# Multidimensional Newton-Raphson method

Multidimensional Newton-Raphson method is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Create a program that finds and outputs the root of a system of nonlinear equations using Newton-Raphson metod.

## C#

For matrix inversion and matrix and vector definitions - see C# source from Gaussian elimination <lang csharp> using System;

namespace Rosetta {

```   internal interface IFun
{
double F(int index, Vector x);
double df(int index, int derivative, Vector x);
double[] weights();
}
```
```   class Newton
{
internal Vector Do(int size, IFun fun, Vector start)
{
Vector X = start.Clone();
Vector F = new Vector(size);
Matrix J = new Matrix(size, size);
Vector D;
do
{
for (int i = 0; i < size; i++)
F[i] = fun.F(i, X);
for (int i = 0; i < size; i++)
for (int j = 0; j < size; j++)
J[i, j] = fun.df(i, j, X);
J.ElimPartial(F);
X -= F;
//need weight vector because different coordinates can diffs by order of magnitudes
} while (F.norm(fun.weights()) > 1e-12);
return X;
}
}
```

} </lang> <lang csharp> using System;

//example from https://eti.pg.edu.pl/documents/176593/26763380/Wykl_AlgorOblicz_7.pdf namespace Rosetta {

```   class Program
{
class Fun: IFun
{
private double[] w = new double[] { 1,1 };
```
```           public double F(int index, Vector x)
{
switch (index)
{
case 0: return Math.Atan(x[0]) - x[1] * x[1] * x[1];
case 1: return 4 * x[0] * x[0] + 9 * x[1] * x[1] - 36;
}
}
```
```           public double df(int index, int derivative, Vector x)
{
switch (index)
{
case 0:
switch (derivative)
{
case 0: return 1 / (1 + x[0] * x[0]);
case 1: return -3*x[1]*x[1];
}
break;
case 1:
switch (derivative)
{
case 0: return 8 * x[0];
case 1: return 18 * x[1];
}
break;
}
}
public double[] weights() { return w; }
}
```
```       static void Main(string[] args)
{
Fun fun = new Fun();
Newton newton = new Newton();
Vector start = new Vector(new double[] { 2.75, 1.25 });
Vector X = newton.Do(2, fun, start);
X.print();
}
}
```

} </lang>

Output:
```
2.54258545959024
1.06149981539336

```

## Go

Translation of: Kotlin

We follow the Kotlin example of coding our own matrix methods rather than using a third party library. <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++ {
return
}
i := r
```
```       for m[i][lead] == 0 {
i++
if rowCount == i {
i = r
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 {
for j := 0; j < colCount; j++ {
m[k][j] -= m[r][j] * mult
}
}
}
}
```

}

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:
```
```      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])
```

}</lang>

Output:
```Approximate solutions are x = 1.2333178,  y = 0.2122450

Approximate solutions are x = 0.8936282,  y = 0.8945270,  z = -0.0400893
```

## 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. <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) {
var i = r
```
```       while (this[i][lead] == 0.0) {
i++
if (rowCount == i) {
i = r
}
}
```
```       val temp = this[i]
this[i] = this[r]
this[r] = temp
```
```       if (this[r][lead] != 0.0) {
for (j in 0 until colCount) this[r][j] /= div
}
```
```       for (k in 0 until rowCount) {
if (k != r) {
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:
```
```      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)
```

}</lang>

Output:
```Approximate solutions are x = 1.2333178,  y = 0.2122450

Approximate solutions are x = 0.8936282,  y = 0.8945270,  z = -0.0400893
```

## zkl

This doesn't use Newton-Raphson (with derivatives) but a hybrid algorithm. <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</lang>

Output:
``` 2.59807621, 1.06365371
L(2.13651e-09,2.94321e-10)
```

A condensed solver (for a different set of functions): <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)</lang>

Output:
```1.00,1.00
```

Another example: <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</lang>

Output:
```  0.89362824,  0.89452701, -0.04008929
L(6.00672e-08,1.0472e-08,9.84017e-09)
```