mirror of
https://github.com/gonum/gonum.git
synced 2025-10-07 08:01:20 +08:00
610 lines
11 KiB
Go
610 lines
11 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 gonum
|
||
|
||
import (
|
||
"math"
|
||
|
||
"gonum.org/v1/gonum/blas"
|
||
"gonum.org/v1/gonum/internal/asm/f64"
|
||
)
|
||
|
||
var _ blas.Float64Level1 = Implementation{}
|
||
|
||
// Dnrm2 computes the Euclidean norm of a vector,
|
||
// sqrt(\sum_i x[i] * x[i]).
|
||
// This function returns 0 if incX is negative.
|
||
func (Implementation) Dnrm2(n int, x []float64, incX int) float64 {
|
||
if incX < 1 {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
return 0
|
||
}
|
||
if incX > 0 && (n-1)*incX >= len(x) {
|
||
panic(badX)
|
||
}
|
||
if n < 2 {
|
||
if n == 1 {
|
||
return math.Abs(x[0])
|
||
}
|
||
if n == 0 {
|
||
return 0
|
||
}
|
||
if n < 1 {
|
||
panic(negativeN)
|
||
}
|
||
}
|
||
var (
|
||
scale float64 = 0
|
||
sumSquares float64 = 1
|
||
)
|
||
if incX == 1 {
|
||
x = x[:n]
|
||
for _, v := range x {
|
||
if v == 0 {
|
||
continue
|
||
}
|
||
absxi := math.Abs(v)
|
||
if math.IsNaN(absxi) {
|
||
return math.NaN()
|
||
}
|
||
if scale < absxi {
|
||
sumSquares = 1 + sumSquares*(scale/absxi)*(scale/absxi)
|
||
scale = absxi
|
||
} else {
|
||
sumSquares = sumSquares + (absxi/scale)*(absxi/scale)
|
||
}
|
||
}
|
||
if math.IsInf(scale, 1) {
|
||
return math.Inf(1)
|
||
}
|
||
return scale * math.Sqrt(sumSquares)
|
||
}
|
||
for ix := 0; ix < n*incX; ix += incX {
|
||
val := x[ix]
|
||
if val == 0 {
|
||
continue
|
||
}
|
||
absxi := math.Abs(val)
|
||
if math.IsNaN(absxi) {
|
||
return math.NaN()
|
||
}
|
||
if scale < absxi {
|
||
sumSquares = 1 + sumSquares*(scale/absxi)*(scale/absxi)
|
||
scale = absxi
|
||
} else {
|
||
sumSquares = sumSquares + (absxi/scale)*(absxi/scale)
|
||
}
|
||
}
|
||
if math.IsInf(scale, 1) {
|
||
return math.Inf(1)
|
||
}
|
||
return scale * math.Sqrt(sumSquares)
|
||
}
|
||
|
||
// Dasum computes the sum of the absolute values of the elements of x.
|
||
// \sum_i |x[i]|
|
||
// Dasum returns 0 if incX is negative.
|
||
func (Implementation) Dasum(n int, x []float64, incX int) float64 {
|
||
var sum float64
|
||
if n < 0 {
|
||
panic(negativeN)
|
||
}
|
||
if incX < 1 {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
return 0
|
||
}
|
||
if incX > 0 && (n-1)*incX >= len(x) {
|
||
panic(badX)
|
||
}
|
||
if incX == 1 {
|
||
x = x[:n]
|
||
for _, v := range x {
|
||
sum += math.Abs(v)
|
||
}
|
||
return sum
|
||
}
|
||
for i := 0; i < n; i++ {
|
||
sum += math.Abs(x[i*incX])
|
||
}
|
||
return sum
|
||
}
|
||
|
||
// Idamax returns the index of an element of x with the largest absolute value.
|
||
// If there are multiple such indices the earliest is returned.
|
||
// Idamax returns -1 if n == 0.
|
||
func (Implementation) Idamax(n int, x []float64, incX int) int {
|
||
if incX < 1 {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
return -1
|
||
}
|
||
if incX > 0 && (n-1)*incX >= len(x) {
|
||
panic(badX)
|
||
}
|
||
if n < 2 {
|
||
if n == 1 {
|
||
return 0
|
||
}
|
||
if n == 0 {
|
||
return -1 // Netlib returns invalid index when n == 0
|
||
}
|
||
if n < 1 {
|
||
panic(negativeN)
|
||
}
|
||
}
|
||
idx := 0
|
||
max := math.Abs(x[0])
|
||
if incX == 1 {
|
||
for i, v := range x[:n] {
|
||
absV := math.Abs(v)
|
||
if absV > max {
|
||
max = absV
|
||
idx = i
|
||
}
|
||
}
|
||
return idx
|
||
}
|
||
ix := incX
|
||
for i := 1; i < n; i++ {
|
||
v := x[ix]
|
||
absV := math.Abs(v)
|
||
if absV > max {
|
||
max = absV
|
||
idx = i
|
||
}
|
||
ix += incX
|
||
}
|
||
return idx
|
||
}
|
||
|
||
// Dswap exchanges the elements of two vectors.
|
||
// x[i], y[i] = y[i], x[i] for all i
|
||
func (Implementation) Dswap(n int, x []float64, incX int, y []float64, incY int) {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
if incY == 0 {
|
||
panic(zeroIncY)
|
||
}
|
||
if n < 1 {
|
||
if n == 0 {
|
||
return
|
||
}
|
||
panic(negativeN)
|
||
}
|
||
if (incX > 0 && (n-1)*incX >= len(x)) || (incX < 0 && (1-n)*incX >= len(x)) {
|
||
panic(badX)
|
||
}
|
||
if (incY > 0 && (n-1)*incY >= len(y)) || (incY < 0 && (1-n)*incY >= len(y)) {
|
||
panic(badY)
|
||
}
|
||
if incX == 1 && incY == 1 {
|
||
x = x[:n]
|
||
for i, v := range x {
|
||
x[i], y[i] = y[i], v
|
||
}
|
||
return
|
||
}
|
||
var ix, iy int
|
||
if incX < 0 {
|
||
ix = (-n + 1) * incX
|
||
}
|
||
if incY < 0 {
|
||
iy = (-n + 1) * incY
|
||
}
|
||
for i := 0; i < n; i++ {
|
||
x[ix], y[iy] = y[iy], x[ix]
|
||
ix += incX
|
||
iy += incY
|
||
}
|
||
}
|
||
|
||
// Dcopy copies the elements of x into the elements of y.
|
||
// y[i] = x[i] for all i
|
||
func (Implementation) Dcopy(n int, x []float64, incX int, y []float64, incY int) {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
if incY == 0 {
|
||
panic(zeroIncY)
|
||
}
|
||
if n < 1 {
|
||
if n == 0 {
|
||
return
|
||
}
|
||
panic(negativeN)
|
||
}
|
||
if (incX > 0 && (n-1)*incX >= len(x)) || (incX < 0 && (1-n)*incX >= len(x)) {
|
||
panic(badX)
|
||
}
|
||
if (incY > 0 && (n-1)*incY >= len(y)) || (incY < 0 && (1-n)*incY >= len(y)) {
|
||
panic(badY)
|
||
}
|
||
if incX == 1 && incY == 1 {
|
||
copy(y[:n], x[:n])
|
||
return
|
||
}
|
||
var ix, iy int
|
||
if incX < 0 {
|
||
ix = (-n + 1) * incX
|
||
}
|
||
if incY < 0 {
|
||
iy = (-n + 1) * incY
|
||
}
|
||
for i := 0; i < n; i++ {
|
||
y[iy] = x[ix]
|
||
ix += incX
|
||
iy += incY
|
||
}
|
||
}
|
||
|
||
// Daxpy adds alpha times x to y
|
||
// y[i] += alpha * x[i] for all i
|
||
func (Implementation) Daxpy(n int, alpha float64, x []float64, incX int, y []float64, incY int) {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
if incY == 0 {
|
||
panic(zeroIncY)
|
||
}
|
||
if n < 1 {
|
||
if n == 0 {
|
||
return
|
||
}
|
||
panic(negativeN)
|
||
}
|
||
if (incX > 0 && (n-1)*incX >= len(x)) || (incX < 0 && (1-n)*incX >= len(x)) {
|
||
panic(badX)
|
||
}
|
||
if (incY > 0 && (n-1)*incY >= len(y)) || (incY < 0 && (1-n)*incY >= len(y)) {
|
||
panic(badY)
|
||
}
|
||
if alpha == 0 {
|
||
return
|
||
}
|
||
if incX == 1 && incY == 1 {
|
||
if len(x) < n {
|
||
panic(badLenX)
|
||
}
|
||
if len(y) < n {
|
||
panic(badLenY)
|
||
}
|
||
f64.AxpyUnitaryTo(y, alpha, x[:n], y)
|
||
return
|
||
}
|
||
var ix, iy int
|
||
if incX < 0 {
|
||
ix = (-n + 1) * incX
|
||
}
|
||
if incY < 0 {
|
||
iy = (-n + 1) * incY
|
||
}
|
||
if ix >= len(x) || ix+(n-1)*incX >= len(x) {
|
||
panic(badLenX)
|
||
}
|
||
if iy >= len(y) || iy+(n-1)*incY >= len(y) {
|
||
panic(badLenY)
|
||
}
|
||
f64.AxpyInc(alpha, x, y, uintptr(n), uintptr(incX), uintptr(incY), uintptr(ix), uintptr(iy))
|
||
}
|
||
|
||
// Drotg computes the plane rotation
|
||
// _ _ _ _ _ _
|
||
// | c s | | a | | r |
|
||
// | -s c | * | b | = | 0 |
|
||
// ‾ ‾ ‾ ‾ ‾ ‾
|
||
// where
|
||
// r = ±√(a^2 + b^2)
|
||
// c = a/r, the cosine of the plane rotation
|
||
// s = b/r, the sine of the plane rotation
|
||
//
|
||
// NOTE: There is a discrepancy between the refence implementation and the BLAS
|
||
// technical manual regarding the sign for r when a or b are zero.
|
||
// Drotg agrees with the definition in the manual and other
|
||
// common BLAS implementations.
|
||
func (Implementation) Drotg(a, b float64) (c, s, r, z float64) {
|
||
if b == 0 && a == 0 {
|
||
return 1, 0, a, 0
|
||
}
|
||
absA := math.Abs(a)
|
||
absB := math.Abs(b)
|
||
aGTb := absA > absB
|
||
r = math.Hypot(a, b)
|
||
if aGTb {
|
||
r = math.Copysign(r, a)
|
||
} else {
|
||
r = math.Copysign(r, b)
|
||
}
|
||
c = a / r
|
||
s = b / r
|
||
if aGTb {
|
||
z = s
|
||
} else if c != 0 { // r == 0 case handled above
|
||
z = 1 / c
|
||
} else {
|
||
z = 1
|
||
}
|
||
return
|
||
}
|
||
|
||
// Drotmg computes the modified Givens rotation. See
|
||
// http://www.netlib.org/lapack/explore-html/df/deb/drotmg_8f.html
|
||
// for more details.
|
||
func (Implementation) Drotmg(d1, d2, x1, y1 float64) (p blas.DrotmParams, rd1, rd2, rx1 float64) {
|
||
var p1, p2, q1, q2, u float64
|
||
|
||
const (
|
||
gam = 4096.0
|
||
gamsq = 16777216.0
|
||
rgamsq = 5.9604645e-8
|
||
)
|
||
|
||
if d1 < 0 {
|
||
p.Flag = blas.Rescaling
|
||
return
|
||
}
|
||
|
||
p2 = d2 * y1
|
||
if p2 == 0 {
|
||
p.Flag = blas.Identity
|
||
rd1 = d1
|
||
rd2 = d2
|
||
rx1 = x1
|
||
return
|
||
}
|
||
p1 = d1 * x1
|
||
q2 = p2 * y1
|
||
q1 = p1 * x1
|
||
|
||
absQ1 := math.Abs(q1)
|
||
absQ2 := math.Abs(q2)
|
||
|
||
if absQ1 < absQ2 && q2 < 0 {
|
||
p.Flag = blas.Rescaling
|
||
return
|
||
}
|
||
|
||
if d1 == 0 {
|
||
p.Flag = blas.Diagonal
|
||
p.H[0] = p1 / p2
|
||
p.H[3] = x1 / y1
|
||
u = 1 + p.H[0]*p.H[3]
|
||
rd1, rd2 = d2/u, d1/u
|
||
rx1 = y1 / u
|
||
return
|
||
}
|
||
|
||
// Now we know that d1 != 0, and d2 != 0. If d2 == 0, it would be caught
|
||
// when p2 == 0, and if d1 == 0, then it is caught above
|
||
|
||
if absQ1 > absQ2 {
|
||
p.H[1] = -y1 / x1
|
||
p.H[2] = p2 / p1
|
||
u = 1 - p.H[2]*p.H[1]
|
||
rd1 = d1
|
||
rd2 = d2
|
||
rx1 = x1
|
||
p.Flag = blas.OffDiagonal
|
||
// u must be greater than zero because |q1| > |q2|, so check from netlib
|
||
// is unnecessary
|
||
// This is left in for ease of comparison with complex routines
|
||
//if u > 0 {
|
||
rd1 /= u
|
||
rd2 /= u
|
||
rx1 *= u
|
||
//}
|
||
} else {
|
||
p.Flag = blas.Diagonal
|
||
p.H[0] = p1 / p2
|
||
p.H[3] = x1 / y1
|
||
u = 1 + p.H[0]*p.H[3]
|
||
rd1 = d2 / u
|
||
rd2 = d1 / u
|
||
rx1 = y1 * u
|
||
}
|
||
|
||
for rd1 <= rgamsq || rd1 >= gamsq {
|
||
if p.Flag == blas.OffDiagonal {
|
||
p.H[0] = 1
|
||
p.H[3] = 1
|
||
p.Flag = blas.Rescaling
|
||
} else if p.Flag == blas.Diagonal {
|
||
p.H[1] = -1
|
||
p.H[2] = 1
|
||
p.Flag = blas.Rescaling
|
||
}
|
||
if rd1 <= rgamsq {
|
||
rd1 *= gam * gam
|
||
rx1 /= gam
|
||
p.H[0] /= gam
|
||
p.H[2] /= gam
|
||
} else {
|
||
rd1 /= gam * gam
|
||
rx1 *= gam
|
||
p.H[0] *= gam
|
||
p.H[2] *= gam
|
||
}
|
||
}
|
||
|
||
for math.Abs(rd2) <= rgamsq || math.Abs(rd2) >= gamsq {
|
||
if p.Flag == blas.OffDiagonal {
|
||
p.H[0] = 1
|
||
p.H[3] = 1
|
||
p.Flag = blas.Rescaling
|
||
} else if p.Flag == blas.Diagonal {
|
||
p.H[1] = -1
|
||
p.H[2] = 1
|
||
p.Flag = blas.Rescaling
|
||
}
|
||
if math.Abs(rd2) <= rgamsq {
|
||
rd2 *= gam * gam
|
||
p.H[1] /= gam
|
||
p.H[3] /= gam
|
||
} else {
|
||
rd2 /= gam * gam
|
||
p.H[1] *= gam
|
||
p.H[3] *= gam
|
||
}
|
||
}
|
||
return
|
||
}
|
||
|
||
// Drot applies a plane transformation.
|
||
// x[i] = c * x[i] + s * y[i]
|
||
// y[i] = c * y[i] - s * x[i]
|
||
func (Implementation) Drot(n int, x []float64, incX int, y []float64, incY int, c float64, s float64) {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
if incY == 0 {
|
||
panic(zeroIncY)
|
||
}
|
||
if n < 1 {
|
||
if n == 0 {
|
||
return
|
||
}
|
||
panic(negativeN)
|
||
}
|
||
if (incX > 0 && (n-1)*incX >= len(x)) || (incX < 0 && (1-n)*incX >= len(x)) {
|
||
panic(badX)
|
||
}
|
||
if (incY > 0 && (n-1)*incY >= len(y)) || (incY < 0 && (1-n)*incY >= len(y)) {
|
||
panic(badY)
|
||
}
|
||
if incX == 1 && incY == 1 {
|
||
x = x[:n]
|
||
for i, vx := range x {
|
||
vy := y[i]
|
||
x[i], y[i] = c*vx+s*vy, c*vy-s*vx
|
||
}
|
||
return
|
||
}
|
||
var ix, iy int
|
||
if incX < 0 {
|
||
ix = (-n + 1) * incX
|
||
}
|
||
if incY < 0 {
|
||
iy = (-n + 1) * incY
|
||
}
|
||
for i := 0; i < n; i++ {
|
||
vx := x[ix]
|
||
vy := y[iy]
|
||
x[ix], y[iy] = c*vx+s*vy, c*vy-s*vx
|
||
ix += incX
|
||
iy += incY
|
||
}
|
||
}
|
||
|
||
// Drotm applies the modified Givens rotation to the 2×n matrix.
|
||
func (Implementation) Drotm(n int, x []float64, incX int, y []float64, incY int, p blas.DrotmParams) {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
if incY == 0 {
|
||
panic(zeroIncY)
|
||
}
|
||
if n <= 0 {
|
||
if n == 0 {
|
||
return
|
||
}
|
||
panic(negativeN)
|
||
}
|
||
if (incX > 0 && (n-1)*incX >= len(x)) || (incX < 0 && (1-n)*incX >= len(x)) {
|
||
panic(badX)
|
||
}
|
||
if (incY > 0 && (n-1)*incY >= len(y)) || (incY < 0 && (1-n)*incY >= len(y)) {
|
||
panic(badY)
|
||
}
|
||
|
||
var h11, h12, h21, h22 float64
|
||
var ix, iy int
|
||
switch p.Flag {
|
||
case blas.Identity:
|
||
return
|
||
case blas.Rescaling:
|
||
h11 = p.H[0]
|
||
h12 = p.H[2]
|
||
h21 = p.H[1]
|
||
h22 = p.H[3]
|
||
case blas.OffDiagonal:
|
||
h11 = 1
|
||
h12 = p.H[2]
|
||
h21 = p.H[1]
|
||
h22 = 1
|
||
case blas.Diagonal:
|
||
h11 = p.H[0]
|
||
h12 = 1
|
||
h21 = -1
|
||
h22 = p.H[3]
|
||
}
|
||
if incX < 0 {
|
||
ix = (-n + 1) * incX
|
||
}
|
||
if incY < 0 {
|
||
iy = (-n + 1) * incY
|
||
}
|
||
if incX == 1 && incY == 1 {
|
||
x = x[:n]
|
||
for i, vx := range x {
|
||
vy := y[i]
|
||
x[i], y[i] = vx*h11+vy*h12, vx*h21+vy*h22
|
||
}
|
||
return
|
||
}
|
||
for i := 0; i < n; i++ {
|
||
vx := x[ix]
|
||
vy := y[iy]
|
||
x[ix], y[iy] = vx*h11+vy*h12, vx*h21+vy*h22
|
||
ix += incX
|
||
iy += incY
|
||
}
|
||
}
|
||
|
||
// Dscal scales x by alpha.
|
||
// x[i] *= alpha
|
||
// Dscal has no effect if incX < 0.
|
||
func (Implementation) Dscal(n int, alpha float64, x []float64, incX int) {
|
||
if incX < 1 {
|
||
if incX == 0 {
|
||
panic(zeroIncX)
|
||
}
|
||
return
|
||
}
|
||
if (n-1)*incX >= len(x) {
|
||
panic(badX)
|
||
}
|
||
if n < 1 {
|
||
if n == 0 {
|
||
return
|
||
}
|
||
panic(negativeN)
|
||
}
|
||
if alpha == 0 {
|
||
if incX == 1 {
|
||
x = x[:n]
|
||
for i := range x {
|
||
x[i] = 0
|
||
}
|
||
return
|
||
}
|
||
for ix := 0; ix < n*incX; ix += incX {
|
||
x[ix] = 0
|
||
}
|
||
return
|
||
}
|
||
if incX == 1 {
|
||
f64.ScalUnitary(alpha, x[:n])
|
||
return
|
||
}
|
||
for ix := 0; ix < n*incX; ix += incX {
|
||
x[ix] *= alpha
|
||
}
|
||
}
|