Multidimensional Newton-Raphson method: Difference between revisions
Multidimensional Newton-Raphson method (view source)
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Line 3:
;Task:
Create a program that finds and outputs the root of a system of nonlinear equations
using Newton-Raphson
<br><br>
=={{header|C sharp|C#}}==
For matrix inversion and matrix and vector definitions - see C# source from [[Gaussian elimination]]
<
using System;
Line 42 ⟶ 43:
}
}
</syntaxhighlight>
<
using System;
Line 99 ⟶ 100:
}
}
</syntaxhighlight>
{{out}}<pre>
2.54258545959024
1.06149981539336
</pre>
=={{header|Go}}==
{{trans|Kotlin}}
<br>
We follow the Kotlin example of coding our own matrix methods rather than using a third party library.
<syntaxhighlight lang="go">package main
import (
"fmt"
"math"
)
type vector = []float64
type matrix []vector
type fun = func(vector) float64
type funs = []fun
type jacobian = []funs
func (m1 matrix) mul(m2 matrix) matrix {
rows1, cols1 := len(m1), len(m1[0])
rows2, cols2 := len(m2), len(m2[0])
if cols1 != rows2 {
panic("Matrices cannot be multiplied.")
}
result := make(matrix, rows1)
for i := 0; i < rows1; i++ {
result[i] = make(vector, cols2)
for j := 0; j < cols2; j++ {
for k := 0; k < rows2; k++ {
result[i][j] += m1[i][k] * m2[k][j]
}
}
}
return result
}
func (m1 matrix) sub(m2 matrix) matrix {
rows, cols := len(m1), len(m1[0])
if rows != len(m2) || cols != len(m2[0]) {
panic("Matrices cannot be subtracted.")
}
result := make(matrix, rows)
for i := 0; i < rows; i++ {
result[i] = make(vector, cols)
for j := 0; j < cols; j++ {
result[i][j] = m1[i][j] - m2[i][j]
}
}
return result
}
func (m matrix) transpose() matrix {
rows, cols := len(m), len(m[0])
trans := make(matrix, cols)
for i := 0; i < cols; i++ {
trans[i] = make(vector, rows)
for j := 0; j < rows; j++ {
trans[i][j] = m[j][i]
}
}
return trans
}
func (m matrix) inverse() matrix {
le := len(m)
for _, v := range m {
if len(v) != le {
panic("Not a square matrix")
}
}
aug := make(matrix, le)
for i := 0; i < le; i++ {
aug[i] = make(vector, 2*le)
copy(aug[i], m[i])
// augment by identity matrix to right
aug[i][i+le] = 1
}
aug.toReducedRowEchelonForm()
inv := make(matrix, le)
// remove identity matrix to left
for i := 0; i < le; i++ {
inv[i] = make(vector, le)
copy(inv[i], aug[i][le:])
}
return inv
}
// note: this mutates the matrix in place
func (m matrix) toReducedRowEchelonForm() {
lead := 0
rowCount, colCount := len(m), len(m[0])
for r := 0; r < rowCount; r++ {
if colCount <= lead {
return
}
i := r
for m[i][lead] == 0 {
i++
if rowCount == i {
i = r
lead++
if colCount == lead {
return
}
}
}
m[i], m[r] = m[r], m[i]
if div := m[r][lead]; div != 0 {
for j := 0; j < colCount; j++ {
m[r][j] /= div
}
}
for k := 0; k < rowCount; k++ {
if k != r {
mult := m[k][lead]
for j := 0; j < colCount; j++ {
m[k][j] -= m[r][j] * mult
}
}
}
lead++
}
}
func solve(fs funs, jacob jacobian, guesses vector) vector {
size := len(fs)
var gu1 vector
gu2 := make(vector, len(guesses))
copy(gu2, guesses)
jac := make(matrix, size)
for i := 0; i < size; i++ {
jac[i] = make(vector, size)
}
tol := 1e-8
maxIter := 12
iter := 0
for {
gu1 = gu2
g := matrix{gu1}.transpose()
t := make(vector, size)
for i := 0; i < size; i++ {
t[i] = fs[i](gu1)
}
f := matrix{t}.transpose()
for i := 0; i < size; i++ {
for j := 0; j < size; j++ {
jac[i][j] = jacob[i][j](gu1)
}
}
g1 := g.sub(jac.inverse().mul(f))
gu2 = make(vector, size)
for i := 0; i < size; i++ {
gu2[i] = g1[i][0]
}
iter++
any := false
for i, v := range gu2 {
if math.Abs(v)-gu1[i] > tol {
any = true
break
}
}
if !any || iter >= maxIter {
break
}
}
return gu2
}
func main() {
/*
solve the two non-linear equations:
y = -x^2 + x + 0.5
y + 5xy = x^2
given initial guesses of x = y = 1.2
Example taken from:
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
Expected results: x = 1.23332, y = 0.2122
*/
f1 := func(x vector) float64 { return -x[0]*x[0] + x[0] + 0.5 - x[1] }
f2 := func(x vector) float64 { return x[1] + 5*x[0]*x[1] - x[0]*x[0] }
fs := funs{f1, f2}
jacob := jacobian{
funs{
func(x vector) float64 { return -2*x[0] + 1 },
func(x vector) float64 { return -1 },
},
funs{
func(x vector) float64 { return 5*x[1] - 2*x[0] },
func(x vector) float64 { return 1 + 5*x[0] },
},
}
guesses := vector{1.2, 1.2}
sol := solve(fs, jacob, guesses)
fmt.Printf("Approximate solutions are x = %.7f, y = %.7f\n", sol[0], sol[1])
/*
solve the three non-linear equations:
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 - 2y^2 - 20z = 0
x^2 - y^2 + z^2 = 0
given initial guesses of x = y = 1.0 and z = 0.0
Example taken from:
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
*/
fmt.Println()
f3 := func(x vector) float64 { return 9*x[0]*x[0] + 36*x[1]*x[1] + 4*x[2]*x[2] - 36 }
f4 := func(x vector) float64 { return x[0]*x[0] - 2*x[1]*x[1] - 20*x[2] }
f5 := func(x vector) float64 { return x[0]*x[0] - x[1]*x[1] + x[2]*x[2] }
fs = funs{f3, f4, f5}
jacob = jacobian{
funs{
func(x vector) float64 { return 18 * x[0] },
func(x vector) float64 { return 72 * x[1] },
func(x vector) float64 { return 8 * x[2] },
},
funs{
func(x vector) float64 { return 2 * x[0] },
func(x vector) float64 { return -4 * x[1] },
func(x vector) float64 { return -20 },
},
funs{
func(x vector) float64 { return 2 * x[0] },
func(x vector) float64 { return -2 * x[1] },
func(x vector) float64 { return 2 * x[2] },
},
}
guesses = vector{1, 1, 0}
sol = solve(fs, jacob, guesses)
fmt.Printf("Approximate solutions are x = %.7f, y = %.7f, z = %.7f\n", sol[0], sol[1], sol[2])
}</syntaxhighlight>
{{out}}
<pre>
Approximate solutions are x = 1.2333178, y = 0.2122450
Approximate solutions are x = 0.8936282, y = 0.8945270, z = -0.0400893
</pre>
=={{header|Julia}}==
NLsolve is a Julia package for nonlinear systems of equations, with the Newton-Raphson method one of the choices for solvers.
<syntaxhighlight lang="julia"># from the NLSolve documentation: to solve
# (x, y) -> ((x+3)*(y^3-7)+18, sin(y*exp(x)-1))
using NLsolve
function f!(F, x)
F[1] = (x[1]+3)*(x[2]^3-7)+18
F[2] = sin(x[2]*exp(x[1])-1)
end
function j!(J, x)
J[1, 1] = x[2]^3-7
J[1, 2] = 3*x[2]^2*(x[1]+3)
u = exp(x[1])*cos(x[2]*exp(x[1])-1)
J[2, 1] = x[2]*u
J[2, 2] = u
end
println(nlsolve(f!, j!, [ 0.1; 1.2], method = :newton))
</syntaxhighlight>{{out}}
<pre>
Results of Nonlinear Solver Algorithm
* Algorithm: Newton with line-search
* Starting Point: [0.1, 1.2]
* Zero: [-3.7818e-16, 1.0]
* Inf-norm of residuals: 0.000000
* Iterations: 4
* Convergence: true
* |x - x'| < 0.0e+00: false
* |f(x)| < 1.0e-08: true
* Function Calls (f): 5
* Jacobian Calls (df/dx): 4
</pre>
=={{header|Kotlin}}==
A straightforward approach multiplying by the inverse of the Jacobian, rather than dividing by f'(x) as one would do in the single dimensional case, which is quick enough here.
As neither the JDK nor the Kotlin Standard Library have matrix functions built in, most of the functions used have been taken from other tasks.
<syntaxhighlight lang="scala">// Version 1.2.31
import kotlin.math.abs
typealias Vector = DoubleArray
typealias Matrix = Array<Vector>
typealias Func = (Vector) -> Double
typealias Funcs = List<Func>
typealias Jacobian = List<Funcs>
operator fun Matrix.times(other: Matrix): Matrix {
val rows1 = this.size
val cols1 = this[0].size
val rows2 = other.size
val cols2 = other[0].size
require(cols1 == rows2)
val result = Matrix(rows1) { Vector(cols2) }
for (i in 0 until rows1) {
for (j in 0 until cols2) {
for (k in 0 until rows2) {
result[i][j] += this[i][k] * other[k][j]
}
}
}
return result
}
operator fun Matrix.minus(other: Matrix): Matrix {
val rows = this.size
val cols = this[0].size
require(rows == other.size && cols == other[0].size)
val result = Matrix(rows) { Vector(cols) }
for (i in 0 until rows) {
for (j in 0 until cols) {
result[i][j] = this[i][j] - other[i][j]
}
}
return result
}
fun Matrix.transpose(): Matrix {
val rows = this.size
val cols = this[0].size
val trans = Matrix(cols) { Vector(rows) }
for (i in 0 until cols) {
for (j in 0 until rows) trans[i][j] = this[j][i]
}
return trans
}
fun Matrix.inverse(): Matrix {
val len = this.size
require(this.all { it.size == len }) { "Not a square matrix" }
val aug = Array(len) { DoubleArray(2 * len) }
for (i in 0 until len) {
for (j in 0 until len) aug[i][j] = this[i][j]
// augment by identity matrix to right
aug[i][i + len] = 1.0
}
aug.toReducedRowEchelonForm()
val inv = Array(len) { DoubleArray(len) }
// remove identity matrix to left
for (i in 0 until len) {
for (j in len until 2 * len) inv[i][j - len] = aug[i][j]
}
return inv
}
fun Matrix.toReducedRowEchelonForm() {
var lead = 0
val rowCount = this.size
val colCount = this[0].size
for (r in 0 until rowCount) {
if (colCount <= lead) return
var i = r
while (this[i][lead] == 0.0) {
i++
if (rowCount == i) {
i = r
lead++
if (colCount == lead) return
}
}
val temp = this[i]
this[i] = this[r]
this[r] = temp
if (this[r][lead] != 0.0) {
val div = this[r][lead]
for (j in 0 until colCount) this[r][j] /= div
}
for (k in 0 until rowCount) {
if (k != r) {
val mult = this[k][lead]
for (j in 0 until colCount) this[k][j] -= this[r][j] * mult
}
}
lead++
}
}
fun solve(funcs: Funcs, jacobian: Jacobian, guesses: Vector): Vector {
val size = funcs.size
var gu1: Vector
var gu2 = guesses.copyOf()
val jac = Matrix(size) { Vector(size) }
val tol = 1.0e-8
val maxIter = 12
var iter = 0
do {
gu1 = gu2
val g = arrayOf(gu1).transpose()
val f = arrayOf(Vector(size) { funcs[it](gu1) }).transpose()
for (i in 0 until size) {
for (j in 0 until size) {
jac[i][j] = jacobian[i][j](gu1)
}
}
val g1 = g - jac.inverse() * f
gu2 = Vector(size) { g1[it][0] }
iter++
}
while (gu2.withIndex().any { iv -> abs(iv.value - gu1[iv.index]) > tol } && iter < maxIter)
return gu2
}
fun main(args: Array<String>) {
/* solve the two non-linear equations:
y = -x^2 + x + 0.5
y + 5xy = x^2
given initial guesses of x = y = 1.2
Example taken from:
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
Expected results: x = 1.23332, y = 0.2122
*/
val f1: Func = { x -> -x[0] * x[0] + x[0] + 0.5 - x[1] }
val f2: Func = { x -> x[1] + 5 * x[0] * x[1] - x[0] * x[0] }
val funcs = listOf(f1, f2)
val jacobian = listOf(
listOf<Func>({ x -> - 2.0 * x[0] + 1.0 }, { _ -> -1.0 }),
listOf<Func>({ x -> 5.0 * x[1] - 2.0 * x[0] }, { x -> 1.0 + 5.0 * x[0] })
)
val guesses = doubleArrayOf(1.2, 1.2)
val (xx, yy) = solve(funcs, jacobian, guesses)
System.out.printf("Approximate solutions are x = %.7f, y = %.7f\n", xx, yy)
/* solve the three non-linear equations:
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 - 2y^2 - 20z = 0
x^2 - y^2 + z^2 = 0
given initial guesses of x = y = 1.0 and z = 0.0
Example taken from:
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
*/
println()
val f3: Func = { x -> 9.0 * x[0] * x[0] + 36.0 * x[1] * x[1] + 4.0 * x[2] * x[2] - 36.0 }
val f4: Func = { x -> x[0] * x[0] - 2.0 * x[1] * x[1] - 20.0 * x[2] }
val f5: Func = { x -> x[0] * x[0] - x[1] * x[1] + x[2] * x[2] }
val funcs2 = listOf(f3, f4, f5)
val jacobian2 = listOf(
listOf<Func>({ x -> 18.0 * x[0] }, { x -> 72.0 * x[1] }, { x -> 8.0 * x[2] }),
listOf<Func>({ x -> 2.0 * x[0] }, { x -> -4.0 * x[1] }, { _ -> -20.0 }),
listOf<Func>({ x -> 2.0 * x[0] }, { x -> -2.0 * x[1] }, { x -> 2.0 * x[2] })
)
val guesses2 = doubleArrayOf(1.0, 1.0, 0.0)
val (xx2, yy2, zz2) = solve(funcs2, jacobian2, guesses2)
System.out.printf("Approximate solutions are x = %.7f, y = %.7f, z = %.7f\n", xx2, yy2, zz2)
}</syntaxhighlight>
{{out}}
<pre>
Approximate solutions are x = 1.2333178, y = 0.2122450
Approximate solutions are x = 0.8936282, y = 0.8945270, z = -0.0400893
</pre>
=={{header|Nim}}==
{{trans|Kotlin}}
<syntaxhighlight lang="nim">import sequtils, strformat, sugar
type
Vector = seq[float]
Matrix = seq[Vector]
Func = (Vector) -> float
Funcs = seq[Func]
Jacobian = seq[Funcs]
func `*`(m1, m2: Matrix): Matrix =
let
rows1 = m1.len
cols1 = m1[0].len
rows2 = m2.len
cols2 = m2[0].len
doAssert cols1 == rows2
result = newSeqWith(rows1, newSeq[float](cols2))
for i in 0..<rows1:
for j in 0..<cols2:
for k in 0..<rows2:
result[i][j] += m1[i][k] * m2[k][j]
func `-`(m1, m2: Matrix): Matrix =
let
rows = m1.len
cols = m1[0].len
doAssert m2.len == rows and m2[0].len == cols
result = newSeqWith(rows, newSeq[float](cols))
for i in 0..<rows:
for j in 0..<cols:
result[i][j] = m1[i][j] - m2[i][j]
func transposed(m: Matrix): Matrix =
let
rows = m.len
cols = m[0].len
result = newSeqWith(cols, newSeq[float](rows))
for i in 0..<cols:
for j in 0..<rows:
result[i][j] = m[j][i]
func toReducedRowEchelonForm(m: var Matrix) =
var lead = 0
let rowCount = m.len
let colCount = m[0].len
for r in 0..<rowCount:
if colCount <= lead: return
var i = r
while m[i][lead] == 0:
inc i
if rowCount == i:
i = r
inc lead
if colCount == lead: return
swap m[i], m[r]
if m[r][lead] != 0:
let divisor = m[r][lead]
for j in 0..<colCount: m[r][j] /= divisor
for k in 0..<rowCount:
if k != r:
let mult = m[k][lead]
for j in 0..<colCount: m[k][j] -= m[r][j] * mult
inc lead
func inverse(m: Matrix): Matrix =
let size = m.len
doAssert m.allIt(it.len == size), "not a square matrix."
var aug = newSeqWith(size, newSeq[float](2 * size))
for i in 0..<size:
for j in 0..<size: aug[i][j] = m[i][j]
# Augment by identity matrix to right.
aug[i][i + size] = 1
aug.toReducedRowEchelonForm()
result = newSeqWith(size, newSeq[float](size))
# Remove identity matrix to left.
for i in 0..<size:
for j in 0..<size: result[i][j] = aug[i][j + size]
proc solve(funcs: Funcs; jacobian: Jacobian; guesses: Vector): Vector =
let size = funcs.len
result = guesses
var jac = newSeqWith(size, newSeq[float](size))
const Tol = 1e-8
let MaxIter = 12
var iter = 1
while true:
let gu = move(result)
let g = transposed(@[gu])
let f = transposed(@[funcs.mapIt(it(gu))])
for i in 0..<size:
for j in 0..<size:
jac[i][j] = jacobian[i][j](gu)
let g1 = g - inverse(jac) * f
result = g1.mapIt(it[0])
inc iter
if iter > MaxIter: break
var exit = true
for idx, val in result:
if abs(val - gu[idx]) > Tol:
exit = false
break
if exit: break
when isMainModule:
#[ Solve the two non-linear equations:
y = -x^2 + x + 0.5
y + 5xy = x^2
given initial guesses of x = y = 1.2
Example taken from:
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
Expected results: x = 1.23332, y = 0.2122
]#
let
f1: Func = (x: Vector) => -x[0] * x[0] + x[0] + 0.5 - x[1]
f2: Func = (x: Vector) => x[1] + 5 * x[0] * x[1] - x[0] * x[0]
funcs1: Funcs = @[f1, f2]
jacobian1: Jacobian = @[@[Func((x: Vector) => - 2 * x[0] + 1),
Func((x: Vector) => -1.0)],
@[Func((x: Vector) => 5 * x[1] - 2 * x[0]),
Func((x: Vector) => 1 + 5 * x[0])]]
guesses1 = @[1.2, 1.2]
sol1 = solve(funcs1, jacobian1, guesses1)
echo &"Approximate solutions are x = {sol1[0]:.7f}, y = {sol1[1]:.7f}"
#[ Solve the three non-linear equations:
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 - 2y^2 - 20z = 0
x^2 - y^2 + z^2 = 0
given initial guesses of x = y = 1.0 and z = 0.0
Example taken from:
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
]#
echo()
let
f3: Func = (x: Vector) => 9 * x[0] * x[0] + 36 * x[1] * x[1] + 4 * x[2] * x[2] - 36
f4: Func = (x: Vector) => x[0] * x[0] - 2 * x[1] * x[1] - 20 * x[2]
f5: Func = (x: Vector) => x[0] * x[0] - x[1] * x[1] + x[2] * x[2]
funcs2: Funcs = @[f3, f4, f5]
jacobian2: Jacobian = @[@[Func((x: Vector) => 18 * x[0]),
Func((x: Vector) => 72 * x[1]),
Func((x: Vector) => 8 * x[2])],
@[Func((x: Vector) => 2 * x[0]),
Func((x: Vector) => -4 * x[1]),
Func((x: Vector) => -20.0)],
@[Func((x: Vector) => 2 * x[0]),
Func((x: Vector) => -2 * x[1]),
Func((x: Vector) => 2 * x[2])]]
guesses2 = @[1.0, 1.0, 0.0]
sol2 = solve(funcs2, jacobian2, guesses2)
echo &"Approximate solutions are x = {sol2[0]:.7f}, y = {sol2[1]:.7f}, z = {sol2[2]:.7f}"</syntaxhighlight>
{{out}}
<pre>Approximate solutions are x = 1.2333178, y = 0.2122450
Approximate solutions are x = 0.8936282, y = 0.8945270, z = -0.0400893</pre>
=={{header|Phix}}==
{{trans|Go}}
Uses code from [[Reduced_row_echelon_form#Phix]],
[[Gauss-Jordan_matrix_inversion#Phix]],
[[Matrix_transposition#Phix]], and
[[Matrix_multiplication#Phix]]<br>
See std distro for a complete runnable version.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #000080;font-style:italic;">-- demo\rosetta\Multidimensional_Newton-Raphson_method.exw</span>
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">size</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">maxIter</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">12</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">iter</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">gu1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">g</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">g1</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">gu2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">jac</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">size</span><span style="color: #0000FF;">),</span><span style="color: #000000;">size</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">tol</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1e-8</span>
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">gu1</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">gu2</span>
<span style="color: #000000;">g</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">matrix_transpose</span><span style="color: #0000FF;">({</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">size</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">t</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">call_func</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],{</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">f</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">matrix_transpose</span><span style="color: #0000FF;">({</span><span style="color: #000000;">t</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">jac</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">call_func</span><span style="color: #0000FF;">(</span><span style="color: #000000;">jacob</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],{</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">g1</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g</span><span style="color: #0000FF;">,</span><span style="color: #000000;">matrix_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">inverse</span><span style="color: #0000FF;">(</span><span style="color: #000000;">jac</span><span style="color: #0000FF;">),</span><span style="color: #000000;">f</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">gu2</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">vslice</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">iter</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">any</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #004600;">true</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sq_gt</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">gu2</span><span style="color: #0000FF;">),</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">),</span><span style="color: #000000;">tol</span><span style="color: #0000FF;">))!=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">any</span> <span style="color: #008080;">or</span> <span style="color: #000000;">iter</span> <span style="color: #0000FF;">>=</span> <span style="color: #000000;">maxIter</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">gu2</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f1</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">-</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">36</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span><span style="color: #0000FF;">-</span><span style="color: #000000;">36</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f4</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">20</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f5</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">y</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">z</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j1</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000080;font-style:italic;">/*v*/</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j4</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j11</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">18</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j12</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">72</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j13</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,?,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j21</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j22</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j23</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000080;font-style:italic;">/*v*/</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">20</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j31</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j32</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j33</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,?,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span>
<span style="color: #000080;font-style:italic;">/*
solve the two non-linear equations:
y = -x^2 + x + 0.5
y + 5xy = x^2
given initial guesses of x = y = 1.2
Example taken from:
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
Expected results: x = 1.23332, y = 0.2122
*/</span>
<span style="color: #000000;">fs</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">f1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f2</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">jacob</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">j1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j2</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">j3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j4</span><span style="color: #0000FF;">}}</span>
<span style="color: #000000;">guesses</span> <span style="color: #0000FF;">:=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1.2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1.2</span><span style="color: #0000FF;">}</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Approximate solutions are x = %.7f, y = %.7f\n\n"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">))</span>
<span style="color: #000080;font-style:italic;">/*
solve the three non-linear equations:
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 - 2y^2 - 20z = 0
x^2 - y^2 + z^2 = 0
given initial guesses of x = y = 1.0 and z = 0.0
Example taken from:
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
*/</span>
<span style="color: #000000;">fs</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">f3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f5</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">jacob</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">j11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j13</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">j21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j22</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j23</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">j31</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j32</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j33</span><span style="color: #0000FF;">}}</span>
<span style="color: #000000;">guesses</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Approximate solutions are x = %.7f, y = %.7f, z = %.7f\n"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Approximate solutions are x = 1.2333178, y = 0.2122450
Approximate solutions are x = 0.8936282, y = 0.8945270, z = -0.04008929
</pre>
=={{header|Raku}}==
(formerly Perl 6)
<syntaxhighlight lang="raku" line># Reference:
# https://github.com/pierre-vigier/Perl6-Math-Matrix
# Mastering Algorithms with Perl
# By Jarkko Hietaniemi, John Macdonald, Jon Orwant
# Publisher: O'Reilly Media, ISBN-10: 1565923987
# https://resources.oreilly.com/examples/9781565923980/blob/master/ch16/solve
use v6;
sub solve_funcs ($funcs, @guesses, $iterations, $epsilon) {
my ($error_value, @values, @delta, @jacobian); my \ε = $epsilon;
for 1 .. $iterations {
for ^+$funcs { @values[$^i] = $funcs[$^i](|@guesses); }
$error_value = 0;
for ^+$funcs { $error_value += @values[$^i].abs }
return @guesses if $error_value ≤ ε;
for ^+$funcs { @delta[$^i] = -@values[$^i] }
@jacobian = jacobian $funcs, @guesses, ε;
@delta = solve_matrix @jacobian, @delta;
loop (my $j = 0, $error_value = 0; $j < +$funcs; $j++) {
$error_value += @delta[$j].abs ;
@guesses[$j] += @delta[$j];
}
return @guesses if $error_value ≤ ε;
}
return @guesses;
}
sub jacobian ($funcs is copy, @points is copy, $epsilon is copy) {
my ($Δ, @P, @M, @plusΔ, @minusΔ);
my Array @jacobian; my \ε = $epsilon;
for ^+@points -> $i {
@plusΔ = @minusΔ = @points;
$Δ = (ε * @points[$i].abs) || ε;
@plusΔ[$i] = @points[$i] + $Δ;
@minusΔ[$i] = @points[$i] - $Δ;
for ^+$funcs { @P[$^k] = $funcs[$^k](|@plusΔ); }
for ^+$funcs { @M[$^k] = $funcs[$^k](|@minusΔ); }
for ^+$funcs -> $j {
@jacobian[$j][$i] = (@P[$j] - @M[$j]) / (2 * $Δ);
}
}
return @jacobian;
}
sub solve_matrix (@matrix_array is copy, @delta is copy) {
# https://github.com/pierre-vigier/Perl6-Math-Matrix/issues/56
{ use Math::Matrix;
my $matrix = Math::Matrix.new(@matrix_array);
my $vector = Math::Matrix.new(@delta.map({.list}));
die "Matrix is not invertible" unless $matrix.is-invertible;
my @result = ( $matrix.inverted dot $vector ).transposed;
return @result.split(" ");
}
}
my $funcs = [
{ 9*$^x² + 36*$^y² + 4*$^z² - 36 }
{ $^x² - 2*$^y² - 20*$^z }
{ $^x² - $^y² + $^z² }
];
my @guesses = (1,1,0);
my @solution = solve_funcs $funcs, @guesses, 20, 1e-8;
say "Solution: ", @solution;
</syntaxhighlight>
{{out}}
<pre>
Solution: [0.8936282344764825 0.8945270103905782 -0.04008928615915281]
</pre>
=={{header|Wren}}==
{{trans|Kotlin}}
{{libheader|Wren-matrix}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./matrix" for Matrix
import "./fmt" for Fmt
var solve = Fn.new { |funcs, jacobian, guesses|
var size = funcs.count
var gu1 = []
var gu2 = guesses.toList
var jac = Matrix.new(size, size)
var tol = 1e-8
var maxIter = 12
var iter = 0
while (true) {
gu1 = gu2
var g = Matrix.new([gu1]).transpose
var v = List.filled(size, 0)
for (i in 0...size) v[i] = funcs[i].call(gu1)
var f = Matrix.new([v]).transpose
for (i in 0...size) {
for (j in 0...size) jac[i, j] = jacobian[i][j].call(gu1)
}
var g1 = g - jac.inverse * f
gu2 = List.filled(size, 0)
for (i in 0...size) gu2[i] = g1[i][0]
iter = iter + 1
if (iter == maxIter) break
if ((0...size).all { |i| (gu2[i] - gu1[i]).abs <= tol }) break
}
return gu2
}
/* solve the two non-linear equations:
y = -x^2 + x + 0.5
y + 5xy = x^2
given initial guesses of x = y = 1.2
Example taken from:
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
Expected results: x = 1.23332, y = 0.2122
*/
var f1 = Fn.new { |x| -x[0] * x[0] + x[0] + 0.5 - x[1] }
var f2 = Fn.new { |x| x[1] + 5 * x[0] * x[1] - x[0] * x[0] }
var funcs = [f1, f2]
var jacobian = [
[ Fn.new { |x| - 2 * x[0] + 1 }, Fn.new { |x| -1 } ],
[ Fn.new { |x| 5 * x[1] - 2 * x[0] }, Fn.new { |x| 1 + 5 * x[0] } ]
]
var guesses = [1.2, 1.2]
var sols = solve.call(funcs, jacobian, guesses)
Fmt.print("Approximate solutions are x = $.7f, y = $.7f", sols[0], sols[1])
/* solve the three non-linear equations:
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 - 2y^2 - 20z = 0
x^2 - y^2 + z^2 = 0
given initial guesses of x = y = 1.0 and z = 0.0
Example taken from:
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
*/
System.print()
var f3 = Fn.new { |x| 9 * x[0] * x[0] + 36 * x[1] *x[1] + 4 * x[2] * x[2] - 36 }
var f4 = Fn.new { |x| x[0] * x[0] - 2 * x[1] * x[1] - 20 * x[2] }
var f5 = Fn.new { |x| x[0] * x[0] - x[1] * x[1] + x[2] * x[2] }
funcs = [f3, f4, f5]
jacobian = [
[ Fn.new { |x| 18 * x[0] }, Fn.new { |x| 72 * x[1] }, Fn.new { |x| 8 * x[2] } ],
[ Fn.new { |x| 2 * x[0] }, Fn.new { |x| -4 * x[1] }, Fn.new { |x| -20 } ],
[ Fn.new { |x| 2 * x[0] }, Fn.new { |x| -2 * x[1] }, Fn.new { |x| 2 * x[2] } ]
]
guesses = [1, 1, 0]
sols = solve.call(funcs, jacobian, guesses)
Fmt.print("Approximate solutions are x = $.7f, y = $.7f, z = $.7f", sols[0], sols[1], sols[2])</syntaxhighlight>
{{out}}
<pre>
Approximate solutions are x = 1.2333178, y = 0.2122450
Approximate solutions are x = 0.8936282, y = 0.8945270, z = -0.0400893
</pre>
=={{header|zkl}}==
This doesn't use Newton-Raphson (with derivatives) but a hybrid algorithm.
<syntaxhighlight lang="zkl">var [const] GSL=Import.lib("zklGSL"); // libGSL (GNU Scientific Library)
// two functions of two variables: f(x,y)=0
fs:=T(fcn(x,y){ x.atan() - y*y*y }, fcn(x,y){ 4.0*x*x + 9*y*y - 36 });
v=GSL.VectorFromData(2.75, 1.25); // an initial guess at the solution
GSL.multiroot_fsolver(fs,v);
v.format(11,8).println(); // answer overwrites initial guess
fs.run(True,v.toList().xplode()).println(); // deltas from zero</syntaxhighlight>
{{out}}
<pre>
2.59807621, 1.06365371
L(2.13651e-09,2.94321e-10)
</pre>
A condensed solver (for a different set of functions):
<syntaxhighlight lang="zkl">v:=GSL.VectorFromData(-10.0, -15.0);
GSL.multiroot_fsolver(T( fcn(x,y){ 1.0 - x }, fcn(x,y){ 10.0*(y - x*x) }),v)
.format().println(); // --> (1,1)</syntaxhighlight>
{{out}}
<pre>
1.00,1.00
</pre>
Another example:
<syntaxhighlight lang="zkl">v:=GSL.VectorFromData(1.0, 1.0, 0.0); // initial guess
fxyzs:=T(
fcn(x,y,z){ x*x*9 + y*y*36 + z*z*4 - 36 }, // 9x^2 + 36y^2 + 4z^2 - 36 = 0
fcn(x,y,z){ x*x - y*y*2 - z*20 }, // x^2 - 2y^2 - 20z = 0
fcn(x,y,z){ x*x - y*y + z*z }); // x^2 - y^2 + z^2 = 0
(v=GSL.multiroot_fsolver(fxyzs,v)).format(12,8).println();
fxyzs.run(True,v.toList().xplode()).println(); // deltas from zero</syntaxhighlight>
{{out}}
<pre>
0.89362824, 0.89452701, -0.04008929
L(6.00672e-08,1.0472e-08,9.84017e-09)
</pre>
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