360 lines
7.5 KiB
Go
360 lines
7.5 KiB
Go
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// Copyright ©2015 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package gonum
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import (
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"math"
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"gonum.org/v1/gonum/blas"
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"gonum.org/v1/gonum/blas/blas64"
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)
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// Dlatrs solves a triangular system of equations scaled to prevent overflow. It
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// solves
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// A * x = scale * b if trans == blas.NoTrans
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// A^T * x = scale * b if trans == blas.Trans
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// where the scale s is set for numeric stability.
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//
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// A is an n×n triangular matrix. On entry, the slice x contains the values of
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// b, and on exit it contains the solution vector x.
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//
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// If normin == true, cnorm is an input and cnorm[j] contains the norm of the off-diagonal
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// part of the j^th column of A. If trans == blas.NoTrans, cnorm[j] must be greater
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// than or equal to the infinity norm, and greater than or equal to the one-norm
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// otherwise. If normin == false, then cnorm is treated as an output, and is set
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// to contain the 1-norm of the off-diagonal part of the j^th column of A.
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//
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// Dlatrs is an internal routine. It is exported for testing purposes.
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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) {
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switch {
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case uplo != blas.Upper && uplo != blas.Lower:
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panic(badUplo)
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case trans != blas.NoTrans && trans != blas.Trans && trans != blas.ConjTrans:
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panic(badTrans)
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case diag != blas.Unit && diag != blas.NonUnit:
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panic(badDiag)
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case n < 0:
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panic(nLT0)
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case lda < max(1, n):
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panic(badLdA)
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}
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// Quick return if possible.
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if n == 0 {
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return 0
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}
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switch {
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case len(a) < (n-1)*lda+n:
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panic(shortA)
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case len(x) < n:
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panic(shortX)
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case len(cnorm) < n:
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panic(shortCNorm)
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}
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upper := uplo == blas.Upper
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nonUnit := diag == blas.NonUnit
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smlnum := dlamchS / dlamchP
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bignum := 1 / smlnum
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scale = 1
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bi := blas64.Implementation()
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if !normin {
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if upper {
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cnorm[0] = 0
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for j := 1; j < n; j++ {
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cnorm[j] = bi.Dasum(j, a[j:], lda)
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}
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} else {
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for j := 0; j < n-1; j++ {
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cnorm[j] = bi.Dasum(n-j-1, a[(j+1)*lda+j:], lda)
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}
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cnorm[n-1] = 0
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}
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}
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// Scale the column norms by tscal if the maximum element in cnorm is greater than bignum.
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imax := bi.Idamax(n, cnorm, 1)
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tmax := cnorm[imax]
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var tscal float64
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if tmax <= bignum {
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tscal = 1
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} else {
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tscal = 1 / (smlnum * tmax)
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bi.Dscal(n, tscal, cnorm, 1)
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}
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// Compute a bound on the computed solution vector to see if bi.Dtrsv can be used.
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j := bi.Idamax(n, x, 1)
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xmax := math.Abs(x[j])
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xbnd := xmax
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var grow float64
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var jfirst, jlast, jinc int
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if trans == blas.NoTrans {
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if upper {
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jfirst = n - 1
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jlast = -1
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jinc = -1
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} else {
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jfirst = 0
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jlast = n
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jinc = 1
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}
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// Compute the growth in A * x = b.
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if tscal != 1 {
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grow = 0
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goto Solve
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}
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if nonUnit {
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grow = 1 / math.Max(xbnd, smlnum)
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xbnd = grow
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for j := jfirst; j != jlast; j += jinc {
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if grow <= smlnum {
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goto Solve
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}
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tjj := math.Abs(a[j*lda+j])
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xbnd = math.Min(xbnd, math.Min(1, tjj)*grow)
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if tjj+cnorm[j] >= smlnum {
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grow *= tjj / (tjj + cnorm[j])
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} else {
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grow = 0
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}
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}
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grow = xbnd
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} else {
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grow = math.Min(1, 1/math.Max(xbnd, smlnum))
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for j := jfirst; j != jlast; j += jinc {
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if grow <= smlnum {
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goto Solve
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}
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grow *= 1 / (1 + cnorm[j])
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}
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}
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} else {
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if upper {
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jfirst = 0
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jlast = n
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jinc = 1
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} else {
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jfirst = n - 1
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jlast = -1
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jinc = -1
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}
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if tscal != 1 {
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grow = 0
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goto Solve
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}
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if nonUnit {
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grow = 1 / (math.Max(xbnd, smlnum))
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xbnd = grow
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for j := jfirst; j != jlast; j += jinc {
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if grow <= smlnum {
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goto Solve
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}
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xj := 1 + cnorm[j]
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grow = math.Min(grow, xbnd/xj)
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tjj := math.Abs(a[j*lda+j])
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if xj > tjj {
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xbnd *= tjj / xj
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}
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}
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grow = math.Min(grow, xbnd)
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} else {
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grow = math.Min(1, 1/math.Max(xbnd, smlnum))
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for j := jfirst; j != jlast; j += jinc {
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if grow <= smlnum {
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goto Solve
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}
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xj := 1 + cnorm[j]
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grow /= xj
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}
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}
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}
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Solve:
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if grow*tscal > smlnum {
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// Use the Level 2 BLAS solve if the reciprocal of the bound on
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// elements of X is not too small.
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bi.Dtrsv(uplo, trans, diag, n, a, lda, x, 1)
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if tscal != 1 {
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bi.Dscal(n, 1/tscal, cnorm, 1)
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}
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return scale
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}
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// Use a Level 1 BLAS solve, scaling intermediate results.
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if xmax > bignum {
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scale = bignum / xmax
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bi.Dscal(n, scale, x, 1)
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xmax = bignum
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}
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if trans == blas.NoTrans {
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for j := jfirst; j != jlast; j += jinc {
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xj := math.Abs(x[j])
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var tjj, tjjs float64
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if nonUnit {
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tjjs = a[j*lda+j] * tscal
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} else {
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tjjs = tscal
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if tscal == 1 {
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goto Skip1
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}
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}
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tjj = math.Abs(tjjs)
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if tjj > smlnum {
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if tjj < 1 {
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if xj > tjj*bignum {
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rec := 1 / xj
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bi.Dscal(n, rec, x, 1)
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scale *= rec
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xmax *= rec
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}
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}
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x[j] /= tjjs
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xj = math.Abs(x[j])
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} else if tjj > 0 {
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if xj > tjj*bignum {
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rec := (tjj * bignum) / xj
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if cnorm[j] > 1 {
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rec /= cnorm[j]
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}
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bi.Dscal(n, rec, x, 1)
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scale *= rec
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xmax *= rec
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}
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x[j] /= tjjs
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xj = math.Abs(x[j])
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} else {
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for i := 0; i < n; i++ {
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x[i] = 0
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}
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x[j] = 1
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xj = 1
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scale = 0
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xmax = 0
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}
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Skip1:
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if xj > 1 {
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rec := 1 / xj
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if cnorm[j] > (bignum-xmax)*rec {
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rec *= 0.5
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bi.Dscal(n, rec, x, 1)
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scale *= rec
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}
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} else if xj*cnorm[j] > bignum-xmax {
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bi.Dscal(n, 0.5, x, 1)
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scale *= 0.5
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}
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if upper {
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if j > 0 {
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bi.Daxpy(j, -x[j]*tscal, a[j:], lda, x, 1)
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i := bi.Idamax(j, x, 1)
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xmax = math.Abs(x[i])
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}
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} else {
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if j < n-1 {
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bi.Daxpy(n-j-1, -x[j]*tscal, a[(j+1)*lda+j:], lda, x[j+1:], 1)
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i := j + bi.Idamax(n-j-1, x[j+1:], 1)
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xmax = math.Abs(x[i])
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}
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}
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}
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} else {
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for j := jfirst; j != jlast; j += jinc {
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xj := math.Abs(x[j])
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uscal := tscal
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rec := 1 / math.Max(xmax, 1)
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var tjjs float64
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if cnorm[j] > (bignum-xj)*rec {
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rec *= 0.5
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if nonUnit {
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tjjs = a[j*lda+j] * tscal
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} else {
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tjjs = tscal
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}
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tjj := math.Abs(tjjs)
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if tjj > 1 {
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rec = math.Min(1, rec*tjj)
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uscal /= tjjs
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}
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if rec < 1 {
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bi.Dscal(n, rec, x, 1)
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scale *= rec
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xmax *= rec
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}
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}
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var sumj float64
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if uscal == 1 {
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if upper {
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sumj = bi.Ddot(j, a[j:], lda, x, 1)
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} else if j < n-1 {
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sumj = bi.Ddot(n-j-1, a[(j+1)*lda+j:], lda, x[j+1:], 1)
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}
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} else {
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if upper {
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for i := 0; i < j; i++ {
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sumj += (a[i*lda+j] * uscal) * x[i]
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}
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} else if j < n {
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for i := j + 1; i < n; i++ {
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sumj += (a[i*lda+j] * uscal) * x[i]
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}
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}
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}
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if uscal == tscal {
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x[j] -= sumj
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xj := math.Abs(x[j])
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var tjjs float64
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if nonUnit {
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tjjs = a[j*lda+j] * tscal
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} else {
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tjjs = tscal
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if tscal == 1 {
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goto Skip2
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}
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}
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tjj := math.Abs(tjjs)
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if tjj > smlnum {
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if tjj < 1 {
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if xj > tjj*bignum {
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rec = 1 / xj
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bi.Dscal(n, rec, x, 1)
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scale *= rec
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xmax *= rec
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}
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}
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x[j] /= tjjs
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} else if tjj > 0 {
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if xj > tjj*bignum {
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rec = (tjj * bignum) / xj
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bi.Dscal(n, rec, x, 1)
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scale *= rec
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xmax *= rec
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}
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x[j] /= tjjs
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} else {
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for i := 0; i < n; i++ {
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x[i] = 0
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}
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x[j] = 1
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scale = 0
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xmax = 0
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}
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} else {
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x[j] = x[j]/tjjs - sumj
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}
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Skip2:
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xmax = math.Max(xmax, math.Abs(x[j]))
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}
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}
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scale /= tscal
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if tscal != 1 {
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bi.Dscal(n, 1/tscal, cnorm, 1)
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}
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return scale
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}
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