mirror of
https://github.com/gonum/gonum.git
synced 2025-10-05 23:26:52 +08:00
351 lines
7.4 KiB
Go
351 lines
7.4 KiB
Go
// Copyright ©2015 The gonum Authors. All rights reserved.
|
||
// Use of this source code is governed by a BSD-style
|
||
// license that can be found in the LICENSE file.
|
||
|
||
package native
|
||
|
||
import (
|
||
"math"
|
||
|
||
"github.com/gonum/blas"
|
||
"github.com/gonum/blas/blas64"
|
||
)
|
||
|
||
// Dlatrs solves a triangular system of equations scaled to prevent overflow. It
|
||
// solves
|
||
// A * x = scale * b if trans == blas.NoTrans
|
||
// A^T * x = scale * b if trans == blas.Trans
|
||
// where the scale s is set for numeric stability.
|
||
//
|
||
// A is an n×n triangular matrix. On entry, the slice x contains the values of
|
||
// of b, and on exit it contains the solution vector x.
|
||
//
|
||
// If normin == true, cnorm is an input and cnorm[j] contains the norm of the off-diagonal
|
||
// part of the j^th column of A. If trans == blas.NoTrans, cnorm[j] must be greater
|
||
// than or equal to the infinity norm, and greater than or equal to the one-norm
|
||
// otherwise. If normin == false, then cnorm is treated as an output, and is set
|
||
// to contain the 1-norm of the off-diagonal part of the j^th column of A.
|
||
//
|
||
// Dlatrs is an internal routine. It is exported for testing purposes.
|
||
func (impl Implementation) Dlatrs(uplo blas.Uplo, trans blas.Transpose, diag blas.Diag, normin bool, n int, a []float64, lda int, x []float64, cnorm []float64) (scale float64) {
|
||
if uplo != blas.Upper && uplo != blas.Lower {
|
||
panic(badUplo)
|
||
}
|
||
if trans != blas.Trans && trans != blas.NoTrans {
|
||
panic(badTrans)
|
||
}
|
||
if diag != blas.Unit && diag != blas.NonUnit {
|
||
panic(badDiag)
|
||
}
|
||
upper := uplo == blas.Upper
|
||
noTrans := trans == blas.NoTrans
|
||
nonUnit := diag == blas.NonUnit
|
||
|
||
if n < 0 {
|
||
panic(nLT0)
|
||
}
|
||
checkMatrix(n, n, a, lda)
|
||
checkVector(n, x, 1)
|
||
checkVector(n, cnorm, 1)
|
||
|
||
if n == 0 {
|
||
return 0
|
||
}
|
||
smlnum := dlamchS / dlamchP
|
||
bignum := 1 / smlnum
|
||
scale = 1
|
||
bi := blas64.Implementation()
|
||
if !normin {
|
||
if upper {
|
||
cnorm[0] = 0
|
||
for j := 1; j < n; j++ {
|
||
cnorm[j] = bi.Dasum(j, a[j:], lda)
|
||
}
|
||
} else {
|
||
for j := 0; j < n-1; j++ {
|
||
cnorm[j] = bi.Dasum(n-j-1, a[(j+1)*lda+j:], lda)
|
||
}
|
||
cnorm[n-1] = 0
|
||
}
|
||
}
|
||
// Scale the column norms by tscal if the maximum element in cnorm is greater than bignum.
|
||
imax := bi.Idamax(n, cnorm, 1)
|
||
tmax := cnorm[imax]
|
||
var tscal float64
|
||
if tmax <= bignum {
|
||
tscal = 1
|
||
} else {
|
||
tscal = 1 / (smlnum * tmax)
|
||
bi.Dscal(n, tscal, cnorm, 1)
|
||
}
|
||
|
||
// Compute a bound on the computed solution vector to see if bi.Dtrsv can be used.
|
||
j := bi.Idamax(n, x, 1)
|
||
xmax := math.Abs(x[j])
|
||
xbnd := xmax
|
||
var grow float64
|
||
var jfirst, jlast, jinc int
|
||
if noTrans {
|
||
if upper {
|
||
jfirst = n - 1
|
||
jlast = -1
|
||
jinc = -1
|
||
} else {
|
||
jfirst = 0
|
||
jlast = n
|
||
jinc = 1
|
||
}
|
||
// Compute the growth in A * x = b.
|
||
if tscal != 1 {
|
||
grow = 0
|
||
goto Solve
|
||
}
|
||
if nonUnit {
|
||
grow = 1 / math.Max(xbnd, smlnum)
|
||
xbnd = grow
|
||
for j := jfirst; j != jlast; j += jinc {
|
||
if grow <= smlnum {
|
||
goto Solve
|
||
}
|
||
tjj := math.Abs(a[j*lda+j])
|
||
xbnd = math.Min(xbnd, math.Min(1, tjj)*grow)
|
||
if tjj+cnorm[j] >= smlnum {
|
||
grow *= tjj / (tjj + cnorm[j])
|
||
} else {
|
||
grow = 0
|
||
}
|
||
}
|
||
grow = xbnd
|
||
} else {
|
||
grow = math.Min(1, 1/math.Max(xbnd, smlnum))
|
||
for j := jfirst; j != jlast; j += jinc {
|
||
if grow <= smlnum {
|
||
goto Solve
|
||
}
|
||
grow *= 1 / (1 + cnorm[j])
|
||
}
|
||
}
|
||
} else {
|
||
if upper {
|
||
jfirst = 0
|
||
jlast = n
|
||
jinc = 1
|
||
} else {
|
||
jfirst = n - 1
|
||
jlast = -1
|
||
jinc = -1
|
||
}
|
||
if tscal != 1 {
|
||
grow = 0
|
||
goto Solve
|
||
}
|
||
if nonUnit {
|
||
grow = 1 / (math.Max(xbnd, smlnum))
|
||
xbnd = grow
|
||
for j := jfirst; j != jlast; j += jinc {
|
||
if grow <= smlnum {
|
||
goto Solve
|
||
}
|
||
xj := 1 + cnorm[j]
|
||
grow = math.Min(grow, xbnd/xj)
|
||
tjj := math.Abs(a[j*lda+j])
|
||
if xj > tjj {
|
||
xbnd *= tjj / xj
|
||
}
|
||
}
|
||
grow = math.Min(grow, xbnd)
|
||
} else {
|
||
grow = math.Min(1, 1/math.Max(xbnd, smlnum))
|
||
for j := jfirst; j != jlast; j += jinc {
|
||
if grow <= smlnum {
|
||
goto Solve
|
||
}
|
||
xj := 1 + cnorm[j]
|
||
grow /= xj
|
||
}
|
||
}
|
||
}
|
||
|
||
Solve:
|
||
if grow*tscal > smlnum {
|
||
// Use the Level 2 BLAS solve if the reciprocal of the bound on
|
||
// elements of X is not too small.
|
||
bi.Dtrsv(uplo, trans, diag, n, a, lda, x, 1)
|
||
if tscal != 1 {
|
||
bi.Dscal(n, 1/tscal, cnorm, 1)
|
||
}
|
||
return scale
|
||
}
|
||
|
||
// Use a Level 1 BLAS solve, scaling intermediate results.
|
||
if xmax > bignum {
|
||
scale = bignum / xmax
|
||
bi.Dscal(n, scale, x, 1)
|
||
xmax = bignum
|
||
}
|
||
if noTrans {
|
||
for j := jfirst; j != jlast; j += jinc {
|
||
xj := math.Abs(x[j])
|
||
var tjj, tjjs float64
|
||
if nonUnit {
|
||
tjjs = a[j*lda+j] * tscal
|
||
} else {
|
||
tjjs = tscal
|
||
if tscal == 1 {
|
||
goto Skip1
|
||
}
|
||
}
|
||
tjj = math.Abs(tjjs)
|
||
if tjj > smlnum {
|
||
if tjj < 1 {
|
||
if xj > tjj*bignum {
|
||
rec := 1 / xj
|
||
bi.Dscal(n, rec, x, 1)
|
||
scale *= rec
|
||
xmax *= rec
|
||
}
|
||
}
|
||
x[j] /= tjjs
|
||
xj = math.Abs(x[j])
|
||
} else if tjj > 0 {
|
||
if xj > tjj*bignum {
|
||
rec := (tjj * bignum) / xj
|
||
if cnorm[j] > 1 {
|
||
rec /= cnorm[j]
|
||
}
|
||
bi.Dscal(n, rec, x, 1)
|
||
scale *= rec
|
||
xmax *= rec
|
||
}
|
||
x[j] /= tjjs
|
||
xj = math.Abs(x[j])
|
||
} else {
|
||
for i := 0; i < n; i++ {
|
||
x[i] = 0
|
||
}
|
||
x[j] = 1
|
||
xj = 1
|
||
scale = 0
|
||
xmax = 0
|
||
}
|
||
Skip1:
|
||
if xj > 1 {
|
||
rec := 1 / xj
|
||
if cnorm[j] > (bignum-xmax)*rec {
|
||
rec *= 0.5
|
||
bi.Dscal(n, rec, x, 1)
|
||
scale *= rec
|
||
}
|
||
} else if xj*cnorm[j] > bignum-xmax {
|
||
bi.Dscal(n, 0.5, x, 1)
|
||
scale *= 0.5
|
||
}
|
||
if upper {
|
||
if j > 0 {
|
||
bi.Daxpy(j, -x[j]*tscal, a[j:], lda, x, 1)
|
||
i := bi.Idamax(j, x, 1)
|
||
xmax = math.Abs(x[i])
|
||
}
|
||
} else {
|
||
if j < n-1 {
|
||
bi.Daxpy(n-j-1, -x[j]*tscal, a[(j+1)*lda+j:], lda, x[j+1:], 1)
|
||
i := j + bi.Idamax(n-j-1, x[j+1:], 1)
|
||
xmax = math.Abs(x[i])
|
||
}
|
||
}
|
||
}
|
||
} else {
|
||
for j := jfirst; j != jlast; j += jinc {
|
||
xj := math.Abs(x[j])
|
||
uscal := tscal
|
||
rec := 1 / math.Max(xmax, 1)
|
||
var tjjs float64
|
||
if cnorm[j] > (bignum-xj)*rec {
|
||
rec *= 0.5
|
||
if nonUnit {
|
||
tjjs = a[j*lda+j] * tscal
|
||
} else {
|
||
tjjs = tscal
|
||
}
|
||
tjj := math.Abs(tjjs)
|
||
if tjj > 1 {
|
||
rec = math.Min(1, rec*tjj)
|
||
uscal /= tjjs
|
||
}
|
||
if rec < 1 {
|
||
bi.Dscal(n, rec, x, 1)
|
||
scale *= rec
|
||
xmax *= rec
|
||
}
|
||
}
|
||
var sumj float64
|
||
if uscal == 1 {
|
||
if upper {
|
||
sumj = bi.Ddot(j, a[j:], lda, x, 1)
|
||
} else if j < n-1 {
|
||
sumj = bi.Ddot(n-j-1, a[(j+1)*lda+j:], lda, x[j+1:], 1)
|
||
}
|
||
} else {
|
||
if upper {
|
||
for i := 0; i < j; i++ {
|
||
sumj += (a[i*lda+j] * uscal) * x[i]
|
||
}
|
||
} else if j < n {
|
||
for i := j + 1; i < n; i++ {
|
||
sumj += (a[i*lda+j] * uscal) * x[i]
|
||
}
|
||
}
|
||
}
|
||
if uscal == tscal {
|
||
x[j] -= sumj
|
||
xj := math.Abs(x[j])
|
||
var tjjs float64
|
||
if nonUnit {
|
||
tjjs = a[j*lda+j] * tscal
|
||
} else {
|
||
tjjs = tscal
|
||
if tscal == 1 {
|
||
goto Skip2
|
||
}
|
||
}
|
||
tjj := math.Abs(tjjs)
|
||
if tjj > smlnum {
|
||
if tjj < 1 {
|
||
if xj > tjj*bignum {
|
||
rec = 1 / xj
|
||
bi.Dscal(n, rec, x, 1)
|
||
scale *= rec
|
||
xmax *= rec
|
||
}
|
||
}
|
||
x[j] /= tjjs
|
||
} else if tjj > 0 {
|
||
if xj > tjj*bignum {
|
||
rec = (tjj * bignum) / xj
|
||
bi.Dscal(n, rec, x, 1)
|
||
scale *= rec
|
||
xmax *= rec
|
||
}
|
||
x[j] /= tjjs
|
||
} else {
|
||
for i := 0; i < n; i++ {
|
||
x[i] = 0
|
||
}
|
||
x[j] = 1
|
||
scale = 0
|
||
xmax = 0
|
||
}
|
||
} else {
|
||
x[j] = x[j]/tjjs - sumj
|
||
}
|
||
Skip2:
|
||
xmax = math.Max(xmax, math.Abs(x[j]))
|
||
}
|
||
}
|
||
scale /= tscal
|
||
if tscal != 1 {
|
||
bi.Dscal(n, 1/tscal, cnorm, 1)
|
||
}
|
||
return scale
|
||
}
|