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// 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 gonum
import (
"math"
"gonum.org/v1/gonum/blas"
"gonum.org/v1/gonum/blas/blas64"
)
// Dlatrs solves a triangular system of equations scaled to prevent overflow. It
// solves
// A * x = scale * b if trans == blas.NoTrans
// Aᵀ * 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
// 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) {
switch {
case uplo != blas.Upper && uplo != blas.Lower:
panic(badUplo)
case trans != blas.NoTrans && trans != blas.Trans && trans != blas.ConjTrans:
panic(badTrans)
case diag != blas.Unit && diag != blas.NonUnit:
panic(badDiag)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
}
// Quick return if possible.
if n == 0 {
return 0
}
switch {
case len(a) < (n-1)*lda+n:
panic(shortA)
case len(x) < n:
panic(shortX)
case len(cnorm) < n:
panic(shortCNorm)
}
upper := uplo == blas.Upper
nonUnit := diag == blas.NonUnit
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 trans == blas.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 trans == blas.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
}