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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 method.

C#

For matrix inversion and matrix and vector definitions - see C# source from Gaussian elimination

` 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;        }           }} `
` using System; //example from https://eti.pg.edu.pl/documents/176593/26763380/Wykl_AlgorOblicz_7.pdfnamespace 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;                }                throw new Exception("bad index");            }             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;                }                throw new Exception("bad index");            }            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();        }    }} `
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.

`package main import (    "fmt"    "math") type vector = []float64type matrix []vectortype fun = func(vector) float64type funs = []funtype 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 placefunc (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])}`
Output:
```Approximate solutions are x = 1.2333178,  y = 0.2122450

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

Julia

NLsolve is a Julia package for nonlinear systems of equations, with the Newton-Raphson method one of the choices for solvers.

`# 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] = uend println(nlsolve(f!, j!, [ 0.1; 1.2], method = :newton)) `
Output:
```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
```

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.

`// Version 1.2.31 import kotlin.math.abs typealias Vector = DoubleArraytypealias Matrix = Array<Vector>typealias Func = (Vector) -> Doubletypealias 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)}`
Output:
```Approximate solutions are x = 1.2333178,  y = 0.2122450

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

Phix

Translation of: Go

Uses code from Reduced_row_echelon_form#Phix, Gauss-Jordan_matrix_inversion#Phix, Matrix_transposition#Phix, and Matrix_multiplication#Phix
See std distro for a complete runnable version.

`-- demo\rosetta\Multidimensional_Newton-Raphson_method.exwfunction solve(sequence fs, jacob, guesses)    integer size := length(fs),            maxIter := 12,            iter := 0    sequence gu1, g, t, f, g1,             gu2 := guesses,             jac := repeat(repeat(0,size),size)    atom tol := 1e-8    while true do        gu1 := gu2        g := matrix_transpose({gu1})        t := repeat(0, size)        for i=1 to size do            t[i] := call_func(fs[i],{gu1})        end for        f := matrix_transpose({t})        for i=1 to size do            for j=1 to size do                jac[i][j] := call_func(jacob[i][j],{gu1})            end for        end for        g1 := sq_sub(g,matrix_mul(inverse(jac),f))        gu2 := vslice(g1,1)        iter += 1        bool any := find(true,sq_gt(sq_sub(sq_abs(gu2),gu1),tol))!=0        if not any or iter >= maxIter then exit end if    end while    return gu2end function function f1(sequence v) atom {x,y} = v return -x*x+x+0.5-y end functionfunction f2(sequence v) atom {x,y} = v return y+5*x*y-x*x end functionfunction f3(sequence v) atom {x,y,z} = v return 9*x*x+36*y*y+4*z*z-36 end functionfunction f4(sequence v) atom {x,y,z} = v return x*x-2*y*y-20*z end functionfunction f5(sequence v) atom {x,y,z} = v return x*x-y*y+z*z end function function j1(sequence v) atom {x} = v return -2*x+1 end functionfunction j2(sequence /*v*/) return -1 end functionfunction j3(sequence v) atom {x,y} = v return 5*y-2*x end functionfunction j4(sequence v) atom {x} = v return 1+5*x end functionfunction j11(sequence v) atom {x} = v return 18*x end functionfunction j12(sequence v) atom {?,y} = v return 72*y end functionfunction j13(sequence v) atom {?,?,z} = v return 8*z end functionfunction j21(sequence v) atom {x} = v return 2*x end functionfunction j22(sequence v) atom {?,y} = v return -4*y end functionfunction j23(sequence /*v*/) return -20 end functionfunction j31(sequence v) atom {x} = v return 2*x end functionfunction j32(sequence v) atom {?,y} = v return -2*y end functionfunction j33(sequence v) atom {?,?,z} = v return 2*z end function procedure main()sequence fs, jacob, guesses    /*       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    */    fs = {routine_id("f1"),routine_id("f2")}    jacob = {{routine_id("j1"),routine_id("j2")},             {routine_id("j3"),routine_id("j4")}}    guesses := {1.2, 1.2}    printf(1,"Approximate solutions are x = %.7f,  y = %.7f\n\n", solve(fs, jacob, guesses))     /*       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    */     fs = {routine_id("f3"), routine_id("f4"), routine_id("f5")}    jacob = {{routine_id("j11"),routine_id("j12"),routine_id("j13")},             {routine_id("j21"),routine_id("j22"),routine_id("j23")},             {routine_id("j31"),routine_id("j32"),routine_id("j33")}}    guesses = {1, 1, 0}    printf(1,"Approximate solutions are x = %.7f,  y = %.7f,  z = %.7f\n", solve(fs, jacob, guesses)) end proceduremain()`
Output:
```Approximate solutions are x = 1.2333178,  y = 0.2122450

Approximate solutions are x = 0.8936282,  y = 0.8945270,  z = -0.04008929
```

Raku

(formerly Perl 6)

`# 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] = [email protected]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; `
Output:
```Solution: [0.8936282344764825 0.8945270103905782 -0.04008928615915281]
```

zkl

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

`var [const] GSL=Import.lib("zklGSL");    // libGSL (GNU Scientific Library)    // two functions of two variables: f(x,y)=0fs:=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 solutionGSL.multiroot_fsolver(fs,v);v.format(11,8).println();		// answer overwrites initial guess fs.run(True,v.toList().xplode()).println();	// deltas from zero`
Output:
``` 2.59807621, 1.06365371
L(2.13651e-09,2.94321e-10)
```

A condensed solver (for a different set of functions):

`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)`
Output:
```1.00,1.00
```

Another example:

`v:=GSL.VectorFromData(1.0, 1.0, 0.0);	// initial guessfxyzs:=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`
Output:
```  0.89362824,  0.89452701, -0.04008929
L(6.00672e-08,1.0472e-08,9.84017e-09)
```