// Row copies the elements in the jth column of the matrix into the slice dst.
// The length of the provided slice must equal the number of columns, unless the
// slice is nil in which case a new slice is first allocated.
func Row(dst []float64, i int, a Matrix) []float64 {
r, c := a.Dims()
if i < 0 || i >= r {
panic(ErrColAccess)
}
if dst == nil {
dst = make([]float64, c)
} else {
if len(dst) != c {
panic(ErrColLength)
}
}
aMat, aTrans := untranspose(a)
if rm, ok := aMat.(RawMatrixer); ok {
m := rm.RawMatrix()
if aTrans {
blas64.Copy(c,
blas64.Vector{Inc: m.Stride, Data: m.Data[i:]},
blas64.Vector{Inc: 1, Data: dst},
)
return dst
}
copy(dst, m.Data[i*m.Stride:i*m.Stride+m.Cols])
return dst
}
for j := 0; j < c; j++ {
dst[j] = a.At(i, j)
}
return dst
}
// CopyVec makes a copy of elements of a into the receiver. It is similar to the
// built-in copy; it copies as much as the overlap between the two vectors and
// returns the number of elements it copied.
func (v *Vector) CopyVec(a *Vector) int {
n := min(v.Len(), a.Len())
if v != a {
blas64.Copy(n, a.mat, v.mat)
}
return n
}
// Col copies the elements in the jth column of the matrix into the slice dst.
// The length of the provided slice must equal the number of rows, unless the
// slice is nil in which case a new slice is first allocated.
func Col(dst []float64, j int, a Matrix) []float64 {
r, c := a.Dims()
if j < 0 || j >= c {
panic(ErrColAccess)
}
if dst == nil {
dst = make([]float64, r)
} else {
if len(dst) != r {
panic(ErrRowLength)
}
}
aMat, aTrans := untranspose(a)
if rm, ok := aMat.(RawMatrixer); ok {
m := rm.RawMatrix()
if aTrans {
copy(dst, m.Data[j*m.Stride:j*m.Stride+m.Cols])
return dst
}
blas64.Copy(r,
blas64.Vector{Inc: m.Stride, Data: m.Data[j:]},
blas64.Vector{Inc: 1, Data: dst},
)
return dst
}
for i := 0; i < r; i++ {
dst[i] = a.At(i, j)
}
return dst
}
// ScaleVec scales the vector a by alpha, placing the result in the receiver.
func (v *Vector) ScaleVec(alpha float64, a *Vector) {
n := a.Len()
if v != a {
v.reuseAs(n)
blas64.Copy(n, a.mat, v.mat)
}
blas64.Scal(n, alpha, v.mat)
}
// Clone makes a copy of a into the receiver, overwriting the previous value of
// the receiver. The clone operation does not make any restriction on shape and
// will not cause shadowing.
//
// See the Cloner interface for more information.
func (m *Dense) Clone(a Matrix) {
r, c := a.Dims()
mat := blas64.General{
Rows: r,
Cols: c,
Stride: c,
}
m.capRows, m.capCols = r, c
aU, trans := untranspose(a)
switch aU := aU.(type) {
case RawMatrixer:
amat := aU.RawMatrix()
mat.Data = make([]float64, r*c)
if trans {
for i := 0; i < r; i++ {
blas64.Copy(c,
blas64.Vector{Inc: amat.Stride, Data: amat.Data[i : i+(c-1)*amat.Stride+1]},
blas64.Vector{Inc: 1, Data: mat.Data[i*c : (i+1)*c]})
}
} else {
for i := 0; i < r; i++ {
copy(mat.Data[i*c:(i+1)*c], amat.Data[i*amat.Stride:i*amat.Stride+c])
}
}
case *Vector:
amat := aU.mat
mat.Data = make([]float64, aU.n)
blas64.Copy(aU.n,
blas64.Vector{Inc: amat.Inc, Data: amat.Data},
blas64.Vector{Inc: 1, Data: mat.Data})
default:
mat.Data = make([]float64, r*c)
w := *m
w.mat = mat
for i := 0; i < r; i++ {
for j := 0; j < c; j++ {
w.set(i, j, a.At(i, j))
}
}
*m = w
return
}
m.mat = mat
}
// CloneVec makes a copy of a into the receiver, overwriting the previous value
// of the receiver.
func (v *Vector) CloneVec(a *Vector) {
if v == a {
return
}
v.n = a.n
v.mat = blas64.Vector{
Inc: 1,
Data: use(v.mat.Data, v.n),
}
blas64.Copy(v.n, a.mat, v.mat)
}
// Clone makes a copy of a into the receiver, overwriting the previous value of
// the receiver. The clone operation does not make any restriction on shape.
//
// See the Cloner interface for more information.
func (m *Dense) Clone(a Matrix) {
r, c := a.Dims()
mat := blas64.General{
Rows: r,
Cols: c,
Stride: c,
}
m.capRows, m.capCols = r, c
aU, trans := untranspose(a)
switch aU := aU.(type) {
case RawMatrixer:
amat := aU.RawMatrix()
// TODO(kortschak): Consider being more precise with determining whether a and m are aliases.
// The current approach is that all RawMatrixers are considered potential aliases.
// Note that below we assume that non-RawMatrixers are not aliases; this is not necessarily
// true, but cases where it is not are not sensible. We should probably fix or document
// this though.
mat.Data = make([]float64, r*c)
if trans {
for i := 0; i < r; i++ {
blas64.Copy(c,
blas64.Vector{Inc: amat.Stride, Data: amat.Data[i : i+(c-1)*amat.Stride+1]},
blas64.Vector{Inc: 1, Data: mat.Data[i*c : (i+1)*c]})
}
} else {
for i := 0; i < r; i++ {
copy(mat.Data[i*c:(i+1)*c], amat.Data[i*amat.Stride:i*amat.Stride+c])
}
}
case Vectorer:
mat.Data = use(m.mat.Data, r*c)
if trans {
for i := 0; i < r; i++ {
aU.Col(mat.Data[i*c:(i+1)*c], i)
}
} else {
for i := 0; i < r; i++ {
aU.Row(mat.Data[i*c:(i+1)*c], i)
}
}
default:
mat.Data = use(m.mat.Data, r*c)
m.mat = mat
for i := 0; i < r; i++ {
for j := 0; j < c; j++ {
m.set(i, j, a.At(i, j))
}
}
return
}
m.mat = mat
}
// SetCol sets the elements of the matrix in the specified column to the values
// of src.
//
// See the VectorSetter interface for more information.
func (m *Dense) SetCol(j int, src []float64) int {
if j >= m.mat.Cols || j < 0 {
panic(ErrColAccess)
}
blas64.Copy(min(len(src), m.mat.Rows),
blas64.Vector{Inc: 1, Data: src},
blas64.Vector{Inc: m.mat.Stride, Data: m.mat.Data[j:]},
)
return min(len(src), m.mat.Rows)
}
// SetCol sets the values in the specified column of the matrix to the values
// in src. len(src) must equal the number of rows in the receiver.
func (m *Dense) SetCol(j int, src []float64) {
if j >= m.mat.Cols || j < 0 {
panic(matrix.ErrColAccess)
}
if len(src) != m.mat.Rows {
panic(matrix.ErrColLength)
}
blas64.Copy(m.mat.Rows,
blas64.Vector{Inc: 1, Data: src},
blas64.Vector{Inc: m.mat.Stride, Data: m.mat.Data[j:]},
)
}
// AddScaledVec adds the vectors a and alpha*b, placing the result in the receiver.
func (v *Vector) AddScaledVec(a *Vector, alpha float64, b *Vector) {
if alpha == 1 {
v.AddVec(a, b)
return
}
if alpha == -1 {
v.SubVec(a, b)
return
}
ar := a.Len()
br := b.Len()
if ar != br {
panic(matrix.ErrShape)
}
v.reuseAs(ar)
if alpha == 0 {
v.CopyVec(a)
return
}
switch {
case v == a && v == b: // v <- v + alpha * v = (alpha + 1) * v
blas64.Scal(ar, alpha+1, v.mat)
case v == a && v != b: // v <- v + alpha * b
blas64.Axpy(ar, alpha, b.mat, v.mat)
case v != a && v == b: // v <- a + alpha * v
if v.mat.Inc == 1 && a.mat.Inc == 1 {
// Fast path for a common case.
v := v.mat.Data
for i, a := range a.mat.Data {
v[i] *= alpha
v[i] += a
}
return
}
blas64.Scal(ar, alpha, v.mat)
blas64.Axpy(ar, 1, a.mat, v.mat)
default: // v <- a + alpha * b
if v.mat.Inc == 1 && a.mat.Inc == 1 && b.mat.Inc == 1 {
// Fast path for a common case.
asm.DaxpyUnitary(alpha, b.mat.Data, a.mat.Data, v.mat.Data)
return
}
blas64.Copy(ar, a.mat, v.mat)
blas64.Axpy(ar, alpha, b.mat, v.mat)
}
}
// Col copies the elements in the jth column of the matrix into the slice dst.
// If the provided slice is nil, a new slice is first allocated.
//
// See the Vectorer interface for more information.
func (m *Dense) Col(dst []float64, j int) []float64 {
if j >= m.mat.Cols || j < 0 {
panic(ErrColAccess)
}
if dst == nil {
dst = make([]float64, m.mat.Rows)
}
dst = dst[:min(len(dst), m.mat.Rows)]
blas64.Copy(len(dst),
blas64.Vector{Inc: m.mat.Stride, Data: m.mat.Data[j:]},
blas64.Vector{Inc: 1, Data: dst},
)
return dst
}
// Copy makes a copy of elements of a into the receiver. It is similar to the
// built-in copy; it copies as much as the overlap between the two matrices and
// returns the number of rows and columns it copied.
//
// See the Copier interface for more information.
func (m *Dense) Copy(a Matrix) (r, c int) {
r, c = a.Dims()
if a == m {
return r, c
}
r = min(r, m.mat.Rows)
c = min(c, m.mat.Cols)
if r == 0 || c == 0 {
return 0, 0
}
aU, trans := untranspose(a)
switch aU := aU.(type) {
case RawMatrixer:
amat := aU.RawMatrix()
if trans {
for i := 0; i < r; i++ {
blas64.Copy(c,
blas64.Vector{Inc: amat.Stride, Data: amat.Data[i : i+(c-1)*amat.Stride+1]},
blas64.Vector{Inc: 1, Data: m.mat.Data[i*m.mat.Stride : i*m.mat.Stride+c]})
}
} else {
for i := 0; i < r; i++ {
copy(m.mat.Data[i*m.mat.Stride:i*m.mat.Stride+c], amat.Data[i*amat.Stride:i*amat.Stride+c])
}
}
case Vectorer:
if trans {
for i := 0; i < r; i++ {
aU.Col(m.mat.Data[i*m.mat.Stride:i*m.mat.Stride+c], i)
}
} else {
for i := 0; i < r; i++ {
aU.Row(m.mat.Data[i*m.mat.Stride:i*m.mat.Stride+c], i)
}
}
default:
for i := 0; i < r; i++ {
for j := 0; j < c; j++ {
m.set(i, j, a.At(i, j))
}
}
}
return r, c
}
// Copy makes a copy of elements of a into the receiver. It is similar to the
// built-in copy; it copies as much as the overlap between the two matrices and
// returns the number of rows and columns it copied. If a aliases the receiver
// and is a transposed Dense or Vector, with a non-unitary increment, Copy will
// panic.
//
// See the Copier interface for more information.
func (m *Dense) Copy(a Matrix) (r, c int) {
r, c = a.Dims()
if a == m {
return r, c
}
r = min(r, m.mat.Rows)
c = min(c, m.mat.Cols)
if r == 0 || c == 0 {
return 0, 0
}
aU, trans := untranspose(a)
switch aU := aU.(type) {
case RawMatrixer:
amat := aU.RawMatrix()
if trans {
if amat.Stride != 1 {
m.checkOverlap(amat)
}
for i := 0; i < r; i++ {
blas64.Copy(c,
blas64.Vector{Inc: amat.Stride, Data: amat.Data[i : i+(c-1)*amat.Stride+1]},
blas64.Vector{Inc: 1, Data: m.mat.Data[i*m.mat.Stride : i*m.mat.Stride+c]})
}
} else {
switch o := offset(m.mat.Data, amat.Data); {
case o < 0:
for i := r - 1; i >= 0; i-- {
copy(m.mat.Data[i*m.mat.Stride:i*m.mat.Stride+c], amat.Data[i*amat.Stride:i*amat.Stride+c])
}
case o > 0:
for i := 0; i < r; i++ {
copy(m.mat.Data[i*m.mat.Stride:i*m.mat.Stride+c], amat.Data[i*amat.Stride:i*amat.Stride+c])
}
default:
// Nothing to do.
}
}
case *Vector:
var n, stride int
amat := aU.mat
if trans {
if amat.Inc != 1 {
m.checkOverlap(aU.asGeneral())
}
n = c
stride = 1
} else {
n = r
stride = m.mat.Stride
}
if amat.Inc == 1 && stride == 1 {
copy(m.mat.Data, amat.Data[:n])
break
}
switch o := offset(m.mat.Data, amat.Data); {
case o < 0:
blas64.Copy(n,
blas64.Vector{Inc: -amat.Inc, Data: amat.Data},
blas64.Vector{Inc: -stride, Data: m.mat.Data})
case o > 0:
blas64.Copy(n,
blas64.Vector{Inc: amat.Inc, Data: amat.Data},
blas64.Vector{Inc: stride, Data: m.mat.Data})
default:
// Nothing to do.
}
default:
for i := 0; i < r; i++ {
for j := 0; j < c; j++ {
m.set(i, j, a.At(i, j))
}
}
}
return r, c
}
// CopyVec makes a copy of elements of a into the receiver. It is similar to the
// built-in copy; it copies as much as the overlap between the two matrices and
// returns the number of rows and columns it copied.
func (v *Vector) CopyVec(a *Vector) (n int) {
n = min(v.Len(), a.Len())
blas64.Copy(n, a.mat, v.mat)
return n
}
// 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
}