// Inverse UNIT diagonal tridiagonal matrix
func unblockedInverseUnitLower(A *matrix.FloatMatrix) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, a10t, a11, A20, a21, A22 matrix.FloatMatrix
err = nil
partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, pTOPLEFT)
for ATL.Rows() < A.Rows() {
repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10t, &a11, nil,
&A20, &a21, &A22, A, 1, pBOTTOMRIGHT)
// -------------------------------------------------
// a21 = -a21
Scale(&a21, -1.0)
// A20 = A20 + a21*a10.t
MVRankUpdate(&A20, &a21, &a10t, 1.0)
// -------------------------------------------------
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
}
return
}
func InverseTrm(A *matrix.FloatMatrix, flags Flags, nb int) (*matrix.FloatMatrix, error) {
var err error = nil
if nb == 0 || A.Cols() < nb {
if flags&UNIT != 0 {
if flags&LOWER != 0 {
err = unblockedInverseUnitLower(A)
} else {
err = unblockedInverseUnitUpper(A)
}
} else {
if flags&LOWER != 0 {
err = unblockedInverseLower(A)
} else {
err = unblockedInverseUpper(A)
}
}
} else {
if flags&LOWER != 0 {
err = blockedInverseLower(A, flags, nb)
} else {
err = blockedInverseUpper(A, flags, nb)
}
}
return A, err
}
// Inverse NON-UNIT diagonal tridiagonal matrix
func unblockedInverseUpper(A *matrix.FloatMatrix) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, a01, A02, a11, a12t, A22 matrix.FloatMatrix
err = nil
partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, pTOPLEFT)
for ATL.Rows() < A.Rows() {
repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
nil, &a11, &a12t,
nil, nil, &A22, A, 1, pBOTTOMRIGHT)
// -------------------------------------------------
aval := a11.Float()
// a12 = -a12/a11
InvScale(&a12t, -aval)
// A02 = A02 + a01*a12
MVRankUpdate(&A02, &a01, &a12t, 1.0)
// a01 = a01/a11
InvScale(&a01, aval)
// a11 = 1.0/a11
a11.SetAt(0, 0, 1.0/aval)
// -------------------------------------------------
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
}
return
}
func blockedInverseUpper(A *matrix.FloatMatrix, flags Flags, nb int) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, A01, A02, A11, A12, A22 matrix.FloatMatrix
err = nil
partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, pTOPLEFT)
for ATL.Rows() < A.Rows() {
repartition2x2to3x3(&ATL,
&A00, &A01, &A02,
nil, &A11, &A12,
nil, nil, &A22, A, nb, pBOTTOMRIGHT)
// -------------------------------------------------
// libflame, variant 1
// A01 = A00*A01
MultTrm(&A01, &A00, 1.0, flags)
// A01 = -A01 / triu(A11)
SolveTrm(&A01, &A11, -1.0, flags|RIGHT)
// A11 = inv(A11)
if flags&UNIT != 0 {
unblockedInverseUnitUpper(&A11)
} else {
unblockedInverseUpper(&A11)
}
// -------------------------------------------------
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &A11, &A22, A, pBOTTOMRIGHT)
}
return
}
func runRefTest(A *matrix.FloatMatrix, ntest, LB int) time.Duration {
var mintime time.Duration
N := A.Cols()
tau := matrix.FloatZeros(N, 1)
fnc := func() {
ERRlapack = lapack.Geqrf(A, tau)
}
A0 := A.Copy()
for n := 0; n < ntest; n++ {
if n > 0 {
// restore original A
A0.CopyTo(A)
tau.Scale(0.0)
}
mperf.FlushCache()
time0 := mperf.Timeit(fnc)
if n == 0 || time0 < mintime {
mintime = time0
}
}
return mintime
}
func TestSolveLeastSquaresQRT(t *testing.T) {
M := 60
N := 40
K := 30
nb := 12
A := matrix.FloatUniform(M, N)
B := matrix.FloatZeros(M, K)
X0 := matrix.FloatUniform(N, K)
// B = A*X0
Mult(B, A, X0, 1.0, 0.0, NOTRANS)
W := matrix.FloatZeros(N, nb)
T := matrix.FloatZeros(N, N)
QR, err := DecomposeQRT(A, T, W, nb)
if err != nil {
t.Logf("decompose error: %v\n", err)
}
// B' = A.-1*B
err = SolveQRT(B, QR, T, W, NOTRANS, nb)
// expect B[0:N, 0:K] == X0, B[N:M, 0:K] == 0.0
var Xref matrix.FloatMatrix
Bref := matrix.FloatZeros(M, K)
Bref.SubMatrix(&Xref, 0, 0, N, K)
Xref.Plus(X0)
Bref.Minus(B)
t.Logf("\nmin ||B - A*X0||\n\twhere B = A*X0\n")
t.Logf("||B - A*X0||_1 ~~ 0.0: %e\n", NormP(Bref, NORM_ONE))
}
func updateBlas(t *testing.T, Y1, Y2, C1, C2, T, W *matrix.FloatMatrix) {
if W.Rows() != C1.Cols() {
panic("W.Rows != C1.Cols")
}
// W = C1.T
ScalePlus(W, C1, 0.0, 1.0, TRANSB)
//fmt.Printf("W = C1.T:\n%v\n", W)
// W = C1.T*Y1
blas.TrmmFloat(Y1, W, 1.0, linalg.OptLower, linalg.OptUnit, linalg.OptRight)
t.Logf("W = C1.T*Y1:\n%v\n", W)
// W = W + C2.T*Y2
blas.GemmFloat(C2, Y2, W, 1.0, 1.0, linalg.OptTransA)
t.Logf("W = W + C2.T*Y2:\n%v\n", W)
// --- here: W == C.T*Y ---
// W = W*T
blas.TrmmFloat(T, W, 1.0, linalg.OptUpper, linalg.OptRight)
t.Logf("W = C.T*Y*T:\n%v\n", W)
// --- here: W == C.T*Y*T ---
// C2 = C2 - Y2*W.T
blas.GemmFloat(Y2, W, C2, -1, 1.0, linalg.OptTransB)
t.Logf("C2 = C2 - Y2*W.T:\n%v\n", C2)
// W = Y1*W.T ==> W.T = W*Y1.T
blas.TrmmFloat(Y1, W, 1.0, linalg.OptLower,
linalg.OptUnit, linalg.OptRight, linalg.OptTrans)
t.Logf("W.T = W*Y1.T:\n%v\n", W)
// C1 = C1 - W.T
ScalePlus(C1, W, 1.0, -1.0, TRANSB)
//fmt.Printf("C1 = C1 - W.T:\n%v\n", C1)
// --- here: C = (I - Y*T*Y.T).T * C ---
}
func runTest(A *matrix.FloatMatrix, ntest, LB int) time.Duration {
var flags matops.Flags
var mintime time.Duration
flags = matops.LOWER
if testUpper {
flags = matops.UPPER
}
fnc := func() {
_, ERRmatops = matops.DecomposeCHOL(A, flags, LB)
}
A0 := A.Copy()
for n := 0; n < ntest; n++ {
if n > 0 {
// restore original A
A0.CopyTo(A)
}
mperf.FlushCache()
time0 := mperf.Timeit(fnc)
if n == 0 || time0 < mintime {
mintime = time0
}
if verbose {
fmt.Printf("%.4f ms\n", time0.Seconds()*1000.0)
}
}
return mintime
}
func setDiagonal(M *matrix.FloatMatrix, srow, scol, erow, ecol int, val float64) {
for i := srow; i < erow; i++ {
if i < ecol {
M.SetAt(i, i, val)
}
}
}
// max |x|
func AMax(X *matrix.FloatMatrix) float64 {
ix := IAMax(X)
if ix == -1 {
return math.NaN()
}
return X.GetIndex(ix)
}
// See function Syrk.
func SyrkFloat(A, C *matrix.FloatMatrix, alpha, beta float64, opts ...linalg.Option) (err error) {
params, e := linalg.GetParameters(opts...)
if e != nil {
err = e
return
}
ind := linalg.GetIndexOpts(opts...)
err = check_level3_func(ind, fsyrk, A, nil, C, params)
if e != nil || err != nil {
return
}
if ind.N == 0 {
return
}
Aa := A.FloatArray()
Ca := C.FloatArray()
uplo := linalg.ParamString(params.Uplo)
trans := linalg.ParamString(params.Trans)
//diag := linalg.ParamString(params.Diag)
dsyrk(uplo, trans, ind.N, ind.K, alpha, Aa[ind.OffsetA:], ind.LDa, beta,
Ca[ind.OffsetC:], ind.LDc)
return
}
func runRefTest(A *matrix.FloatMatrix, ntest, LB int) time.Duration {
var mintime time.Duration
lopt := linalg.OptLower
if testUpper {
lopt = linalg.OptUpper
}
fnc := func() {
ERRlapack = lapack.Potrf(A, lopt)
}
A0 := A.Copy()
for n := 0; n < ntest; n++ {
if n > 0 {
// restore original A
A0.CopyTo(A)
}
mperf.FlushCache()
time0 := mperf.Timeit(fnc)
if n == 0 || time0 < mintime {
mintime = time0
}
}
return mintime
}
/*
Copy x to y using packed storage.
The vector x is an element of S, with the 's' components stored in
unpacked storage. On return, x is copied to y with the 's' components
stored in packed storage and the off-diagonal entries scaled by
sqrt(2).
*/
func pack(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, opts ...la_.Option) (err error) {
/*DEBUGGED*/
err = nil
mnl := la_.GetIntOpt("mnl", 0, opts...)
offsetx := la_.GetIntOpt("offsetx", 0, opts...)
offsety := la_.GetIntOpt("offsety", 0, opts...)
nlq := mnl + dims.At("l")[0] + dims.Sum("q")
blas.Copy(x, y, &la_.IOpt{"n", nlq}, &la_.IOpt{"offsetx", offsetx},
&la_.IOpt{"offsety", offsety})
iu, ip := offsetx+nlq, offsety+nlq
for _, n := range dims.At("s") {
for k := 0; k < n; k++ {
blas.Copy(x, y, &la_.IOpt{"n", n - k}, &la_.IOpt{"offsetx", iu + k*(n+1)},
&la_.IOpt{"offsety", ip})
y.SetIndex(ip, (y.GetIndex(ip) / math.Sqrt(2.0)))
ip += n - k
}
iu += n * n
}
np := dims.SumPacked("s")
blas.ScalFloat(y, math.Sqrt(2.0), &la_.IOpt{"n", np}, &la_.IOpt{"offset", offsety + nlq})
return
}
func trmvTest(t *testing.T, A *matrix.FloatMatrix, flags Flags, nb int) bool {
N := A.Cols()
//S := 0
//E := A.Cols()
X0 := matrix.FloatWithValue(A.Rows(), 1, 2.0)
X1 := X0.Copy()
trans := linalg.OptNoTrans
if flags&TRANS != 0 {
trans = linalg.OptTrans
}
diag := linalg.OptNonUnit
if flags&UNIT != 0 {
diag = linalg.OptUnit
}
uplo := linalg.OptUpper
if flags&LOWER != 0 {
uplo = linalg.OptLower
}
blas.TrmvFloat(A, X0, uplo, diag, trans)
Ar := A.FloatArray()
Xr := X1.FloatArray()
if nb == 0 {
DTrimvUnblkMV(Xr, Ar, flags, 1, A.LeadingIndex(), N)
}
result := X0.AllClose(X1)
t.Logf(" X0 == X1: %v\n", result)
if !result && A.Rows() < 8 {
t.Logf(" BLAS TRMV X0:\n%v\n", X0)
t.Logf(" DTrmv X1:\n%v\n", X1)
}
return result
}
// In-place version of pack(), which also accepts matrix arguments x.
// The columns of x are elements of S, with the 's' components stored
// in unpacked storage. On return, the 's' components are stored in
// packed storage and the off-diagonal entries are scaled by sqrt(2).
//
func pack2(x *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int) (err error) {
if len(dims.At("s")) == 0 {
return nil
}
const sqrt2 = 1.41421356237309504880
iu := mnl + dims.Sum("l", "q")
ip := iu
row := matrix.FloatZeros(1, x.Cols())
//fmt.Printf("x.size = %d %d\n", x.Rows(), x.Cols())
for _, n := range dims.At("s") {
for k := 0; k < n; k++ {
cnt := n - k
row = x.GetRow(iu+(n+1)*k, row)
//fmt.Printf("%02d: %v\n", iu+(n+1)*k, x.FloatArray())
x.SetRow(ip, row)
for i := 1; i < n-k; i++ {
row = x.GetRow(iu+(n+1)*k+i, row)
//fmt.Printf("%02d: %v\n", iu+(n+1)*k+i, x.FloatArray())
x.SetRow(ip+i, row.Scale(sqrt2))
}
ip += cnt
}
iu += n * n
}
return nil
}
/*
* ( a11 a12 ) ( 1 0 )( d1 0 )( l l21.t )
* ( a21 A22 ) ( l21 L22 )( 0 A22 )( 0 L22.t )
*
* a11 = d1
* a21 = l21*d1 => l21 = a21/d1
* A22 = l21*d1*l21.t + L22*D2*L22.t => L22 = A22 - l21*d1*l21t
*/
func unblkLowerLDLnoPiv(A *matrix.FloatMatrix) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, a10, a11, A20, a21, A22 matrix.FloatMatrix
err = nil
partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, pTOPLEFT)
for ATL.Rows() < A.Rows() {
repartition2x2to3x3(&ATL,
&A00, nil, nil,
&a10, &a11, nil,
&A20, &a21, &A22, A, 1, pBOTTOMRIGHT)
// --------------------------------------------------------
// d11 = a11; no-op
// A22 = A22 - l21*d11*l21.T = A22 - a21*a21.T/a11; triangular update
err = MVUpdateTrm(&A22, &a21, &a21, -1.0/a11.Float(), LOWER)
// l21 = a21/a11
InvScale(&a21, a11.Float())
// ---------------------------------------------------------
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
}
return
}
// See function Trsm.
func TrsmFloat(A, B *matrix.FloatMatrix, alpha float64, opts ...linalg.Option) (err error) {
params, e := linalg.GetParameters(opts...)
if e != nil {
err = e
return
}
ind := linalg.GetIndexOpts(opts...)
err = check_level3_func(ind, ftrsm, A, B, nil, params)
if err != nil {
return
}
if ind.N == 0 || ind.M == 0 {
return
}
Aa := A.FloatArray()
Ba := B.FloatArray()
uplo := linalg.ParamString(params.Uplo)
transA := linalg.ParamString(params.TransA)
side := linalg.ParamString(params.Side)
diag := linalg.ParamString(params.Diag)
dtrsm(side, uplo, transA, diag, ind.M, ind.N, alpha,
Aa[ind.OffsetA:], ind.LDa, Ba[ind.OffsetB:], ind.LDb)
return
}
// blocked LU decomposition w/o pivots, FLAME LU nopivots variant 5
func blockedLUnoPiv(A *matrix.FloatMatrix, nb int) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, A01, A02, A10, A11, A12, A20, A21, A22 matrix.FloatMatrix
err = nil
partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, pTOPLEFT)
for ATL.Rows() < A.Rows() {
repartition2x2to3x3(&ATL,
&A00, &A01, &A02,
&A10, &A11, &A12,
&A20, &A21, &A22, A, nb, pBOTTOMRIGHT)
// A00 = LU(A00)
unblockedLUnoPiv(&A11)
// A12 = trilu(A00)*A12.-1 (TRSM)
SolveTrm(&A12, &A11, 1.0, LEFT|LOWER|UNIT)
// A21 = A21.-1*triu(A00) (TRSM)
SolveTrm(&A21, &A11, 1.0, RIGHT|UPPER)
// A22 = A22 - A21*A12
Mult(&A22, &A21, &A12, -1.0, 1.0, NOTRANS)
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &A11, &A22, A, pBOTTOMRIGHT)
}
return
}
/*
Matrix-vector multiplication.
A is a matrix or spmatrix of size (m, n) where
N = dims['l'] + sum(dims['q']) + sum( k**2 for k in dims['s'] )
representing a mapping from R^n to S.
If trans is 'N':
y := alpha*A*x + beta * y (trans = 'N').
x is a vector of length n. y is a vector of length N.
If trans is 'T':
y := alpha*A'*x + beta * y (trans = 'T').
x is a vector of length N. y is a vector of length n.
The 's' components in S are stored in unpacked 'L' storage.
*/
func sgemv(A, x, y *matrix.FloatMatrix, alpha, beta float64, dims *sets.DimensionSet, opts ...la_.Option) error {
m := dims.Sum("l", "q") + dims.SumSquared("s")
n := la_.GetIntOpt("n", -1, opts...)
if n == -1 {
n = A.Cols()
}
trans := la_.GetIntOpt("trans", int(la_.PNoTrans), opts...)
offsetX := la_.GetIntOpt("offsetx", 0, opts...)
offsetY := la_.GetIntOpt("offsety", 0, opts...)
offsetA := la_.GetIntOpt("offseta", 0, opts...)
if trans == int(la_.PTrans) && alpha != 0.0 {
trisc(x, dims, offsetX)
//fmt.Printf("trisc x=\n%v\n", x.ConvertToString())
}
//fmt.Printf("alpha=%.4f beta=%.4f m=%d n=%d\n", alpha, beta, m, n)
//fmt.Printf("A=\n%v\nx=\n%v\ny=\n%v\n", A, x.ConvertToString(), y.ConvertToString())
err := blas.GemvFloat(A, x, y, alpha, beta, &la_.IOpt{"trans", trans},
&la_.IOpt{"n", n}, &la_.IOpt{"m", m}, &la_.IOpt{"offseta", offsetA},
&la_.IOpt{"offsetx", offsetX}, &la_.IOpt{"offsety", offsetY})
//fmt.Printf("gemv y=\n%v\n", y.ConvertToString())
if trans == int(la_.PTrans) && alpha != 0.0 {
triusc(x, dims, offsetX)
}
return err
}
func (p *acenterProg) F2(x, z *matrix.FloatMatrix) (f, Df, H *matrix.FloatMatrix, err error) {
f, Df, err = p.F1(x)
u := matrix.Pow(x, 2.0).Scale(-1.0).Add(1.0)
z0 := z.GetIndex(0)
u2 := matrix.Pow(u, 2.0)
hd := matrix.Div(matrix.Add(u2, 1.0), u2).Scale(2 * z0)
H = matrix.FloatDiagonal(hd.NumElements(), hd.FloatArray()...)
return
}
/*
* Compute an LU factorization of a general M-by-N matrix without pivoting.
*
* Arguments:
* A On entry, the M-by-N matrix to be factored. On exit the factors
* L and U from factorization A = P*L*U, the unit diagonal elements
* of L are not stored.
*
* nb Blocking factor for blocked invocations. If bn == 0 or
* min(M,N) < nb unblocked algorithm is used.
*
* Returns:
* LU factorization and error indicator.
*
* Compatible with lapack.DGETRF
*/
func DecomposeLUnoPiv(A *matrix.FloatMatrix, nb int) (*matrix.FloatMatrix, error) {
var err error
mlen := imin(A.Rows(), A.Cols())
if mlen <= nb || nb == 0 {
err = unblockedLUnoPiv(A)
} else {
err = blockedLUnoPiv(A, nb)
}
return A, err
}
func sinv(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int) (err error) {
/*DEBUGGED*/
err = nil
// For the nonlinear and 'l' blocks:
//
// yk o\ xk = yk .\ xk.
ind := mnl + dims.At("l")[0]
blas.Tbsv(y, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"ldA", 1})
// For the 'q' blocks:
//
// [ l0 -l1' ]
// yk o\ xk = 1/a^2 * [ ] * xk
// [ -l1 (a*I + l1*l1')/l0 ]
//
// where yk = (l0, l1) and a = l0^2 - l1'*l1.
for _, m := range dims.At("q") {
aa := blas.Nrm2Float(y, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offset", ind + 1})
ee := y.GetIndex(ind)
aa = (ee + aa) * (ee - aa)
cc := x.GetIndex(ind)
dd := blas.DotFloat(x, y, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offsetx", ind + 1},
&la_.IOpt{"offsety", ind + 1})
x.SetIndex(ind, cc*ee-dd)
blas.ScalFloat(x, aa/ee, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offset", ind + 1})
blas.AxpyFloat(y, x, dd/ee-cc, &la_.IOpt{"n", m - 1},
&la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1})
blas.ScalFloat(x, 1.0/aa, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind})
ind += m
}
// For the 's' blocks:
//
// yk o\ xk = xk ./ gamma
//
// where gammaij = .5 * (yk_i + yk_j).
ind2 := ind
for _, m := range dims.At("s") {
for j := 0; j < m; j++ {
u := matrix.FloatVector(y.FloatArray()[ind2+j : ind2+m])
u.Add(y.GetIndex(ind2 + j))
u.Scale(0.5)
blas.Tbsv(u, x, &la_.IOpt{"n", m - j}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1},
&la_.IOpt{"offsetx", ind + j*(m+1)})
}
ind += m * m
ind2 += m
}
return
}
/*
Returns sqrt(x' * J * x) where J = [1, 0; 0, -I], for a vector
x in a second order cone.
*/
func jnrm2(x *matrix.FloatMatrix, n, offset int) float64 {
/*DEBUGGED*/
if n <= 0 {
n = x.NumElements()
}
if offset < 0 {
offset = 0
}
a := blas.Nrm2Float(x, &la_.IOpt{"n", n - 1}, &la_.IOpt{"offset", offset + 1})
fst := x.GetIndex(offset)
return math.Sqrt(fst-a) * math.Sqrt(fst+a)
}
// See function Scal.
func ScalFloat(X *matrix.FloatMatrix, alpha float64, opts ...linalg.Option) (err error) {
ind := linalg.GetIndexOpts(opts...)
err = check_level1_func(ind, fscal, X, nil)
if err != nil {
return
}
if ind.Nx == 0 {
return
}
Xa := X.FloatArray()
dscal(ind.Nx, alpha, Xa[ind.OffsetX:], ind.IncX)
return
}
func saveData(A *matrix.FloatMatrix) {
var fd *os.File
if fileName == "" {
fileName = testName + ".dat"
}
fd, err := os.Create(fileName)
if err != nil {
fmt.Fprintf(os.Stderr, "create error: %v\n", err)
return
}
io.WriteString(fd, A.ToString("%14e"))
fd.Close()
}
/*
* ( A11 a12 ) ( U11 u12 )( D1 0 )( U11.t 0 )
* ( a21 a22 ) ( 0 1 )( 0 d2 )( u12.t 1 )
*
* a22 = d2
* a01 = u12*d2 => u12 = a12/d2
* A11 = u12*d2*u12.t + U11*D1*U11.t => U11 = A11 - u12*d2*u12.t
*/
func unblkUpperLDLnoPiv(A *matrix.FloatMatrix) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, a01, A02, a11, a12, A22 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, 1, pTOPLEFT)
// --------------------------------------------------------
// A00 = A00 - u01*d11*u01.T = A00 - a01*a01.T/a11; triangular update
err = MVUpdateTrm(&A00, &a01, &a01, -1.0/a11.Float(), UPPER)
// u01 = a01/a11
InvScale(&a01, a11.Float())
// ---------------------------------------------------------
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, pTOPLEFT)
}
return
}
func createGpProg(K []int, F, g *matrix.FloatMatrix) *gpConvexProg {
gp := &gpConvexProg{mnl: len(K) - 1, l: F.Rows(), n: F.Cols()}
gp.ind = make([][2]int, len(K))
s := 0
for i := 0; i < len(K); i++ {
gp.ind[i][0] = s
gp.ind[i][1] = s + K[i]
s += K[i]
}
gp.F = F
gp.g = g
gp.maxK = maxdim(K)
return gp
}
/* From LAPACK/dlarfg.f
*
* DLARFG 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 matrix.
*
* Otherwise 1 <= tau <= 2.
*/
func computeHouseholder(a11, x, tau *matrix.FloatMatrix, flags Flags) {
// norm_x2 = ||x||_2
norm_x2 := Norm2(x)
if norm_x2 == 0.0 {
//a11.SetAt(0, 0, -a11.GetAt(0, 0))
tau.SetAt(0, 0, 0.0)
return
}
alpha := a11.GetAt(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)
InvScale(x, alpha-beta)
tau.SetAt(0, 0, (beta-alpha)/beta)
a11.SetAt(0, 0, beta)
}
func solveMVTest(t *testing.T, A, X0 *matrix.FloatMatrix, flags Flags, bN, bNB int) {
X1 := X0.Copy()
uplo := linalg.OptUpper
diag := linalg.OptNonUnit
if flags&LOWER != 0 {
uplo = linalg.OptLower
}
if flags&UNIT != 0 {
diag = linalg.OptUnit
}
blas.TrsvFloat(A, X0, uplo, diag)
Ar := A.FloatArray()
Xr := X1.FloatArray()
if bN == bNB {
DSolveUnblkMV(Xr, Ar, flags, 1, A.LeadingIndex(), bN)
} else {
DSolveBlkMV(Xr, Ar, flags, 1, A.LeadingIndex(), bN, bNB)
}
ok := X1.AllClose(X0)
t.Logf("X1 == X0: %v\n", ok)
if !ok && bN < 8 {
t.Logf("A=\n%v\n", A)
t.Logf("X0=\n%v\n", X0)
t.Logf("blas: X0\n%v\n", X0)
t.Logf("X1:\n%v\n", X1)
}
}
// unblocked LU decomposition w/o pivots, FLAME LU nopivots variant 5
func unblockedLUnoPiv(A *matrix.FloatMatrix) (err error) {
var ATL, ATR, ABL, ABR matrix.FloatMatrix
var A00, a01, A02, a10, a11, a12, A20, a21, A22 matrix.FloatMatrix
err = nil
partition2x2(
&ATL, &ATR,
&ABL, &ABR, A, 0, 0, pTOPLEFT)
for ATL.Rows() < A.Rows() {
repartition2x2to3x3(&ATL,
&A00, &a01, &A02,
&a10, &a11, &a12,
&A20, &a21, &A22, A, 1, pBOTTOMRIGHT)
// a21 = a21/a11
//a21.Scale(1.0/a11.Float())
InvScale(&a21, a11.Float())
// A22 = A22 - a21*a12
err = MVRankUpdate(&A22, &a21, &a12, -1.0)
continue3x3to2x2(
&ATL, &ATR,
&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
}
return
}