Vnorm)
	return &s
}
Exemple #2
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// LQ computes an LQ Decomposition for an m-by-n matrix a with m <= n by Householder
// reflections. The LQ decomposition is an m-by-n orthogonal matrix q and an m-by-m
// lower triangular matrix l so that a = l.q. LQ will panic with ErrShape if m > n.
//
// The LQ decomposition always exists, even if the matrix does not have full rank,
// so LQ will never fail unless m > n. The primary use of the LQ decomposition is
// in the least squares solution of non-square systems of simultaneous linear equations.
// This will fail if LQIsFullRank() returns false. The matrix a is overwritten by the
// decomposition.
func LQ(a *Dense) LQFactor {
	// Initialize.
	m, n := a.Dims()
	if m > n {
		panic(ErrShape)
	}

	lq := *a

	lDiag := make([]float64, m)
	projs := NewVector(m, nil)

	// Main loop.
	for k := 0; k < m; k++ {
		hh := lq.RawRowView(k)[k:]
		norm := blas64.Nrm2(len(hh), blas64.Vector{Inc: 1, Data: hh})
		lDiag[k] = norm

		if norm != 0 {
			hhNorm := (norm * math.Sqrt(1-hh[0]/norm))
			if hhNorm == 0 {
				hh[0] = 0
			} else {
				// Form k-th Householder vector.
				s := 1 / hhNorm
				hh[0] -= norm
				blas64.Scal(len(hh), s, blas64.Vector{Inc: 1, Data: hh})

				// Apply transformation to remaining columns.
				if k < m-1 {
					a = lq.View(k+1, k, m-k-1, n-k).(*Dense)
					projs = projs.ViewVec(0, m-k-1)
					projs.MulVec(a, false, NewVector(len(hh), hh))

					for j := 0; j < m-k-1; j++ {
						dst := a.RawRowView(j)
						blas64.Axpy(len(dst), -projs.at(j),
							blas64.Vector{Inc: 1, Data: hh},
							blas64.Vector{Inc: 1, Data: dst},
						)
					}
				}
			}
		}
	}
	*a = lq

	return LQFactor{a, lDiag}
}
Exemple #3
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// Norm returns the specified (induced) norm of the matrix a. See
// https://en.wikipedia.org/wiki/Matrix_norm for the definition of an induced norm.
//
// Valid norms are:
//    1 - The maximum absolute column sum
//    2 - Frobenius norm, the square root of the sum of the squares of the elements.
//  Inf - The maximum absolute row sum.
// Norm will panic with ErrNormOrder if an illegal norm order is specified and
// with matrix.ErrShape if the matrix has zero size.
func Norm(a Matrix, norm float64) float64 {
	r, c := a.Dims()
	if r == 0 || c == 0 {
		panic(matrix.ErrShape)
	}
	aU, aTrans := untranspose(a)
	var work []float64
	switch rma := aU.(type) {
	case RawMatrixer:
		rm := rma.RawMatrix()
		n := normLapack(norm, aTrans)
		if n == lapack.MaxColumnSum {
			work = make([]float64, rm.Cols)
		}
		return lapack64.Lange(n, rm, work)
	case RawTriangular:
		rm := rma.RawTriangular()
		n := normLapack(norm, aTrans)
		if n == lapack.MaxRowSum || n == lapack.MaxColumnSum {
			work = make([]float64, rm.N)
		}
		return lapack64.Lantr(n, rm, work)
	case RawSymmetricer:
		rm := rma.RawSymmetric()
		n := normLapack(norm, aTrans)
		if n == lapack.MaxRowSum || n == lapack.MaxColumnSum {
			work = make([]float64, rm.N)
		}
		return lapack64.Lansy(n, rm, work)
	case *Vector:
		rv := rma.RawVector()
		switch norm {
		default:
			panic("unreachable")
		case 1:
			if aTrans {
				imax := blas64.Iamax(rma.n, rv)
				return math.Abs(rma.At(imax, 0))
			}
			return blas64.Asum(rma.n, rv)
		case 2:
			return blas64.Nrm2(rma.n, rv)
		case math.Inf(1):
			if aTrans {
				return blas64.Asum(rma.n, rv)
			}
			imax := blas64.Iamax(rma.n, rv)
			return math.Abs(rma.At(imax, 0))
		}
	}
	switch norm {
	default:
		panic("unreachable")
	case 1:
		var max float64
		for j := 0; j < c; j++ {
			var sum float64
			for i := 0; i < r; i++ {
				sum += math.Abs(a.At(i, j))
			}
			if sum > max {
				max = sum
			}
		}
		return max
	case 2:
		var sum float64
		for i := 0; i < r; i++ {
			for j := 0; j < c; j++ {
				v := a.At(i, j)
				sum += v * v
			}
		}
		return math.Sqrt(sum)
	case math.Inf(1):
		var max float64
		for i := 0; i < r; i++ {
			var sum float64
			for j := 0; j < c; j++ {
				sum += math.Abs(a.At(i, j))
			}
			if sum > max {
				max = sum
			}
		}
		return max
	}
}
Exemple #4
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func Dgelq2Test(t *testing.T, impl Dgelq2er) {
	for c, test := range []struct {
		m, n, lda int
	}{
		{1, 1, 0},
		{2, 2, 0},
		{3, 2, 0},
		{2, 3, 0},
		{1, 12, 0},
		{2, 6, 0},
		{3, 4, 0},
		{4, 3, 0},
		{6, 2, 0},
		{1, 12, 0},
		{1, 1, 20},
		{2, 2, 20},
		{3, 2, 20},
		{2, 3, 20},
		{1, 12, 20},
		{2, 6, 20},
		{3, 4, 20},
		{4, 3, 20},
		{6, 2, 20},
		{1, 12, 20},
	} {
		n := test.n
		m := test.m
		lda := test.lda
		if lda == 0 {
			lda = test.n
		}
		k := min(m, n)
		tau := make([]float64, k)
		for i := range tau {
			tau[i] = rand.Float64()
		}
		work := make([]float64, m)
		for i := range work {
			work[i] = rand.Float64()
		}
		a := make([]float64, m*lda)
		for i := 0; i < m*lda; i++ {
			a[i] = rand.Float64()
		}
		aCopy := make([]float64, len(a))
		copy(aCopy, a)
		impl.Dgelq2(m, n, a, lda, tau, work)

		Q := constructQ("LQ", m, n, a, lda, tau)

		// Check that Q is orthonormal
		for i := 0; i < Q.Rows; i++ {
			nrm := blas64.Nrm2(Q.Cols, blas64.Vector{Inc: 1, Data: Q.Data[i*Q.Stride:]})
			if math.Abs(nrm-1) > 1e-14 {
				t.Errorf("Q not normal. Norm is %v", nrm)
			}
			for j := 0; j < i; j++ {
				dot := blas64.Dot(Q.Rows,
					blas64.Vector{Inc: 1, Data: Q.Data[i*Q.Stride:]},
					blas64.Vector{Inc: 1, Data: Q.Data[j*Q.Stride:]},
				)
				if math.Abs(dot) > 1e-14 {
					t.Errorf("Q not orthogonal. Dot is %v", dot)
				}
			}
		}

		L := blas64.General{
			Rows:   m,
			Cols:   n,
			Stride: n,
			Data:   make([]float64, m*n),
		}
		for i := 0; i < m; i++ {
			for j := 0; j <= min(i, n-1); j++ {
				L.Data[i*L.Stride+j] = a[i*lda+j]
			}
		}

		ans := blas64.General{
			Rows:   m,
			Cols:   n,
			Stride: lda,
			Data:   make([]float64, m*lda),
		}
		copy(ans.Data, aCopy)
		blas64.Gemm(blas.NoTrans, blas.NoTrans, 1, L, Q, 0, ans)
		if !floats.EqualApprox(aCopy, ans.Data, 1e-14) {
			t.Errorf("Case %v, LQ mismatch. Want %v, got %v.", c, aCopy, ans.Data)
		}
	}
}
Exemple #5
0
func Dgeqr2Test(t *testing.T, impl Dgeqr2er) {
	for c, test := range []struct {
		m, n, lda int
	}{
		{1, 1, 0},
		{2, 2, 0},
		{3, 2, 0},
		{2, 3, 0},
		{1, 12, 0},
		{2, 6, 0},
		{3, 4, 0},
		{4, 3, 0},
		{6, 2, 0},
		{12, 1, 0},
		{1, 1, 20},
		{2, 2, 20},
		{3, 2, 20},
		{2, 3, 20},
		{1, 12, 20},
		{2, 6, 20},
		{3, 4, 20},
		{4, 3, 20},
		{6, 2, 20},
		{12, 1, 20},
	} {
		n := test.n
		m := test.m
		lda := test.lda
		if lda == 0 {
			lda = test.n
		}
		a := make([]float64, m*lda)
		for i := range a {
			a[i] = rand.Float64()
		}
		aCopy := make([]float64, len(a))
		k := min(m, n)
		tau := make([]float64, k)
		for i := range tau {
			tau[i] = rand.Float64()
		}
		work := make([]float64, n)
		for i := range work {
			work[i] = rand.Float64()
		}
		copy(aCopy, a)
		impl.Dgeqr2(m, n, a, lda, tau, work)

		// Test that the QR factorization has completed successfully. Compute
		// Q based on the vectors.
		q := constructQ("QR", m, n, a, lda, tau)

		// Check that q is orthonormal
		for i := 0; i < m; i++ {
			nrm := blas64.Nrm2(m, blas64.Vector{1, q.Data[i*m:]})
			if math.Abs(nrm-1) > 1e-14 {
				t.Errorf("Case %v, q not normal", c)
			}
			for j := 0; j < i; j++ {
				dot := blas64.Dot(m, blas64.Vector{1, q.Data[i*m:]}, blas64.Vector{1, q.Data[j*m:]})
				if math.Abs(dot) > 1e-14 {
					t.Errorf("Case %v, q not orthogonal", i)
				}
			}
		}
		// Check that A = Q * R
		r := blas64.General{
			Rows:   m,
			Cols:   n,
			Stride: n,
			Data:   make([]float64, m*n),
		}
		for i := 0; i < m; i++ {
			for j := i; j < n; j++ {
				r.Data[i*n+j] = a[i*lda+j]
			}
		}
		atmp := blas64.General{
			Rows:   m,
			Cols:   n,
			Stride: lda,
			Data:   make([]float64, m*lda),
		}
		copy(atmp.Data, a)
		blas64.Gemm(blas.NoTrans, blas.NoTrans, 1, q, r, 0, atmp)
		if !floats.EqualApprox(atmp.Data, aCopy, 1e-14) {
			t.Errorf("Q*R != a")
		}
	}
}
Exemple #6
0
func DlangeTest(t *testing.T, impl Dlanger) {
	for _, test := range []struct {
		m, n, lda int
	}{
		{4, 3, 0},
		{3, 4, 0},
		{4, 3, 100},
		{3, 4, 100},
	} {
		m := test.m
		n := test.n
		lda := test.lda
		if lda == 0 {
			lda = n
		}
		a := make([]float64, m*lda)
		for i := range a {
			a[i] = (rand.Float64() - 0.5)
		}
		work := make([]float64, n)
		for i := range work {
			work[i] = rand.Float64()
		}
		aCopy := make([]float64, len(a))
		copy(aCopy, a)

		// Test MaxAbs norm.
		norm := impl.Dlange(lapack.MaxAbs, m, n, a, lda, work)
		var ans float64
		for i := 0; i < m; i++ {
			idx := blas64.Iamax(n, blas64.Vector{1, aCopy[i*lda:]})
			ans = math.Max(ans, math.Abs(a[i*lda+idx]))
		}
		// Should be strictly equal because there is no floating point summation error.
		if ans != norm {
			t.Errorf("MaxAbs mismatch. Want %v, got %v.", ans, norm)
		}

		// Test MaxColumnSum norm.
		norm = impl.Dlange(lapack.MaxColumnSum, m, n, a, lda, work)
		ans = 0
		for i := 0; i < n; i++ {
			sum := blas64.Asum(m, blas64.Vector{lda, aCopy[i:]})
			ans = math.Max(ans, sum)
		}
		if math.Abs(norm-ans) > 1e-14 {
			t.Errorf("MaxColumnSum mismatch. Want %v, got %v.", ans, norm)
		}

		// Test MaxRowSum norm.
		norm = impl.Dlange(lapack.MaxRowSum, m, n, a, lda, work)
		ans = 0
		for i := 0; i < m; i++ {
			sum := blas64.Asum(n, blas64.Vector{1, aCopy[i*lda:]})
			ans = math.Max(ans, sum)
		}
		if math.Abs(norm-ans) > 1e-14 {
			t.Errorf("MaxRowSum mismatch. Want %v, got %v.", ans, norm)
		}

		// Test Frobenius norm
		norm = impl.Dlange(lapack.NormFrob, m, n, a, lda, work)
		ans = 0
		for i := 0; i < m; i++ {
			sum := blas64.Nrm2(n, blas64.Vector{1, aCopy[i*lda:]})
			ans += sum * sum
		}
		ans = math.Sqrt(ans)
		if math.Abs(norm-ans) > 1e-14 {
			t.Errorf("NormFrob mismatch. Want %v, got %v.", ans, norm)
		}
	}
}
Exemple #7
0
// SymRankOne performs a rank-1 update of the original matrix A and refactorizes
// its Cholesky factorization, storing the result into the reciever. That is, if
// in the original Cholesky factorization
//  U^T * U = A,
// in the updated factorization
//  U'^T * U' = A + alpha * x * x^T = A'.
//
// Note that when alpha is negative, the updating problem may be ill-conditioned
// and the results may be inaccurate, or the updated matrix A' may not be
// positive definite and not have a Cholesky factorization. SymRankOne returns
// whether the updated matrix A' is positive definite.
//
// SymRankOne updates a Cholesky factorization in O(n²) time. The Cholesky
// factorization computation from scratch is O(n³).
func (c *Cholesky) SymRankOne(orig *Cholesky, alpha float64, x *Vector) (ok bool) {
	if !orig.valid() {
		panic(badCholesky)
	}
	n := orig.Size()
	if x.Len() != n {
		panic(matrix.ErrShape)
	}
	if orig != c {
		if c.isZero() {
			c.chol = NewTriDense(n, matrix.Upper, nil)
		} else if c.chol.mat.N != n {
			panic(matrix.ErrShape)
		}
		c.chol.Copy(orig.chol)
	}

	if alpha == 0 {
		return true
	}

	// Algorithms for updating and downdating the Cholesky factorization are
	// described, for example, in
	// - J. J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart: LINPACK
	//   Users' Guide. SIAM (1979), pages 10.10--10.14
	// or
	// - P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders: Methods for
	//   modifying matrix factorizations. Mathematics of Computation 28(126)
	//   (1974), Method C3 on page 521
	//
	// The implementation is based on LINPACK code
	// http://www.netlib.org/linpack/dchud.f
	// http://www.netlib.org/linpack/dchdd.f
	// and
	// https://icl.cs.utk.edu/lapack-forum/viewtopic.php?f=2&t=2646
	//
	// According to http://icl.cs.utk.edu/lapack-forum/archives/lapack/msg00301.html
	// LINPACK is released under BSD license.
	//
	// See also:
	// - M. A. Saunders: Large-scale Linear Programming Using the Cholesky
	//   Factorization. Technical Report Stanford University (1972)
	//   http://i.stanford.edu/pub/cstr/reports/cs/tr/72/252/CS-TR-72-252.pdf
	// - Matthias Seeger: Low rank updates for the Cholesky decomposition.
	//   EPFL Technical Report 161468 (2004)
	//   http://infoscience.epfl.ch/record/161468

	work := make([]float64, n)
	blas64.Copy(n, x.RawVector(), blas64.Vector{1, work})

	if alpha > 0 {
		// Compute rank-1 update.
		if alpha != 1 {
			blas64.Scal(n, math.Sqrt(alpha), blas64.Vector{1, work})
		}
		umat := c.chol.mat
		stride := umat.Stride
		for i := 0; i < n; i++ {
			// Compute parameters of the Givens matrix that zeroes
			// the i-th element of x.
			c, s, r, _ := blas64.Rotg(umat.Data[i*stride+i], work[i])
			if r < 0 {
				// Multiply by -1 to have positive diagonal
				// elemnts.
				r *= -1
				c *= -1
				s *= -1
			}
			umat.Data[i*stride+i] = r
			if i < n-1 {
				// Multiply the extended factorization matrix by
				// the Givens matrix from the left. Only
				// the i-th row and x are modified.
				blas64.Rot(n-i-1,
					blas64.Vector{1, umat.Data[i*stride+i+1 : i*stride+n]},
					blas64.Vector{1, work[i+1 : n]},
					c, s)
			}
		}
		c.updateCond(-1)
		return true
	}

	// Compute rank-1 downdate.
	alpha = math.Sqrt(-alpha)
	if alpha != 1 {
		blas64.Scal(n, alpha, blas64.Vector{1, work})
	}
	// Solve U^T * p = x storing the result into work.
	ok = lapack64.Trtrs(blas.Trans, c.chol.RawTriangular(), blas64.General{
		Rows:   n,
		Cols:   1,
		Stride: 1,
		Data:   work,
	})
	if !ok {
		// The original matrix is singular. Should not happen, because
		// the factorization is valid.
		panic(badCholesky)
	}
	norm := blas64.Nrm2(n, blas64.Vector{1, work})
	if norm >= 1 {
		// The updated matrix is not positive definite.
		return false
	}
	norm = math.Sqrt((1 + norm) * (1 - norm))
	cos := make([]float64, n)
	sin := make([]float64, n)
	for i := n - 1; i >= 0; i-- {
		// Compute parameters of Givens matrices that zero elements of p
		// backwards.
		cos[i], sin[i], norm, _ = blas64.Rotg(norm, work[i])
		if norm < 0 {
			norm *= -1
			cos[i] *= -1
			sin[i] *= -1
		}
	}
	umat := c.chol.mat
	stride := umat.Stride
	for i := n - 1; i >= 0; i-- {
		// Apply Givens matrices to U.
		// TODO(vladimir-ch): Use workspace to avoid modifying the
		// receiver in case an invalid factorization is created.
		blas64.Rot(n-i, blas64.Vector{1, work[i:n]}, blas64.Vector{1, umat.Data[i*stride+i : i*stride+n]}, cos[i], sin[i])
		if umat.Data[i*stride+i] == 0 {
			// The matrix is singular (may rarely happen due to
			// floating-point effects?).
			ok = false
		} else if umat.Data[i*stride+i] < 0 {
			// Diagonal elements should be positive. If it happens
			// that on the i-th row the diagonal is negative,
			// multiply U from the left by an identity matrix that
			// has -1 on the i-th row.
			blas64.Scal(n-i, -1, blas64.Vector{1, umat.Data[i*stride+i : i*stride+n]})
		}
	}
	if ok {
		c.updateCond(-1)
	} else {
		c.Reset()
	}
	return ok
}