/*
* Compute QR factorization of a M-by-N matrix A using compact WY transformation: A = Q * R,
* where Q = I - Y*T*Y.T, T is block reflector and Y holds elementary reflectors as lower
* trapezoidal matrix saved below diagonal elements of the matrix A.
*
* Arguments:
* A On entry, the M-by-N matrix A. On exit, the elements on and above
* the diagonal contain the min(M,N)-by-N upper trapezoidal matrix R.
* The elements below the diagonal with the matrix 'T', represent
* the ortogonal matrix Q as product of elementary reflectors.
*
* T On exit, the K block reflectors which, together with trilu(A) represent
* the ortogonal matrix Q as Q = I - Y*T*Y.T where Y = trilu(A).
* K is ceiling(N/LB) where LB is blocking size from used blocking configuration.
* The matrix T is LB*N augmented matrix of K block reflectors,
* T = [T(0) T(1) .. T(K-1)]. Block reflector T(n) is LB*LB matrix, expect
* reflector T(K-1) that is IB*IB matrix where IB = min(LB, K % LB)
*
* W Workspace, required size returned by QRTFactorWork().
*
* conf Optional blocking configuration. If not provided then default configuration
* is used.
*
* Returns:
* Error indicator.
*
* QRTFactor is compatible with lapack.DGEQRT
*/
func QRTFactor(A, T, W *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
ok := false
rsize := 0
if m(A) < n(A) {
return gomas.NewError(gomas.ESIZE, "QRTFactor")
}
wsz := QRTFactorWork(A, conf)
if W == nil || W.Len() < wsz {
return gomas.NewError(gomas.EWORK, "QRTFactor", wsz)
}
tr, tc := T.Size()
if conf.LB == 0 || conf.LB > n(A) {
ok = tr == tc && tr == n(A)
rsize = n(A) * n(A)
} else {
ok = tr == conf.LB && tc == n(A)
rsize = conf.LB * n(A)
}
if !ok {
return gomas.NewError(gomas.ESMALL, "QRTFactor", rsize)
}
if conf.LB == 0 || n(A) <= conf.LB {
err = unblockedQRT(A, T, W)
} else {
Wrk := cmat.MakeMatrix(n(A), conf.LB, W.Data())
err = blockedQRT(A, T, Wrk, conf)
}
return err
}
/*
* Solve a system of linear equations A*X = B or A.T*X = B with general N-by-N
* matrix A using the LU factorization computed by LUFactor().
*
* Arguments:
* B On entry, the right hand side matrix B. On exit, the solution matrix X.
*
* A The factor L and U from the factorization A = P*L*U as computed by
* LUFactor()
*
* pivots The pivot indices from LUFactor().
*
* flags The indicator of the form of the system of equations.
* If flags&TRANSA then system is transposed. All other values
* indicate non transposed system.
*
* Compatible with lapack.DGETRS.
*/
func LUSolve(B, A *cmat.FloatMatrix, pivots Pivots, flags int, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.DefaultConf()
if len(confs) > 0 {
conf = confs[0]
}
ar, ac := A.Size()
br, _ := B.Size()
if ar != ac {
return gomas.NewError(gomas.ENOTSQUARE, "SolveLU")
}
if br != ac {
return gomas.NewError(gomas.ESIZE, "SolveLU")
}
if pivots != nil {
applyPivots(B, pivots)
}
if flags&gomas.TRANSA != 0 {
// transposed X = A.-1*B == (L.T*U.T).-1*B == U.-T*(L.-T*B)
blasd.SolveTrm(B, A, 1.0, gomas.LOWER|gomas.UNIT|gomas.TRANSA, conf)
blasd.SolveTrm(B, A, 1.0, gomas.UPPER|gomas.TRANSA, conf)
} else {
// non-transposed X = A.-1*B == (L*U).-1*B == U.-1*(L.-1*B)
blasd.SolveTrm(B, A, 1.0, gomas.LOWER|gomas.UNIT, conf)
blasd.SolveTrm(B, A, 1.0, gomas.UPPER, conf)
}
return err
}
func MVMult(Y, A, X *cmat.FloatMatrix, alpha, beta float64, bits int, confs ...*gomas.Config) *gomas.Error {
ok := true
yr, yc := Y.Size()
ar, ac := A.Size()
xr, xc := X.Size()
if ar*ac == 0 {
return nil
}
if yr != 1 && yc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVMult")
}
if xr != 1 && xc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVMult")
}
nx := X.Len()
ny := Y.Len()
if bits&gomas.TRANSA != 0 {
bits |= gomas.TRANS
}
if bits&gomas.TRANS != 0 {
ok = ny == ac && nx == ar
} else {
ok = ny == ar && nx == ac
}
if !ok {
return gomas.NewError(gomas.ESIZE, "MVMult")
}
if beta != 1.0 {
vscal(Y, beta, ny)
}
gemv(Y, A, X, alpha, beta, bits, 0, nx, 0, ny)
return nil
}
/*
* Generate the M by N matrix Q with orthogonal rows which
* are defined as the first M rows of the product of K first elementary
* reflectors.
*
* Arguments
* A On entry, the elementary reflectors as returned by LQFactor().
* stored right of diagonal of the M by N matrix A.
* On exit, the orthogonal matrix Q
*
* tau Scalar coefficents of elementary reflectors
*
* W Workspace
*
* K The number of elementary reflector whose product define the matrix Q
*
* conf Optional blocking configuration.
*
* Compatible with lapackd.ORGLQ.
*/
func LQBuild(A, tau, W *cmat.FloatMatrix, K int, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
if K <= 0 || K > n(A) {
return gomas.NewError(gomas.EVALUE, "LQBuild", K)
}
wsz := wsBuildLQ(A, 0)
if W == nil || W.Len() < wsz {
return gomas.NewError(gomas.EWORK, "LQBuild", wsz)
}
// adjust blocking factor for workspace size
lb := estimateLB(A, W.Len(), wsBuildLQ)
//lb = imin(lb, conf.LB)
lb = conf.LB
if lb == 0 || m(A) <= lb {
unblkBuildLQ(A, tau, W, m(A)-K, n(A)-K, true)
} else {
var Twork, Wrk cmat.FloatMatrix
Twork.SetBuf(lb, lb, lb, W.Data())
Wrk.SetBuf(m(A)-lb, lb, m(A)-lb, W.Data()[Twork.Len():])
blkBuildLQ(A, tau, &Twork, &Wrk, K, lb, conf)
}
return err
}
/*
* Symmetric matrix-vector multiplication. Y = beta*Y + alpha*A*X
*/
func MVMultSym(Y, A, X *cmat.FloatMatrix, alpha, beta float64, bits int, confs ...*gomas.Config) *gomas.Error {
ok := true
yr, yc := Y.Size()
ar, ac := A.Size()
xr, xc := X.Size()
if ar*ac == 0 {
return nil
}
if yr != 1 && yc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVMultSym")
}
if xr != 1 && xc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVMultSym")
}
nx := X.Len()
ny := Y.Len()
ok = ny == ar && nx == ac && ac == ar
if !ok {
return gomas.NewError(gomas.ESIZE, "MVMultSym")
}
if beta != 1.0 {
vscal(Y, beta, ny)
}
symv(Y, A, X, alpha, bits, nx)
return nil
}
/*
* Compute RQ factorization of a M-by-N matrix A: A = R*Q
*
* Arguments:
* A On entry, the M-by-N matrix A, M <= N. On exit, upper triangular matrix R
* and the orthogonal matrix Q as product of elementary reflectors.
*
* tau On exit, the scalar factors of the elementary reflectors.
*
* W Workspace, M-by-nb matrix used for work space in blocked invocations.
*
* conf The blocking configuration. If nil then default blocking configuration
* is used. Member conf.LB defines blocking size of blocked algorithms.
* If it is zero then unblocked algorithm is used.
*
* Returns:
* Error indicator.
*
* Additional information
*
* Ortogonal matrix Q is product of elementary reflectors H(k)
*
* Q = H(0)H(1),...,H(K-1), where K = min(M,N)
*
* Elementary reflector H(k) is stored on row k of A right of the diagonal with
* implicit unit value on diagonal entry. The vector TAU holds scalar factors of
* the elementary reflectors.
*
* Contents of matrix A after factorization is as follow:
*
* ( v0 v0 r r r r ) M=4, N=6
* ( v1 v1 v1 r r r )
* ( v2 v2 v2 v2 r r )
* ( v3 v3 v3 v3 v3 r )
*
* where l is element of L, vk is element of H(k).
*
* RQFactor is compatible with lapack.DGERQF
*/
func RQFactor(A, tau, W *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
// must have: M <= N
if m(A) > n(A) {
return gomas.NewError(gomas.ESIZE, "RQFactor")
}
wsmin := wsLQ(A, 0)
if W == nil || W.Len() < wsmin {
return gomas.NewError(gomas.EWORK, "RQFactor", wsmin)
}
lb := estimateLB(A, W.Len(), wsRQ)
lb = imin(lb, conf.LB)
if lb == 0 || m(A) <= lb {
unblockedRQ(A, tau, W)
} else {
var Twork, Wrk cmat.FloatMatrix
// block reflector T in first LB*LB elements in workspace
// the rest, m(A)-LB*LB, is workspace for intermediate matrix operands
Twork.SetBuf(lb, lb, lb, W.Data())
Wrk.SetBuf(m(A)-lb, lb, m(A)-lb, W.Data()[Twork.Len():])
blockedRQ(A, tau, &Twork, &Wrk, lb, conf)
}
return err
}
/*
* Multiply and replace C with Q*C or Q.T*C where Q is a real orthogonal matrix
* defined as the product of k elementary reflectors and block reflector T
*
* Q = H(1) H(2) . . . H(k)
*
* as returned by DecomposeQRT().
*
* Arguments:
* C On entry, the M-by-N matrix C. On exit C is overwritten by Q*C or Q.T*C.
*
* A QR factorization as returned by QRTFactor() where the lower trapezoidal
* part holds the elementary reflectors.
*
* T The block reflector computed from elementary reflectors as returned by
* DecomposeQRT() or computed from elementary reflectors and scalar coefficients
* by BuildT()
*
* W Workspace, size as returned by QRTMultWork()
*
* conf Blocking configuration
*
* flags Indicators. Valid indicators LEFT, RIGHT, TRANS, NOTRANS
*
* Preconditions:
* a. cols(A) == cols(T),
* columns A define number of elementary reflector, must match order of block reflector.
* b. if conf.LB == 0, cols(T) == rows(T)
* unblocked invocation, block reflector T is upper triangular
* c. if conf.LB != 0, rows(T) == conf.LB
* blocked invocation, T is sequence of triangular block reflectors of order LB
* d. if LEFT, rows(C) >= cols(A) && cols(C) >= rows(A)
*
* e. if RIGHT, cols(C) >= cols(A) && rows(C) >= rows(A)
*
* Compatible with lapack.DGEMQRT
*/
func QRTMult(C, A, T, W *cmat.FloatMatrix, flags int, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
wsz := QRTMultWork(C, T, flags, conf)
if W == nil || W.Len() < wsz {
return gomas.NewError(gomas.EWORK, "QRTMult", wsz)
}
ok := false
switch flags & gomas.RIGHT {
case gomas.RIGHT:
ok = n(C) >= m(A)
default:
ok = m(C) >= n(A)
}
if !ok {
return gomas.NewError(gomas.ESIZE, "QRTMult")
}
var Wrk cmat.FloatMatrix
if flags&gomas.RIGHT != 0 {
Wrk.SetBuf(m(C), conf.LB, m(C), W.Data())
blockedMultQTRight(C, A, T, &Wrk, flags, conf)
} else {
Wrk.SetBuf(n(C), conf.LB, n(C), W.Data())
blockedMultQTLeft(C, A, T, &Wrk, flags, conf)
}
return err
}
/*
* Compute
* B = B*diag(D).-1 flags & RIGHT == true
* B = diag(D).-1*B flags & LEFT == true
*
* If flags is LEFT (RIGHT) then element-wise divides columns (rows) of B with vector D.
*
* Arguments:
* B M-by-N matrix if flags&RIGHT == true or N-by-M matrix if flags&LEFT == true
*
* D N element column or row vector or N-by-N matrix
*
* flags Indicator bits, LEFT or RIGHT
*/
func SolveDiag(B, D *cmat.FloatMatrix, flags int, confs ...*gomas.Config) *gomas.Error {
var c, d0 cmat.FloatMatrix
var d *cmat.FloatMatrix
conf := gomas.CurrentConf(confs...)
d = D
if !D.IsVector() {
d0.Diag(D)
d = &d0
}
dn := d0.Len()
br, bc := B.Size()
switch flags & (gomas.LEFT | gomas.RIGHT) {
case gomas.LEFT:
if br != dn {
return gomas.NewError(gomas.ESIZE, "SolveDiag")
}
// scale rows;
for k := 0; k < dn; k++ {
c.Row(B, k)
blasd.InvScale(&c, d.GetAt(k), conf)
}
case gomas.RIGHT:
if bc != dn {
return gomas.NewError(gomas.ESIZE, "SolveDiag")
}
// scale columns
for k := 0; k < dn; k++ {
c.Column(B, k)
blasd.InvScale(&c, d.GetAt(k), conf)
}
}
return nil
}
// Solve secular function arising in symmetric eigenproblems. On exit 'Y' contains new
// eigenvalues and 'V' the rank-one update vector corresponding new eigenvalues.
// The matrix Qd holds for each eigenvalue then computed deltas as row vectors.
// On entry 'D' holds original eigenvalues and 'Z' is the rank-one update vector.
func TRDSecularSolveAll(y, v, Qd, d, z *cmat.FloatMatrix, rho float64, confs ...*gomas.Config) (err *gomas.Error) {
var delta cmat.FloatMatrix
var lmbda float64
var e, ei int
ei = 0
err = nil
if y.Len() != d.Len() || z.Len() != d.Len() || m(Qd) != n(Qd) || m(Qd) != d.Len() {
err = gomas.NewError(gomas.ESIZE, "TRDSecularSolveAll")
return
}
for i := 0; i < d.Len(); i++ {
delta.Row(Qd, i)
lmbda, e = trdsecRoot(d, z, &delta, i, rho)
if e < 0 && ei == 0 {
ei = -(i + 1)
}
y.SetAt(i, lmbda)
}
if ei == 0 {
trdsecUpdateVecDelta(v, Qd, d, rho)
} else {
err = gomas.NewError(gomas.ECONVERGE, "TRDSecularSolveAll", ei)
}
return
}
/*
* Compute QL factorization of a M-by-N matrix A: A = Q * L.
*
* Arguments:
* A On entry, the M-by-N matrix A, M >= N. On exit, lower triangular matrix L
* and the orthogonal matrix Q as product of elementary reflectors.
*
* tau On exit, the scalar factors of the elemenentary reflectors.
*
* W Workspace, N-by-nb matrix used for work space in blocked invocations.
*
* conf The blocking configuration. If nil then default blocking configuration
* is used. Member conf.LB defines blocking size of blocked algorithms.
* If it is zero then unblocked algorithm is used.
*
* Returns:
* Error indicator.
*
* Additional information
*
* Ortogonal matrix Q is product of elementary reflectors H(k)
*
* Q = H(K-1)...H(1)H(0), where K = min(M,N)
*
* Elementary reflector H(k) is stored on column k of A above the diagonal with
* implicit unit value on diagonal entry. The vector TAU holds scalar factors
* of the elementary reflectors.
*
* Contents of matrix A after factorization is as follow:
*
* ( v0 v1 v2 v3 ) for M=6, N=4
* ( v0 v1 v2 v3 )
* ( l v1 v2 v3 )
* ( l l v2 v3 )
* ( l l l v3 )
* ( l l l l )
*
* where l is element of L, vk is element of H(k).
*
* DecomposeQL is compatible with lapack.DGEQLF
*/
func QLFactor(A, tau, W *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
var tauh cmat.FloatMatrix
conf := gomas.CurrentConf(confs...)
if m(A) < n(A) {
return gomas.NewError(gomas.ESIZE, "QLFactor")
}
wsmin := wsQL(A, 0)
if W == nil || W.Len() < wsmin {
return gomas.NewError(gomas.EWORK, "QLFactor", wsmin)
}
if tau.Len() < n(A) {
return gomas.NewError(gomas.ESIZE, "QLFactor")
}
tauh.SubMatrix(tau, 0, 0, n(A), 1)
lb := estimateLB(A, W.Len(), wsQL)
lb = imin(lb, conf.LB)
if lb == 0 || n(A) <= lb {
unblockedQL(A, &tauh, W)
} else {
var Twork, Wrk cmat.FloatMatrix
// block reflector T in first LB*LB elements in workspace
// the rest, n(A)-LB*LB, is workspace for intermediate matrix operands
Twork.SetBuf(conf.LB, conf.LB, -1, W.Data())
Wrk.SetBuf(n(A)-conf.LB, conf.LB, -1, W.Data()[Twork.Len():])
blockedQL(A, &tauh, &Twork, &Wrk, lb, conf)
}
return err
}
func EigenSym(D, A, W *cmat.FloatMatrix, bits int, confs ...*gomas.Config) (err *gomas.Error) {
var sD, sE, E, tau, Wred cmat.FloatMatrix
var vv *cmat.FloatMatrix
err = nil
vv = nil
conf := gomas.CurrentConf(confs...)
if m(A) != n(A) || D.Len() != m(A) {
err = gomas.NewError(gomas.ESIZE, "EigenSym")
return
}
if bits&gomas.WANTV != 0 && W.Len() < 3*n(A) {
err = gomas.NewError(gomas.EWORK, "EigenSym")
return
}
if bits&(gomas.LOWER|gomas.UPPER) == 0 {
bits = bits | gomas.LOWER
}
ioff := 1
if bits&gomas.LOWER != 0 {
ioff = -1
}
E.SetBuf(n(A)-1, 1, n(A)-1, W.Data())
tau.SetBuf(n(A), 1, n(A), W.Data()[n(A)-1:])
wrl := W.Len() - 2*n(A) - 1
Wred.SetBuf(wrl, 1, wrl, W.Data()[2*n(A)-1:])
// reduce to tridiagonal
if err = TRDReduce(A, &tau, &Wred, bits, conf); err != nil {
err.Update("EigenSym")
return
}
sD.Diag(A)
sE.Diag(A, ioff)
blasd.Copy(D, &sD)
blasd.Copy(&E, &sE)
if bits&gomas.WANTV != 0 {
if err = TRDBuild(A, &tau, &Wred, n(A), bits, conf); err != nil {
err.Update("EigenSym")
return
}
vv = A
}
// resize workspace
wrl = W.Len() - n(A) - 1
Wred.SetBuf(wrl, 1, wrl, W.Data()[n(A)-1:])
if err = TRDEigen(D, &E, vv, &Wred, bits, conf); err != nil {
err.Update("EigenSym")
return
}
return
}
// Computes eigenvectors corresponding the updated eigenvalues and rank-one update vector.
// The matrix Qd holds precomputed deltas as returned by TRDSecularSolveAll(). If Qd is nil or
// Qd same as the matrix Q then computation is in-place and Q is assumed to hold precomputed
// deltas. On exit, Q holds the column eigenvectors.
func TRDSecularEigen(Q, v, Qd *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
if m(Q) != n(Q) || (Qd != nil && (m(Qd) != n(Qd) || m(Qd) != m(Q))) {
return gomas.NewError(gomas.ESIZE, "TRDSecularEigen")
}
if m(Q) != v.Len() {
return gomas.NewError(gomas.ESIZE, "TRDSecularEigen")
}
if Qd == nil || Qd == Q {
trdsecEigenBuildInplace(Q, v)
} else {
trdsecEigenBuild(Q, v, Qd)
}
return nil
}
func Swap(X, Y *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
xr, xc := X.Size()
yr, yc := Y.Size()
if xr != 1 && xc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "Swap")
}
if yr != 1 && yc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "Swap")
}
if X.Len() != Y.Len() {
return gomas.NewError(gomas.ESIZE, "Swap")
}
vswap(X, Y, X.Len())
return nil
}
/*
* Reduce general matrix A to upper Hessenberg form H by similiarity
* transformation H = Q.T*A*Q.
*
* Arguments:
* A On entry, the general matrix A. On exit, the elements on and
* above the first subdiagonal contain the reduced matrix H.
* The elements below the first subdiagonal with the vector tau
* represent the ortogonal matrix A as product of elementary reflectors.
*
* tau On exit, the scalar factors of the elementary reflectors.
*
* W Workspace, as defined by HessReduceWork()
*
* conf The blocking configration.
*
* HessReduce is compatible with lapack.DGEHRD.
*/
func HessReduce(A, tau, W *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
wmin := m(A)
wopt := HessReduceWork(A, conf)
wsz := W.Len()
if wsz < wmin {
return gomas.NewError(gomas.EWORK, "ReduceHess", wmin)
}
// use blocked version if workspace big enough for blocksize 4
lb := conf.LB
if wsz < wopt {
lb = estimateLB(A, wsz, wsHess)
}
if lb == 0 || n(A) <= lb {
unblkHessGQvdG(A, tau, W, 0)
} else {
// blocked version
var W0 cmat.FloatMatrix
// shape workspace for blocked algorithm
W0.SetBuf(m(A)+lb, lb, m(A)+lb, W.Data())
blkHessGQvdG(A, tau, &W0, lb, conf)
}
return err
}
/*
* Reduce a general M-by-N matrix A to upper or lower bidiagonal form B
* by an ortogonal transformation A = Q*B*P.T, B = Q.T*A*P
*
*
* Arguments
* A On entry, the real M-by-N matrix. On exit the upper/lower
* bidiagonal matrix and ortogonal matrices Q and P.
*
* tauq Scalar factors for elementary reflector forming the
* ortogonal matrix Q.
*
* taup Scalar factors for elementary reflector forming the
* ortogonal matrix P.
*
* W Workspace needed for reduction.
*
* conf Current blocking configuration. Optional.
*
*
* Details
*
* Matrices Q and P are products of elementary reflectors H(k) and G(k)
*
* If M > N:
* Q = H(1)*H(2)*...*H(N) and P = G(1)*G(2)*...*G(N-1)
*
* where H(k) = 1 - tauq*u*u.T and G(k) = 1 - taup*v*v.T
*
* Elementary reflector H(k) are stored on columns of A below the diagonal with
* implicit unit value on diagonal entry. Vector TAUQ holds corresponding scalar
* factors. Reflector G(k) are stored on rows of A right of first superdiagonal
* with implicit unit value on superdiagonal. Corresponding scalar factors are
* stored on vector TAUP.
*
* If M < N:
* Q = H(1)*H(2)*...*H(N-1) and P = G(1)*G(2)*...*G(N)
*
* where H(k) = 1 - tauq*u*u.T and G(k) = 1 - taup*v*v.T
*
* Elementary reflector H(k) are stored on columns of A below the first sub diagonal
* with implicit unit value on sub diagonal entry. Vector TAUQ holds corresponding
* scalar factors. Reflector G(k) are sotre on rows of A right of diagonal with
* implicit unit value on superdiagonal. Corresponding scalar factors are stored
* on vector TAUP.
*
* Contents of matrix A after reductions are as follows.
*
* M = 6 and N = 5: M = 5 and N = 6:
*
* ( d e v1 v1 v1 ) ( d v1 v1 v1 v1 v1 )
* ( u1 d e v2 v2 ) ( e d v2 v2 v2 v2 )
* ( u1 u2 d e v3 ) ( u1 e d v3 v3 v3 )
* ( u1 u2 u3 d e ) ( u1 u2 e d v4 v4 )
* ( u1 u2 u3 u4 d ) ( u1 u2 u3 e d v5 )
* ( u1 u2 u3 u4 u5 )
*/
func BDReduce(A, tauq, taup, W *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
_ = conf
wmin := wsBired(A, 0)
wsz := W.Len()
if wsz < wmin {
return gomas.NewError(gomas.EWORK, "ReduceBidiag", wmin)
}
lb := conf.LB
wneed := wsBired(A, lb)
if wneed > wsz {
lb = estimateLB(A, wsz, wsBired)
}
if m(A) >= n(A) {
if lb > 0 && n(A) > lb {
blkBidiagLeft(A, tauq, taup, W, lb, conf)
} else {
unblkReduceBidiagLeft(A, tauq, taup, W)
}
} else {
if lb > 0 && m(A) > lb {
blkBidiagRight(A, tauq, taup, W, lb, conf)
} else {
unblkReduceBidiagRight(A, tauq, taup, W)
}
}
return err
}
/*
* Compute LDL^T factorization of real symmetric matrix.
*
* Computes of a real symmetric matrix A using Bunch-Kauffman pivoting method.
* The form of factorization is
*
* A = L*D*L.T or A = U*D*U.T
*
* where L (or U) is product of permutation and unit lower (or upper) triangular matrix
* and D is block diagonal symmetric matrix with 1x1 and 2x2 blocks.
*
* Arguments
* A On entry, the N-by-N symmetric matrix A. If flags bit LOWER (or UPPER) is set then
* lower (or upper) triangular matrix and strictly upper (or lower) part is not
* accessed. On exit, the block diagonal matrix D and lower (or upper) triangular
* product matrix L (or U).
*
* W Workspace, size as returned by WorksizeBK().
*
* ipiv Pivot vector. On exit details of interchanges and the block structure of D. If
* ipiv[k] > 0 then D[k,k] is 1x1 and rows and columns k and ipiv[k]-1 were changed.
* If ipiv[k] == ipiv[k+1] < 0 then D[k,k] is 2x2. If A is lower then rows and
* columns k+1 and ipiv[k]-1 were changed. And if A is upper then rows and columns
* k and ipvk[k]-1 were changed.
*
* flags Indicator bits, LOWER or UPPER.
*
* confs Optional blocking configuration. If not provided then default blocking
* as returned by DefaultConf() is used.
*
* Unblocked algorithm is used if blocking configuration LB is zero or if N < LB.
*
* Compatible with lapack.SYTRF.
*/
func BKFactor(A, W *cmat.FloatMatrix, ipiv Pivots, flags int, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
for k, _ := range ipiv {
ipiv[k] = 0
}
wsz := BKFactorWork(A, conf)
if W.Len() < wsz {
return gomas.NewError(gomas.EWORK, "DecomposeBK", wsz)
}
var Wrk cmat.FloatMatrix
if n(A) < conf.LB || conf.LB == 0 {
// make workspace rows(A)*2 matrix
Wrk.SetBuf(m(A), 2, m(A), W.Data())
if flags&gomas.LOWER != 0 {
err, _ = unblkDecompBKLower(A, &Wrk, ipiv, conf)
} else if flags&gomas.UPPER != 0 {
err, _ = unblkDecompBKUpper(A, &Wrk, ipiv, conf)
}
} else {
// make workspace rows(A)*(LB+1) matrix
Wrk.SetBuf(m(A), conf.LB+1, m(A), W.Data())
if flags&gomas.LOWER != 0 {
err = blkDecompBKLower(A, &Wrk, &ipiv, conf)
} else if flags&gomas.UPPER != 0 {
err = blkDecompBKUpper(A, &Wrk, &ipiv, conf)
}
}
return err
}
func MVSolveTrm(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, "MVSolveTrm")
}
nx := X.Len()
if ac != nx || ar != ac {
return gomas.NewError(gomas.ESIZE, "MVSolveTrm")
}
trsv(X, A, alpha, bits, nx)
return nil
}
func MVUpdateSym(A, X *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, "MVUpdateSym")
}
nx := X.Len()
if ac != nx || ar != ac {
return gomas.NewError(gomas.ESIZE, "MVUpdateSym")
}
updtrmv(A, X, X, alpha, bits, nx, nx)
return nil
}
/*
* Multiply and replace C with Q*C or Q.T*C where Q is a real orthogonal matrix
* defined as the product of k elementary reflectors.
*
* Q = H(0) H(1) . . . H(K-1)
*
* as returned by QRFactor().
*
* Arguments:
* C On entry, the M-by-N matrix C or if flag bit RIGHT is set then N-by-M matrix
* On exit C is overwritten by Q*C or Q.T*C. If bit RIGHT is set then C is
* overwritten by C*Q or C*Q.T
*
* A QR factorization as returned by QRFactor() where the lower trapezoidal
* part holds the elementary reflectors.
*
* tau The scalar factors of the elementary reflectors.
*
* W Workspace matrix, required size is returned by WorksizeMultQ().
*
* flags Indicators. Valid indicators LEFT, RIGHT, TRANS
*
* conf Blocking configuration. Field LB defines block size. If it is zero
* unblocked invocation is assumed. Actual blocking size is adjusted
* to available workspace size and minimum of configured block size and
* block size implied by workspace is used.
*
* Compatible with lapack.DORMQR
*/
func QRMult(C, A, tau, W *cmat.FloatMatrix, flags int, confs ...*gomas.Config) *gomas.Error {
var err *gomas.Error = nil
conf := gomas.CurrentConf(confs...)
// default to multiply from left if side not defined
if flags&(gomas.LEFT|gomas.RIGHT) == 0 {
flags = flags | gomas.LEFT
}
// n(A) is number of elementary reflectors defining the Q matrix
ok := false
wsizer := wsMultQLeft
switch flags & gomas.RIGHT {
case gomas.RIGHT:
ok = n(C) == m(A)
wsizer = wsMultQRight
default:
ok = m(C) == m(A)
}
if !ok {
return gomas.NewError(gomas.ESIZE, "QRMult")
}
// minimum workspace size
wsz := wsizer(C, 0)
if W == nil || W.Len() < wsz {
return gomas.NewError(gomas.EWORK, "QRMult", wsz)
}
// estimate blocking factor for current workspace
lb := estimateLB(C, W.Len(), wsizer)
lb = imin(lb, conf.LB)
if lb == 0 || n(A) <= lb {
if flags&gomas.RIGHT != 0 {
unblockedMultQRight(C, A, tau, W, flags)
} else {
unblockedMultQLeft(C, A, tau, W, flags)
}
} else {
if flags&gomas.RIGHT != 0 {
blockedMultQRight(C, A, tau, W, flags, lb, conf)
} else {
blockedMultQLeft(C, A, tau, W, flags, lb, conf)
}
}
return err
}
/*
* Build block reflector T from Householder elementary reflectors stored in TriLU(A)
* and scalar factors 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
*
* Compatible with lapack.DLAFRT
*/
func QRReflector(T, A, tau *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
if n(T) < n(A) || m(T) < n(A) {
return gomas.NewError(gomas.ESIZE, "QRReflector")
}
unblkQRBlockReflector(T, A, tau)
return nil
}
func Copy(A, B *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
ar, ac := A.Size()
br, bc := B.Size()
avec := ar == 1 || ac == 1
bvec := br == 1 || bc == 1
if avec && bvec {
if A.Len() < B.Len() {
return gomas.NewError(gomas.ESIZE, "Copy")
}
vcopy(A, B, A.Len())
return nil
}
if ar != br || ac != bc {
return gomas.NewError(gomas.ESIZE, "Copy")
}
mcopy(A, B, ar, ac)
return nil
}
func Transpose(A, B *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
ar, ac := A.Size()
br, bc := B.Size()
if ar != bc || ac != br {
return gomas.NewError(gomas.ESIZE, "Transpose")
}
mtranspose(A, B, br, bc)
return nil
}
/*
* Build block reflector T from Householder elementary reflectors stored in A
* and scalar factors in tau.
*
* Q = I - Y*T*Y.T; Householder H = I - tau*v*v.T
*
* T = | T 0 | z = -tau*T*Yt*v
* | z c | c = tau
*
* Compatible with lapack.DLAFRT
*/
func RQReflector(T, A, tau *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
if n(T) < m(A) || m(T) < m(A) {
return gomas.NewError(gomas.ESIZE, "BuildLQT")
}
unblkBlockReflectorRQ(T, A, tau)
return nil
}
func Axpby(Y, X *cmat.FloatMatrix, alpha, beta float64, confs ...*gomas.Config) *gomas.Error {
if X.Len() == 0 || Y.Len() == 0 {
return nil
}
xr, xc := X.Size()
if xr != 1 && xc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "Axpby")
}
yr, yc := Y.Size()
if yr != 1 && yc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "Axpby")
}
nx := X.Len()
if nx != Y.Len() {
return gomas.NewError(gomas.ESIZE, "Axpby")
}
axpby(Y, X, alpha, beta, nx)
return nil
}
/*
* Triangular matrix multiplication.
*/
func MultTrm(B, A *cmat.FloatMatrix, alpha float64, bits int, confs ...*gomas.Config) *gomas.Error {
conf := gomas.DefaultConf()
if len(confs) > 0 {
conf = confs[0]
}
if B.Len() == 0 || A.Len() == 0 {
return nil
}
ok := true
ar, ac := A.Size()
br, bc := B.Size()
P := ac
E := bc
switch {
case bits&gomas.RIGHT != 0:
ok = bc == ar && ar == ac
E = br
case bits&gomas.LEFT != 0:
fallthrough
default:
ok = ac == br && ar == ac
}
if !ok {
return gomas.NewError(gomas.ESIZE, "MultTrm")
}
// single threaded
if conf.NProc == 1 || conf.WB <= 0 || E < conf.WB/2 {
trmm(B, A, alpha, bits, P, 0, E, conf)
return nil
}
// parallelized
wait := make(chan int, 4)
_, nN := blocking(0, E, conf.WB/2)
nT := 0
for j := 0; j < nN; j++ {
jS := blockIndex(j, nN, conf.WB/2, E)
jL := blockIndex(j+1, nN, conf.WB/2, E)
task := func(q chan int) {
trmm(B, A, alpha, bits, P, jS, jL, conf)
q <- 1
}
conf.Sched.Schedule(gomas.NewTask(task, wait))
nT += 1
}
for nT > 0 {
<-wait
nT -= 1
}
return nil
}
/*
* Build block reflector for QL factorized matrix.
*/
func QLReflector(T, A, tau *cmat.FloatMatrix, confs ...*gomas.Config) *gomas.Error {
var tauh cmat.FloatMatrix
if n(T) < n(A) || m(T) < n(A) {
return gomas.NewError(gomas.ESIZE, "QLReflector")
}
tauh.SubMatrix(tau, 0, 0, imin(m(A), n(A)), 1)
unblkQLBlockReflector(T, A, &tauh)
return nil
}
func Update2Sym(Cc, A, B *cmat.FloatMatrix, alpha, beta float64, bits int, confs ...*gomas.Config) *gomas.Error {
conf := gomas.DefaultConf()
if len(confs) > 0 {
conf = confs[0]
}
ok := true
cr, cc := Cc.Size()
ar, ac := A.Size()
br, bc := B.Size()
if cr*cc == 0 {
return nil
}
P := ac
E := cr
if bits&gomas.TRANS != 0 && bits&gomas.TRANSA == 0 {
bits |= gomas.TRANSA
}
switch {
case bits&gomas.TRANSA != 0:
ok = cr == cc && cr == ac && bc == ac && br == ar
P = ar
default:
ok = cr == cc && cr == ar && br == ar && bc == ac
}
if !ok {
return gomas.NewError(gomas.ESIZE, "Update2Sym")
}
if conf.NProc == 1 || conf.WB <= 0 || E <= conf.WB {
syr2k(Cc, A, B, alpha, beta, bits, P, 0, E, conf)
return nil
}
// parallelized
wait := make(chan int, 4)
_, nN := blocking(0, E, conf.WB)
nT := 0
for j := 0; j < nN; j++ {
jS := blockIndex(j, nN, conf.WB, E)
jE := blockIndex(j+1, nN, conf.WB, E)
task := func(q chan int) {
syr2k(Cc, A, B, alpha, beta, bits, P, jS, jE, conf)
q <- 1
}
conf.Sched.Schedule(gomas.NewTask(task, wait))
nT += 1
}
for nT > 0 {
<-wait
nT -= 1
}
return nil
}
// Solve secular function arising in symmetric eigenproblems. On exit 'Y' contains new
// eigenvalues On entry 'D' holds original eigenvalues and 'Z' is the rank-one update vector.
// Parameter 'delta' is workspace needed for computation.
func TRDSecularSolve(Y, D, Z, delta *cmat.FloatMatrix, rho float64, confs ...*gomas.Config) (err *gomas.Error) {
var lmbda float64
var e, ei int
ei = 0
err = nil
if delta.Len() != D.Len() || Y.Len() != D.Len() || Z.Len() != D.Len() {
err = gomas.NewError(gomas.ESIZE, "TRDSecularSolve")
return
}
for i := 0; i < D.Len(); i++ {
lmbda, e = trdsecRoot(D, Z, delta, i, rho)
if e < 0 && ei == 0 {
ei = -(i + 1)
}
Y.SetAt(i, lmbda)
}
if ei != 0 {
err = gomas.NewError(gomas.ECONVERGE, "TRDSecularSolve", ei)
}
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
}
func MVUpdateTrm(A, X, Y *cmat.FloatMatrix, alpha float64, bits int, confs ...*gomas.Config) *gomas.Error {
ar, ac := A.Size()
yr, yc := Y.Size()
xr, xc := X.Size()
if ar*ac == 0 {
return nil
}
if yr != 1 && yc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVUpdateTrm")
}
if xr != 1 && xc != 1 {
return gomas.NewError(gomas.ENEED_VECTOR, "MVUpdateTrm")
}
nx := X.Len()
ny := Y.Len()
if ac != ny || ar != nx {
return gomas.NewError(gomas.ESIZE, "MVUpdateTrm")
}
updtrmv(A, X, Y, alpha, bits, ny, nx)
return nil
}