/*
* 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
}
// unblocked LU decomposition w/o pivots, FLAME LU nopivots variant 5
func unblockedLUnoPiv(A *cmat.FloatMatrix, conf *gomas.Config) *gomas.Error {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a10, a11, a12, A20, a21, A22 cmat.FloatMatrix
var err *gomas.Error = nil
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
for m(&ATL) < m(A) {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
&a10, &a11, &a12,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
// a21 = a21/a11
blasd.InvScale(&a21, a11.Get(0, 0))
// A22 = A22 - a21*a12
blasd.MVUpdate(&A22, &a21, &a12, -1.0)
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
}
return err
}
// Compute eigenvector corresponding precomputed deltas
func trdsecEigenVecDelta(qi, delta, z *cmat.FloatMatrix) {
var dk, zk float64
for k := 0; k < delta.Len(); k++ {
zk = z.GetAt(k)
dk = delta.GetAt(k)
qi.SetAt(k, zk/dk)
}
s := blasd.Nrm2(qi)
blasd.InvScale(qi, s)
}
func TestDVecScal(t *testing.T) {
const N = 911
X := cmat.NewMatrix(N, 1)
Y := cmat.NewMatrix(N, 1)
zeromean := cmat.NewFloatUniformSource(2.0, 0.5)
X.SetFrom(zeromean)
Y.Copy(X)
// B = A*B
blasd.Scale(X, 2.0)
blasd.InvScale(X, 2.0)
ok := X.AllClose(Y)
t.Logf("X = InvScale(Scale(X, 2.0), 2.0) : %v\n", ok)
}
func trdsecEigenBuildInplace(Q, z *cmat.FloatMatrix) {
var QTL, QBR, Q00, q11, q12, q21, Q22, qi cmat.FloatMatrix
var zk0, zk1, dk0, dk1 float64
util.Partition2x2(
&QTL, nil,
nil, &QBR /**/, Q, 0, 0, util.PTOPLEFT)
for m(&QBR) > 0 {
util.Repartition2x2to3x3(&QTL,
&Q00, nil, nil,
nil, &q11, &q12,
nil, &q21, &Q22 /**/, Q, 1, util.PBOTTOMRIGHT)
//---------------------------------------------------------------
k := m(&Q00)
zk0 = z.GetAt(k)
dk0 = q11.Get(0, 0)
q11.Set(0, 0, zk0/dk0)
for i := 0; i < q12.Len(); i++ {
zk1 = z.GetAt(k + i + 1)
dk0 = q12.GetAt(i)
dk1 = q21.GetAt(i)
q12.SetAt(i, zk0/dk1)
q21.SetAt(i, zk1/dk0)
}
//---------------------------------------------------------------
util.Continue3x3to2x2(
&QTL, nil,
nil, &QBR /**/, &Q00, &q11, &Q22 /**/, Q, util.PBOTTOMRIGHT)
}
// scale column eigenvectors
for k := 0; k < z.Len(); k++ {
qi.Column(Q, k)
t := blasd.Nrm2(&qi)
blasd.InvScale(&qi, t)
}
}
func test1(N int, beta float64, t *testing.T) {
var sI cmat.FloatMatrix
if N&0x1 != 0 {
N = N + 1
}
D := cmat.NewMatrix(N, 1)
Z := cmat.NewMatrix(N, 1)
Y := cmat.NewMatrix(N, 1)
V := cmat.NewMatrix(N, 1)
Q := cmat.NewMatrix(N, N)
I := cmat.NewMatrix(N, N)
D.SetAt(0, 1.0)
Z.SetAt(0, 2.0)
for i := 1; i < N-1; i++ {
if i < N/2 {
D.SetAt(i, 2.0-float64(N/2-i)*beta)
} else {
D.SetAt(i, 2.0+float64(i+1-N/2)*beta)
}
Z.SetAt(i, beta)
}
D.SetAt(N-1, 10.0/3.0)
Z.SetAt(N-1, 2.0)
w := blasd.Nrm2(Z)
blasd.InvScale(Z, w)
rho := 1.0 / (w * w)
lapackd.TRDSecularSolveAll(Y, V, Q, D, Z, rho)
lapackd.TRDSecularEigen(Q, V, nil)
blasd.Mult(I, Q, Q, 1.0, 0.0, gomas.TRANSA)
sI.Diag(I)
sI.Add(-1.0)
nrm := lapackd.NormP(I, lapackd.NORM_ONE)
t.Logf("N=%d, beta=%e ||I - Q.T*Q||_1: %e\n", N, beta, nrm)
}
func unblockedLowerCHOL(A *cmat.FloatMatrix, flags int, nr int) (err *gomas.Error) {
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10, a11, A20, a21, A22 cmat.FloatMatrix
err = nil
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
for m(&ATL) < m(A) {
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, nil,
&A20, &a21, &A22, A, 1, util.PBOTTOMRIGHT)
// a11 = sqrt(a11)
aval := a11.Get(0, 0)
if aval < 0.0 {
if err == nil {
err = gomas.NewError(gomas.ENEGATIVE, "DecomposeCHOL", m(&ATL)+nr)
}
} else {
a11.Set(0, 0, math.Sqrt(aval))
}
// a21 = a21/a11
blasd.InvScale(&a21, a11.Get(0, 0))
// A22 = A22 - a21*a21' (SYR)
blasd.MVUpdateSym(&A22, &a21, -1.0, gomas.LOWER)
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
}
return
}
/* From LAPACK/dlarfg.f
*
* Generates a real elementary reflector H of order n, such
* that
*
* H * ( alpha ) = ( beta ), H**T * H = I.
* ( x ) ( 0 )
*
* where alpha and beta are scalars, and x is an (n-1)-element real
* vector. H is represented in the form
*
* H = I - tau * ( 1 ) * ( 1 v**T ) ,
* ( v )
*
* where tau is a real scalar and v is a real (n-1)-element
* vector.
*
* If the elements of x are all zero, then tau = 0 and H is taken to be
* the unit cmat.
*
* Otherwise 1 <= tau <= 2.
*/
func computeHouseholder(a11, x, tau *cmat.FloatMatrix) {
// norm_x2 = ||x||_2
norm_x2 := blasd.Nrm2(x)
if norm_x2 == 0.0 {
tau.Set(0, 0, 0.0)
return
}
alpha := a11.Get(0, 0)
sign := 1.0
if math.Signbit(alpha) {
sign = -1.0
}
// beta = -(alpha / |alpha|) * ||alpha x||
// = -sign(alpha) * sqrt(alpha**2, norm_x2**2)
beta := -sign * sqrtX2Y2(alpha, norm_x2)
// x = x /(a11 - beta)
blasd.InvScale(x, alpha-beta)
tau.Set(0, 0, (beta-alpha)/beta)
a11.Set(0, 0, beta)
}
/*
* Unblocked solve A*X = B for Bunch-Kauffman factorized symmetric real matrix.
*/
func unblkSolveBKUpper(B, A *cmat.FloatMatrix, p Pivots, phase int, conf *gomas.Config) *gomas.Error {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12t, A22 cmat.FloatMatrix
var Aref *cmat.FloatMatrix
var BT, BB, B0, b1, B2, Bx cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
var aStart, aDir, bStart, bDir util.Direction
var nc int
np := 0
if phase == 2 {
aStart = util.PTOPLEFT
aDir = util.PBOTTOMRIGHT
bStart = util.PTOP
bDir = util.PBOTTOM
nc = 1
Aref = &ABR
} else {
aStart = util.PBOTTOMRIGHT
aDir = util.PTOPLEFT
bStart = util.PBOTTOM
bDir = util.PTOP
nc = m(A)
Aref = &ATL
}
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, aStart)
util.Partition2x1(
&BT,
&BB, B, 0, bStart)
partitionPivot2x1(
&pT,
&pB, p, 0, bStart)
// phase 1:
// - solve U*D*X = B, overwriting B with X
// - looping from BOTTOM to TOP
// phase 1:
// - solve U*X = B, overwriting B with X
// - looping from TOP to BOTTOM
for n(Aref) > 0 {
// see if next diagonal block is 1x1 or 2x2
np = 1
if p[nc-1] < 0 {
np = 2
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12t,
nil, nil, &A22 /**/, A, np, aDir)
util.Repartition2x1to3x1(&BT,
&B0,
&b1,
&B2 /**/, B, np, bDir)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, p, np, bDir)
// ------------------------------------------------------------
switch phase {
case 1:
// computes D.-1*(U.-1*B);
// b1 is current row, last row of BT
if np == 1 {
if p1[0] != nc {
// swap rows on top part of B
swapRows(&BT, m(&BT)-1, p1[0]-1)
}
// B0 = B0 - a01*b1
blasd.MVUpdate(&B0, &a01, &b1, -1.0)
// b1 = b1/d1
blasd.InvScale(&b1, a11.Get(0, 0))
nc -= 1
} else if np == 2 {
if p1[0] != -nc {
// swap rows on top part of B
swapRows(&BT, m(&BT)-2, -p1[0]-1)
}
b := a11.Get(0, 1)
apb := a11.Get(0, 0) / b
dpb := a11.Get(1, 1) / b
// (a/b)*(d/b)-1.0 == (a*d - b^2)/b^2
scale := apb*dpb - 1.0
scale *= b
// B0 = B0 - a01*b1
blasd.Mult(&B0, &a01, &b1, -1.0, 1.0, gomas.NONE, conf)
// b1 = a11.-1*b1.T
//(2x2 block, no subroutine for doing this in-place)
for k := 0; k < n(&b1); k++ {
s0 := b1.Get(0, k)
s1 := b1.Get(1, k)
b1.Set(0, k, (dpb*s0-s1)/scale)
b1.Set(1, k, (apb*s1-s0)/scale)
}
nc -= 2
}
case 2:
// compute X = U.-T*B
if np == 1 {
blasd.MVMult(&b1, &B0, &a01, -1.0, 1.0, gomas.TRANS)
if p1[0] != nc {
// swap rows on bottom part of B
util.Merge2x1(&Bx, &B0, &b1)
swapRows(&Bx, m(&Bx)-1, p1[0]-1)
}
nc += 1
} else if np == 2 {
blasd.Mult(&b1, &a01, &B0, -1.0, 1.0, gomas.TRANSA, conf)
if p1[0] != -nc {
// swap rows on bottom part of B
util.Merge2x1(&Bx, &B0, &b1)
swapRows(&Bx, m(&Bx)-2, -p1[0]-1)
}
nc += 2
}
}
// ------------------------------------------------------------
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, aDir)
util.Continue3x1to2x1(
&BT,
&BB, &B0, &b1, B, bDir)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, p, bDir)
}
return err
}
func unblkBoundedBKUpper(A, wrk *cmat.FloatMatrix, p *Pivots, ncol int, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12, A22, a11inv cmat.FloatMatrix
var w00, w01, w11 cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
if ncol > n(A) {
ncol = n(A)
}
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, m(wrk)-2, 0, 2, 2)
a11inv.Set(0, 1, -1.0)
a11inv.Set(1, 0, -1.0)
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
partitionPivot2x1(
&pT,
&pB, *p, 0, util.PBOTTOM)
for n(&ATL) > 0 && nc < ncol {
util.Partition2x2(
&w00, &w01,
nil, &w11, wrk, nc, nc, util.PBOTTOMRIGHT)
r, np := findAndBuildBKPivotUpper(&ATL, &ATR, &w00, &w01, nc)
if np > ncol-nc {
// next pivot does not fit into ncol columns,
// return with number of factorized columns
return err, nc
}
cwrk.SubMatrix(&w00, 0, n(&w00)-np, m(&ATL), np)
if r != -1 {
// pivoting needed; do swaping here
k := m(&ATL) - np
applyBKPivotSymUpper(&ATL, k, r)
// swap right hand rows to get correct updates
swapRows(&ATR, k, r)
swapRows(&w01, k, r)
if np == 2 && r != k {
/* for 2x2 blocks we need diagonal pivots.
* [r, r] | [ r,-1]
* a11 == ---------------- 2-by-2 pivot, swapping [1,0] and [r,0]
* [-1,r] | [-1,-1]
*/
t0 := w00.Get(k, -1)
tr := w00.Get(r, -1)
w00.Set(k, -1, tr)
w00.Set(r, -1, t0)
t0 = w00.Get(k, -2)
tr = w00.Get(r, -2)
w00.Set(k, -2, tr)
w00.Set(r, -2, t0)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22 /**/, A, np, util.PTOPLEFT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, *p, np, util.PTOP)
// ------------------------------------------------------------
wlc := n(&w00) - np
cwrk.SubMatrix(&w00, 0, wlc, m(&a01), n(&a01))
if np == 1 {
//
a11.Set(0, 0, w00.Get(m(&a01), wlc))
// a21 = a21/a11
blasd.Copy(&a01, &cwrk)
blasd.InvScale(&a01, a11.Get(0, 0))
// store pivot point relative to original matrix
if r == -1 {
p1[0] = m(&ATL)
} else {
p1[0] = r + 1
}
} else if np == 2 {
/* a | b d/b | -1
* w00 == ------ == a11 --> a11.-1 == -------- * scale
* . | d -1 | a/b
*/
a := w00.Get(m(&ATL)-2, -2)
b := w00.Get(m(&ATL)-2, -1)
d := w00.Get(m(&ATL)-1, -1)
a11inv.Set(0, 0, d/b)
a11inv.Set(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
// a01 = a01*a11.-1
blasd.Mult(&a01, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
a11.Set(0, 0, a)
a11.Set(0, 1, b)
a11.Set(1, 1, d)
// store pivot point relative to original matrix
p1[0] = -(r + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, *p, util.PTOP)
}
return err, nc
}
/*
* Unblocked Bunch-Kauffman LDL factorization.
*
* Corresponds lapack.DSYTF2
*/
func unblkDecompBKUpper(A, wrk *cmat.FloatMatrix, p Pivots, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a11, a12, A22, a11inv cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PBOTTOMRIGHT)
partitionPivot2x1(
&pT,
&pB, p, 0, util.PBOTTOM)
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, 0, n(wrk)-2, 2, 2)
a11inv.Set(1, 0, -1.0)
a11inv.Set(0, 1, -1.0)
for n(&ATL) > 0 {
nr := m(&ATL) - 1
r, np := findBKPivotUpper(&ATL)
if r != -1 {
cwrk.SubMatrix(&ATL, 0, n(&ATL)-np, m(&ATL), np)
// pivoting needed; do swaping here
applyBKPivotSymUpper(&ATL, m(&ATL)-np, r)
if np == 2 {
/* [r, r] | [r, nr]
* a11 == ---------------- 2-by-2 pivot, swapping [nr-1,nr] and [r,nr]
* [nr,r] | [nr,nr] (nr is the current diagonal entry)
*/
t := ATL.Get(nr-1, nr)
ATL.Set(nr-1, nr, ATL.Get(r, nr))
ATL.Set(r, nr, t)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12,
nil, nil, &A22 /**/, A, np, util.PTOPLEFT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, p, np, util.PTOP)
// ------------------------------------------------------------
if np == 1 {
// A00 = A00 - a01*a01.T/a11
blasd.MVUpdateTrm(&A00, &a01, &a01, -1.0/a11.Get(0, 0), gomas.UPPER)
// a01 = a01/a11
blasd.InvScale(&a01, a11.Get(0, 0))
// store pivot point relative to original matrix
if r == -1 {
p1[0] = m(&ATL)
} else {
p1[0] = r + 1
}
} else if np == 2 {
/* see comments on unblkDecompBKLower() */
a := a11.Get(0, 0)
b := a11.Get(0, 1)
d := a11.Get(1, 1)
a11inv.Set(0, 0, d/b)
a11inv.Set(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
cwrk.SubMatrix(wrk, 2, 0, m(&a01), n(&a01))
blasd.Copy(&cwrk, &a01)
// a01 = a01*a11.-1
blasd.Mult(&a01, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
// A00 = A00 - a01*a11.-1*a01.T = A00 - a01*cwrk.T
blasd.UpdateTrm(&A00, &a01, &cwrk, -1.0, 1.0, gomas.UPPER|gomas.TRANSB, conf)
// store pivot point relative to original matrix
p1[0] = -(r + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PTOPLEFT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, p, util.PTOP)
}
return err, nc
}
/*
* 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 *cmat.FloatMatrix, p *Pivots, ncol int, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10t, a11, A20, a21, A22, a11inv cmat.FloatMatrix
var w00, w10, w11 cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
if ncol > n(A) {
ncol = n(A)
}
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, 0, n(wrk)-2, 2, 2)
a11inv.Set(1, 0, -1.0)
a11inv.Set(0, 1, -1.0)
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
partitionPivot2x1(
&pT,
&pB, *p, 0, util.PTOP)
for n(&ABR) > 0 && nc < ncol {
util.Partition2x2(
&w00, nil,
&w10, &w11, wrk, nc, nc, util.PTOPLEFT)
r, np := findAndBuildBKPivotLower(&ABL, &ABR, &w10, &w11, nc)
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 != 0 && r != np-1 {
// pivoting needed; do swaping here
applyBKPivotSymLower(&ABR, np-1, r)
// swap left hand rows to get correct updates
swapRows(&ABL, np-1, r)
swapRows(&w10, np-1, r)
if np == 2 {
/*
* [0,0] | [r,0]
* a11 == ------------- 2-by-2 pivot, swapping [1,0] and [r,0]
* [r,0] | [r,r]
*/
t0 := w11.Get(1, 0)
tr := w11.Get(r, 0)
w11.Set(1, 0, tr)
w11.Set(r, 0, t0)
// interchange diagonal entries on w11[:,1]
t0 = w11.Get(1, 1)
tr = w11.Get(r, 1)
w11.Set(1, 1, tr)
w11.Set(r, 1, t0)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10t, &a11, nil,
&A20, &a21, &A22 /**/, A, np, util.PBOTTOMRIGHT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, *p, np, util.PBOTTOM)
// ------------------------------------------------------------
if np == 1 {
//
cwrk.SubMatrix(&w11, np, 0, m(&a21), np)
a11.Set(0, 0, w11.Get(0, 0))
// a21 = a21/a11
blasd.Copy(&a21, &cwrk)
blasd.InvScale(&a21, a11.Get(0, 0))
// store pivot point relative to original matrix
p1[0] = r + m(&ATL) + 1
} else if np == 2 {
/*
* See comments for this block in unblkDecompBKLower().
*/
a := w11.Get(0, 0)
b := w11.Get(1, 0)
d := w11.Get(1, 1)
a11inv.Set(0, 0, d/b)
a11inv.Set(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.SubMatrix(&w11, np, 0, m(&a21), np)
// a21 = a21*a11.-1
blasd.Mult(&a21, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
a11.Set(0, 0, a)
a11.Set(1, 0, b)
a11.Set(1, 1, d)
// store pivot point relative to original matrix
p1[0] = -(r + m(&ATL) + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, *p, util.PBOTTOM)
}
return err, nc
}
/*
* Unblocked Bunch-Kauffman LDL factorization.
*
* Corresponds lapack.DSYTF2
*/
func unblkDecompBKLower(A, wrk *cmat.FloatMatrix, p Pivots, conf *gomas.Config) (*gomas.Error, int) {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a10t, a11, A20, a21, A22, a11inv cmat.FloatMatrix
var cwrk cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
nc := 0
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
partitionPivot2x1(
&pT,
&pB, p, 0, util.PTOP)
// permanent working space for symmetric inverse of a11
a11inv.SubMatrix(wrk, 0, n(wrk)-2, 2, 2)
a11inv.Set(1, 0, -1.0)
a11inv.Set(0, 1, -1.0)
for n(&ABR) > 0 {
r, np := findBKPivotLower(&ABR)
if r != 0 && r != np-1 {
// pivoting needed; do swaping here
applyBKPivotSymLower(&ABR, np-1, r)
if np == 2 {
/* [0,0] | [r,0]
* a11 == ------------- 2-by-2 pivot, swapping [1,0] and [r,0]
* [r,0] | [r,r]
*/
t := ABR.Get(1, 0)
ABR.Set(1, 0, ABR.Get(r, 0))
ABR.Set(r, 0, t)
}
}
// repartition according the pivot size
util.Repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10t, &a11, nil,
&A20, &a21, &A22 /**/, A, np, util.PBOTTOMRIGHT)
repartPivot2x1to3x1(&pT,
&p0,
&p1,
&p2 /**/, p, np, util.PBOTTOM)
// ------------------------------------------------------------
if np == 1 {
// A22 = A22 - a21*a21.T/a11
blasd.MVUpdateTrm(&A22, &a21, &a21, -1.0/a11.Get(0, 0), gomas.LOWER)
// a21 = a21/a11
blasd.InvScale(&a21, a11.Get(0, 0))
// store pivot point relative to original matrix
p1[0] = r + m(&ATL) + 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.Get(0, 0)
b := a11.Get(1, 0)
d := a11.Get(1, 1)
a11inv.Set(0, 0, d/b)
a11inv.Set(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
cwrk.SubMatrix(wrk, 2, 0, m(&a21), n(&a21))
blasd.Copy(&cwrk, &a21)
// a21 = a21*a11.-1
blasd.Mult(&a21, &cwrk, &a11inv, scale, 0.0, gomas.NONE, conf)
// A22 = A22 - a21*a11.-1*a21.T = A22 - a21*cwrk.T
blasd.UpdateTrm(&A22, &a21, &cwrk, -1.0, 1.0, gomas.LOWER|gomas.TRANSB, conf)
// store pivot point relative to original matrix
p1[0] = -(r + m(&ATL) + 1)
p1[1] = p1[0]
}
// ------------------------------------------------------------
nc += np
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, p, util.PBOTTOM)
}
return err, nc
}
// unblocked LU decomposition with pivots: FLAME LU variant 3; Left-looking
func unblockedLUpiv(A *cmat.FloatMatrix, p *Pivots, offset int, conf *gomas.Config) *gomas.Error {
var err *gomas.Error = nil
var ATL, ATR, ABL, ABR cmat.FloatMatrix
var A00, a01, A02, a10, a11, a12, A20, a21, A22 cmat.FloatMatrix
var AL, AR, A0, a1, A2, aB1, AB0 cmat.FloatMatrix
var pT, pB, p0, p1, p2 Pivots
err = nil
util.Partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, util.PTOPLEFT)
util.Partition1x2(
&AL, &AR, A, 0, util.PLEFT)
partitionPivot2x1(
&pT,
&pB, *p, 0, util.PTOP)
for m(&ATL) < m(A) && n(&ATL) < n(A) {
util.Repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
&a10, &a11, &a12,
&A20, &a21, &A22 /**/, A, 1, util.PBOTTOMRIGHT)
util.Repartition1x2to1x3(&AL,
&A0, &a1, &A2 /**/, A, 1, util.PRIGHT)
repartPivot2x1to3x1(&pT,
&p0, &p1, &p2 /**/, *p, 1, util.PBOTTOM)
// apply previously computed pivots on current column
applyPivots(&a1, p0)
// a01 = trilu(A00) \ a01 (TRSV)
blasd.MVSolveTrm(&a01, &A00, 1.0, gomas.LOWER|gomas.UNIT)
// a11 = a11 - a10 *a01
aval := a11.Get(0, 0) - blasd.Dot(&a10, &a01)
a11.Set(0, 0, aval)
// a21 = a21 -A20*a01
blasd.MVMult(&a21, &A20, &a01, -1.0, 1.0, gomas.NONE)
// pivot index on current column [a11, a21].T
aB1.Column(&ABR, 0)
p1[0] = pivotIndex(&aB1)
// pivots to current column
applyPivots(&aB1, p1)
// a21 = a21 / a11
if aval == 0.0 {
if err == nil {
ij := m(&ATL) + p1[0] - 1
err = gomas.NewError(gomas.ESINGULAR, "DecomposeLU", ij)
}
} else {
blasd.InvScale(&a21, a11.Get(0, 0))
}
// apply pivots to previous columns
AB0.SubMatrix(&ABL, 0, 0)
applyPivots(&AB0, p1)
// scale last pivots to origin matrix row numbers
p1[0] += m(&ATL)
util.Continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, util.PBOTTOMRIGHT)
util.Continue1x3to1x2(
&AL, &AR, &A0, &a1, A, util.PRIGHT)
contPivot3x1to2x1(
&pT,
&pB, p0, p1, *p, util.PBOTTOM)
}
if n(&ATL) < n(A) {
applyPivots(&ATR, *p)
blasd.SolveTrm(&ATR, &ATL, 1.0, gomas.LEFT|gomas.UNIT|gomas.LOWER, conf)
}
return err
}