I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

Gradient descent 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.

Gradient descent (also known as steepest descent) is a first-order iterative optimization algorithm for finding the minimum of a function which is described in this Wikipedia article.

Use this algorithm to search for minimum values of the bi-variate function:

```  f(x, y) = (x - 1)(x - 1)e^(-y^2) + y(y+2)e^(-2x^2)
```

around x = 0.1 and y = -1.

This book excerpt shows sample C# code for solving this task.

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
Translation of: Go
modified to use the actual gradient function -
Translation of: Fortran

THe results agree with the Fortran sample and the Julia sample to 6 places.

`PROC steepest descent = ( REF[]LONG REAL x, LONG REAL alphap, tolerance )VOID:BEGIN    LONG REAL alpha := alphap;    LONG REAL g0    := g( x ); # Initial estimate of result. #    # Calculate initial gradient. #    [ LWB x : UPB x ]LONG REAL fi  := grad g( x );    # Calculate initial norm. #    LONG REAL del g := 0.0;    FOR i FROM LWB x TO UPB x DO        del g +:= fi[ i ] * fi[ i ]    OD;    del g       := long sqrt( del g );    LONG REAL b := alpha / del g;    # Iterate until value is <= tolerance. #    WHILE del g > tolerance DO        # Calculate next value. #        FOR i FROM LWB x TO UPB x DO            x[i] -:= b * fi[i]        OD;        # Calculate next gradient. #        fi := grad g( x );        # Calculate next norm. #        del g := 0;        FOR i FROM LWB x TO UPB x DO            del g +:= fi[ i ] * fi[ i ]        OD;        del g := long sqrt( del g );        IF del g > 0 THEN            b     := alpha / del g;             # Calculate next value. #            LONG REAL g1 := g( x );            # Adjust parameter. #            IF g1 > g0 THEN                alpha /:= 2            ELSE                g0 := g1            FI        FI    ODEND # steepest descent # ;# calculates the gradient of g(p).                                                     ## The derivitives wrt x and y are (as in the Fortran sample ):                         ## g' wrt x = 2( x - 1 )e^( - ( y^2 ) ) - 4xe^( -2( x^2) )y( y + 2 )                    # # g' wrt y = ( -2( x-1 )^2ye^( - (y^2) )) + e^(-2( x^2 ) )( y + 2 ) + e^( -2( x^2 ) )y #PROC grad g = ( []LONG REAL p )[]LONG REAL:BEGIN    [ LWB p : UPB p ]LONG REAL z;    LONG REAL x = p[ 0 ];    LONG REAL y = p[ 1 ];    z[ 0 ] := 2 * ( x - 1 ) * long exp( - ( y * y ) )            - 4 * x * long exp( -2 * ( x * x ) ) * y * ( y + 2 );    z[ 1 ] := ( -2 * ( x - 1 ) * ( x - 1 ) * y ) * long exp( - y * y )            + long exp( -2 * x * x ) * ( y + 2 )            + long exp( -2 * x * x ) * y;    zEND # grad g # ;# Function for which minimum is to be found. ## g( x, y ) = ( ( x - 1 )^2 )e^( - ( x^2 ) ) + y( y + 2 )e^( - 2(x^2)) #PROC g = ( []LONG REAL x )LONG REAL:    ( x[ 0 ] - 1 )  * ( x[ 0 ] - 1 )  * long exp( - x[ 1 ] * x[ 1 ] ) + x[ 1 ] * ( x[ 1 ] + 2 )  * long exp( - 2 * x[ 0 ] * x[ 0 ] )  ;BEGIN    LONG REAL          tolerance := 0.0000006;    LONG REAL          alpha     := 0.1;    [ 0 : 1 ]LONG REAL x         := ( []LONG REAL( 0.1, -1 ) )[ AT 0 ]; # Initial guess of location of minimum. #    steepest descent( x, alpha, tolerance );    print( ( "Testing steepest descent method:", newline ) );    print( ( "The minimum is at x[0] = ", fixed( x[ 0 ], -10, 6 ), ", x[1] = ", fixed( x[ 1 ], -10, 6 ), newline ) )END`
Output:
```Testing steepest descent method:
The minimum is at x[0] =   0.107627, x[1] =  -1.223260
```

## ALGOL W

Translation of: ALGOL 68
which is a
Translation of: Go
with the gradient function from
Translation of: Fortran

The results agree (to 6 places) with the Fortran and Julia samples.

`begin    procedure steepestDescent ( long real array x ( * ); long real value alphap, tolerance ) ;    begin        long real array fi ( 0 :: 1 );        long real alpha, g0, g1, delG, b;        alpha := alphap;        g0    := g( x ); % Initial estimate of result. %        % Calculate initial gradient. %        gradG( fi, x );        % Calculate initial norm. %        delG  := 0.0;        for i := 0 until 1 do delG := delG + ( fi( i ) * fi( i ) );        delG  := longSqrt( delG );        b     := alpha / delG;        % Iterate until value is <= tolerance. %        while delG > tolerance do begin            % Calculate next value. %            for i := 0 until 1 do x(i) := x( i ) - ( b * fi(i) );            % Calculate next gradient. %            gradG( fi, x );            % Calculate next norm. %            delG  := 0;            for i := 0 until 1 do delG := delg + ( fi( i ) * fi( i ) );            delG  := longSqrt( delG );            if delG > 0 then begin                b     := alpha / delG;                 % Calculate next value. %                g1 := g( x );                % Adjust parameter. %                if g1 > g0                then alpha := alpha / 2                else g0 := g1            end if_delG_gt_0        end while_delG_gt_tolerance    end steepestDescent ;    % Provides a rough calculation of gradient g(x). %    procedure gradG ( long real array z, p ( * ) ) ;    begin        long real x, y;        x := p( 0 );        y := p( 1 );        z( 0 ) := 2 * ( x - 1 ) * longExp( - ( y * y ) )                - 4 * x * longExp( -2 * ( x * x ) ) * y * ( y + 2 );        z( 1 ) := ( -2 * ( x - 1 ) * ( x - 1 ) * y ) * longExp( - y * y )                + longExp( -2 * x * x ) * ( y + 2 )                + longExp( -2 * x * x ) * y    end gradG ;    % Function for which minimum is to be found. %    long real procedure g ( long real array x ( * ) ) ;          ( x( 0 ) - 1 ) * ( x( 0 ) - 1 ) * longExp( - x( 1 ) * x( 1 ) )        + x( 1 ) * ( x( 1 ) + 2 ) * longExp( - 2 * x( 0 ) * x( 0 ) )        ;    begin        long real alpha, tolerance;        long real array x ( 0 :: 1 );        x( 0 ) :=  0.1; % Initial guess of location of minimum. %        x( 1 ) := -1;               tolerance := 0.0000006;        alpha     := 0.1;        steepestDescent( x, alpha, tolerance );        r_format := "A"; r_w := 11; r_d := 7; s_w := 0; % output formatting %        write( "Testing steepest descent method:" );        write( "The minimum is at x(0) = ", x( 0 ), ", x(1) = ", x( 1 ) )    endend.`
Output:
```Testing steepest descent method:
The minimum is at x(0) =   0.1076268, x(1) =  -1.2232596
```

## Fortran

Compiler: gfortran 8.3.0
The way a FORTRAN programmer would do this would be to automatically differentiate the function using the diff command in Maxima:

```(%i3) (x-1)*(x-1)*exp(-y^2)+y*(y+2)*exp(-2*x^2);
2          2
2   - y      - 2 x
(%o3)                (x - 1)  %e     + %e       y (y + 2)
(%i4) diff(%o3,x);
2              2
- y          - 2 x
(%o4)              2 (x - 1) %e     - 4 x %e       y (y + 2)
(%i5) diff(%o3,y);
2           2                  2
2     - y       - 2 x              - 2 x
(%o5)       (- 2 (x - 1)  y %e    ) + %e       (y + 2) + %e       y
```

and then have it automatically turned into statements with the fortran command:

```(%i6) fortran(%o4);
2*(x-1)*exp(-y**2)-4*x*exp(-2*x**2)*y*(y+2)
(%o6)                                done
(%i7) fortran(%o5);
(-2*(x-1)**2*y*exp(-y**2))+exp(-2*x**2)*(y+2)+exp(-2*x**2)*y
(%o7)                                done
```

The optimization subroutine GD sets the reverse communication variable IFLAG. This allows the evaluation of the gradient to be done separately.

`      SUBROUTINE EVALFG (N, X, F, G)       IMPLICIT NONE       INTEGER N       DOUBLE PRECISION X(N), F, G(N)       F = (X(1) - 1.D0)**2 * EXP(-X(2)**2) +      \$      X(2) * (X(2) + 2.D0) * EXP(-2.D0 * X(1)**2)       G(1) = 2.D0 * (X(1) - 1.D0) * EXP(-X(2)**2) - 4.D0 * X(1) *      \$        EXP(-2.D0 * X(1)**2) * X(2) * (X(2) + 2.D0)       G(2) = (-2.D0 * (X(1) - 1.D0)**2 * X(2) * EXP(-X(2)**2)) +      \$        EXP(-2.D0 * X(1)**2) * (X(2) + 2.D0) +      \$        EXP(-2.D0 * X(1)**2) * X(2)       RETURN      END *-----------------------------------------------------------------------* gd - Gradient descent* G must be set correctly at the initial point X.**___Name______Type________In/Out___Description_________________________*   N         Integer     In       Number of Variables.*   X(N)      Double      Both     Variables*   G(N)      Double      Both     Gradient*   TOL       Double      In       Relative convergence tolerance*   IFLAG     Integer     Out      Reverse Communication Flag*                                    on output:  0  done*                                                1  compute G and call again*-----------------------------------------------------------------------      SUBROUTINE GD (N, X, G, TOL, IFLAG)       IMPLICIT NONE       INTEGER N, IFLAG       DOUBLE PRECISION X(N), G(N), TOL       DOUBLE PRECISION ETA       PARAMETER (ETA = 0.3D0)     ! Learning rate       INTEGER I       DOUBLE PRECISION GNORM      ! norm of gradient        GNORM = 0.D0                ! convergence test       DO I = 1, N         GNORM = GNORM + G(I)**2       END DO       GNORM = SQRT(GNORM)       IF (GNORM < TOL) THEN         IFLAG = 0         RETURN                     ! success       END IF        DO I = 1, N                 ! take step         X(I) = X(I) - ETA * G(I)       END DO       IFLAG = 1       RETURN                      ! let main program evaluate G      END  ! of gd       PROGRAM GDDEMO       IMPLICIT NONE       INTEGER N       PARAMETER (N = 2)       INTEGER ITER, J, IFLAG       DOUBLE PRECISION X(N), F, G(N), TOL        X(1) = -0.1D0         ! initial values       X(2) = -1.0D0       TOL = 1.D-15       CALL EVALFG (N, X, F, G)       IFLAG = 0       DO J = 1, 1 000 000         CALL GD (N, X, G, TOL, IFLAG)         IF (IFLAG .EQ. 1) THEN           CALL EVALFG (N, X, F, G)         ELSE           ITER = J           GO TO 50         END IF       END DO       STOP 'too many iterations!'   50   PRINT '(A, I7, A, F19.15, A, F19.15, A, F19.15)',      \$          'After ', ITER, ' steps, found minimum at x=',      \$           X(1), ' y=', X(2), ' of f=', F       STOP 'program complete'      END `
Output:
```After      31 steps, found minimum at x=  0.107626843548372 y= -1.223259663839920 of f= -0.750063420551493
STOP program complete
```

## Go

This is a translation of the C# code in the book excerpt linked to above and hence also of the first Typescript example below.

However, since it was originally written, I've substituted Fortran's gradient function for the original one (see Talk page) which now gives results which agree (to 6 decimal places) with those of the Fortran, Julia, Algol 68 and Algol W solutions. As a number of other solutions are based on this one, I suggest their authors update them accordingly.

`package main import (    "fmt"    "math") func steepestDescent(x []float64, alpha, tolerance float64) {    n := len(x)    g0 := g(x) // Initial estimate of result.     // Calculate initial gradient.    fi := gradG(x)     // Calculate initial norm.    delG := 0.0    for i := 0; i < n; i++ {        delG += fi[i] * fi[i]    }    delG = math.Sqrt(delG)    b := alpha / delG     // Iterate until value is <= tolerance.    for delG > tolerance {        // Calculate next value.        for i := 0; i < n; i++ {            x[i] -= b * fi[i]        }         // Calculate next gradient.        fi = gradG(x)         // Calculate next norm.        delG = 0        for i := 0; i < n; i++ {            delG += fi[i] * fi[i]        }        delG = math.Sqrt(delG)        b = alpha / delG         // Calculate next value.        g1 := g(x)         // Adjust parameter.        if g1 > g0 {            alpha /= 2        } else {            g0 = g1        }    }} // Provides a rough calculation of gradient g(p).func gradG(p []float64) []float64 {    z := make([]float64, len(p))    x := p[0]    y := p[1]    z[0] = 2*(x-1)*math.Exp(-y*y) - 4*x*math.Exp(-2*x*x)*y*(y+2)    z[1] = -2*(x-1)*(x-1)*y*math.Exp(-y*y) + math.Exp(-2*x*x)*(y+2) + math.Exp(-2*x*x)*y    return z} // Function for which minimum is to be found.func g(x []float64) float64 {    return (x[0]-1)*(x[0]-1)*        math.Exp(-x[1]*x[1]) + x[1]*(x[1]+2)*        math.Exp(-2*x[0]*x[0])} func main() {    tolerance := 0.0000006    alpha := 0.1    x := []float64{0.1, -1} // Initial guess of location of minimum.     steepestDescent(x, alpha, tolerance)    fmt.Println("Testing steepest descent method:")    fmt.Printf("The minimum is at x = %f, y = %f for which f(x, y) = %f.\n", x[0], x[1], g(x))}`
Output:
```Testing steepest descent method:
The minimum is at x = 0.107627, y = -1.223260 for which f(x, y) = -0.750063.
```

## Julia

`using Optim, Base.MathConstants f(x) = (x[1] - 1) * (x[1] - 1) * e^(-x[2]^2) + x[2] * (x[2] + 2) * e^(-2 * x[1]^2) println(optimize(f, [0.1, -1.0], GradientDescent())) `
Output:
```Results of Optimization Algorithm
* Algorithm: Gradient Descent
* Starting Point: [0.1,-1.0]
* Minimizer: [0.107626844383003,-1.2232596628723371]
* Minimum: -7.500634e-01
* Iterations: 14
* Convergence: true
* |x - x'| ≤ 0.0e+00: false
|x - x'| = 2.97e-09
* |f(x) - f(x')| ≤ 0.0e+00 |f(x)|: true
|f(x) - f(x')| = 0.00e+00 |f(x)|
* |g(x)| ≤ 1.0e-08: true
|g(x)| = 2.54e-09
* Stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 35
* Gradient Calls: 35
```

## Perl

Calculate with `bignum` for numerical stability.

Translation of: Raku
`use strict;use warnings;use bignum; sub steepestDescent {    my(\$alpha, \$tolerance, @x) = @_;    my \$N = @x;    my \$h = \$tolerance;    my \$g0 = g(@x) ;    # Initial estimate of result.     my @fi = gradG(\$h, @x) ;    #  Calculate initial gradient     # Calculate initial norm.    my \$delG = 0;    for (0..\$N-1) { \$delG += \$fi[\$_]**2 }    my \$b = \$alpha / sqrt(\$delG);     while ( \$delG > \$tolerance ) {   # Iterate until value is <= tolerance.       #  Calculate next value.       for (0..\$N-1) { \$x[\$_] -= \$b * \$fi[\$_] }       \$h /= 2;        @fi = gradG(\$h, @x);    # Calculate next gradient.       # Calculate next norm.       \$delG = 0;       for (0..\$N-1) { \$delG += \$fi[\$_]**2 }       \$b = \$alpha / sqrt(\$delG);        my \$g1 = g(@x);   # Calculate next value.        \$g1 > \$g0 ? (\$alpha /= 2) : (\$g0 = \$g1);  # Adjust parameter.    }    @x} # Provides a rough calculation of gradient g(x).sub gradG {    my(\$h, @x) = @_;    my \$N = @x;    my @y = @x;    my \$g0 = g(@x);    my @z;    for (0..\$N-1) { \$y[\$_] += \$h ; \$z[\$_] = (g(@y) - \$g0) / \$h }    return @z} # Function for which minimum is to be found.sub g { my(@x) = @_; (\$x[0]-1)**2 * exp(-\$x[1]**2) + \$x[1]*(\$x[1]+2) * exp(-2*\$x[0]**2) }; my \$tolerance = 0.0000001;my \$alpha     = 0.01;my @x = <0.1 -1>; # Initial guess of location of minimum. printf "The minimum is at x[0] = %.6f, x[1] = %.6f", steepestDescent(\$alpha, \$tolerance, @x);`
Output:
`The minimum is at x[0] = 0.107653, x[1] = -1.223370`

## Phix

Translation of: Go
`-- Function for which minimum is to be found.function g(sequence x)    atom {x0,x1} = x    return (x0-1)*(x0-1)*exp(-x1*x1) +                x1*(x1+2)*exp(-2*x0*x0)end function -- Provides a rough calculation of gradient g(x).function gradG(sequence p)    atom {x,y} = p    p[1] = 2*(x-1)*exp(-y*y) - 4*x*exp(-2*x*x)*y*(y+2)    p[2] = -2*(x-1)*(x-1)*y*exp(-y*y) + exp(-2*x*x)*(y+2) + exp(-2*x*x)*y    return pend function function steepestDescent(sequence x, atom alpha, tolerance)    integer n = length(x)    atom g0 = g(x) -- Initial estimate of result.     -- Calculate initial gradient.    sequence fi = gradG(x)     -- Calculate initial norm.    atom delG = sqrt(sum(sq_mul(fi,fi))),         b = alpha / delG     -- Iterate until value is <= tolerance.    while delG>tolerance do        -- Calculate next value.        x = sq_sub(x,sq_mul(b,fi))         -- Calculate next gradient.        fi = gradG(x)         -- Calculate next norm.        delG = sqrt(sum(sq_mul(fi,fi)))        b = alpha / delG         -- Calculate next value.        atom g1 = g(x)         -- Adjust parameter.        if g1>g0 then            alpha /= 2        else            g0 = g1        end if    end while    return xend function constant tolerance = 0.0000001, alpha = 0.1sequence x = steepestDescent({0.1,-1}, alpha, tolerance)printf(1,"Testing steepest descent method:\n")printf(1,"The minimum is at x = %.13f, y = %.13f for which f(x, y) = %.16f\n", {x[1], x[2], g(x)})`
Output:

Results now match (at least) Algol 68/W, Fortran, Go, Julia, Raku, REXX, and Wren [to 6dp or better anyway].
Note that specifying a tolerance < 1e-7 causes an infinite loop on Phix, whereas REXX copes with a much smaller tolerance.
Results on 32/64 bit Phix agree to 13dp, which I therefore choose to show in full here (but otherwise would not really trust).

```Testing steepest descent method:
The minimum is at x = 0.1076268243295, y = -1.2232596548816 for which f(x, y) = -0.7500634205514924
```

## Racket

Translation of: Go

Note the different implementation of `grad`. I believe that the vector should be reset and only the partial derivative in a particular dimension is to be used. For this reason, I've _yet another_ result!

I could have used ∇ and Δ in the variable names, but it looked too confusing, so I've gone with grad- and del-

`#lang racket (define (apply-vector f v)  (apply f (vector->list v))) ;; Provides a rough calculation of gradient g(v).(define ((grad/del f) v δ #:fv (fv (apply-vector f v)))  (define dim (vector-length v))  (define tmp (vector-copy v))  (define grad (for/vector #:length dim ((i dim)                            (v_i v))              (vector-set! tmp i (+ v_i δ))              (define ∂f/∂v_i (/ (- (apply-vector f tmp) fv) δ))              (vector-set! tmp i v_i)              ∂f/∂v_i))  (values grad (sqrt (for/sum ((∂_i grad)) (sqr ∂_i))))) (define (steepest-descent g x α tolerance)  (define grad/del-g (grad/del g))   (define (loop x δ α gx grad-gx del-gx b)    (cond      [(<= del-gx tolerance) x]      [else        (define δ´ (/ δ 2))        (define x´ (vector-map + (vector-map (curry * (- b)) grad-gx) x))        (define gx´ (apply-vector g x´))        (define-values (grad-gx´ del-gx´) (grad/del-g x´ δ´ #:fv gx´))        (define b´ (/ α del-gx´))        (if (> gx´ gx)            (loop x´ δ´ (/ α 2) gx  grad-gx´ del-gx´ b´)            (loop x´ δ´ α       gx´ grad-gx´ del-gx´ b´))]))   (define gx (apply-vector g x))  (define δ tolerance)  (define-values (grad-gx del-gx) (grad/del-g x δ #:fv gx))  (loop x δ α gx grad-gx del-gx (/ α del-gx))) (define (Gradient-descent)  (steepest-descent    (λ (x y)       (+ (* (- x 1) (- x 1) (exp (- (sqr y))))        (* y (+ y 2) (exp (- (* 2 (sqr x)))))))    #(0.1 -1.) 0.1 0.0000006)) (module+ main  (Gradient-descent)) `
Output:
`'#(0.10760797905122492 -1.2232993981966753)`

## Raku

(formerly Perl 6)

Translation of: Go
`# 20200904 Updated Raku programming solution sub steepestDescent(@x, \$alpha is copy, \$h) {    my \$g0 = g |@x ; # Initial estimate of result.    my @fi = gradG |@x ; #  Calculate initial gradient    my \$b = \$alpha / my \$delG = sqrt ( sum @fi»² ) ;  # Calculate initial norm.    while ( \$delG > \$h ) {   # Iterate until value is <= tolerance.       @x «-»= \$b «*« @fi; #  Calculate next value.       @fi = gradG |@x ; # Calculate next gradient and next value       \$b = \$alpha / (\$delG = sqrt( sum @fi»² ));  # Calculate next norm.       my \$g1 = g |@x ;       \$g1 > \$g0 ?? ( \$alpha /= 2 ) !! ( \$g0 = \$g1 )   # Adjust parameter.   }} sub gradG(\x,\y) { # gives a rough calculation of gradient g(x).   2*(x-1)*exp(-y²) - 4*x*exp(-2*x²)*y*(y+2) , -2*(x-1)²*y*exp(-y²) + exp(-2*x²)*(2*y+2)} # Function for which minimum is to be found.sub g(\x,\y) { (x-1)² * exp(-y²) + y*(y+2) * exp(-2*x²) } my \$tolerance = 0.0000006 ; my \$alpha = 0.1; my @x = 0.1, -1; # Initial guess of location of minimum. steepestDescent(@x, \$alpha, \$tolerance); say "Testing steepest descent method:";say "The minimum is at x[0] = ", @x[0], ", x[1] = ", @x[1]; `
Output:
```Testing steepest descent method:
The minimum is at x[0] = 0.10762682432947938, x[1] = -1.2232596548816097
```

## REXX

The   tolerance   can be much smaller;   a tolerance of   1e-200   was tested.   It works, but causes the program to execute a bit slower, but still sub-second execution time.

`/*REXX pgm searches for minimum values of the bi─variate function (AKA steepest descent)*/numeric digits (length( e() ) - length(.) ) % 2  /*use half of number decimal digs in E.*/tolerance=  1e-30                                /*use a much smaller tolerance for REXX*/     alpha=  0.1      x.0=  0.1;     x.1= -1say center(' testing for the steepest descent method ', 79, "═")call steepestD                                   /* ┌──◄── # digs past dec. point ─►───┐*/say 'The minimum is at:     x[0]='      format(x.0,,9)    "     x[1]="     format(x.1,,9)exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/e:     return 2.718281828459045235360287471352662497757247093699959574966967627724g:     return (x.0-1)**2  *  exp(- (x.1**2) )    +    x.1 * (x.1 + 2)  *  exp(-2 * x.0**2)/*──────────────────────────────────────────────────────────────────────────────────────*/gradG: x= x.0;               y= x.1              /*define X and Y  from the  X  array.  */       xm= x-1;              eny2= exp(-y*y);   enx2= exp(-2 * x**2)        /*shortcuts.*/       z.0=  2 * xm        * eny2   -   4 * x * enx2 * y * (y+2)       z.1= -2 * xm**2 * y * eny2   +           enx2     * (y+2)   +   enx2 * y       return                                    /*a rough calculation of the gradient. *//*──────────────────────────────────────────────────────────────────────────────────────*/steepestD: g0= g()                               /*the initial estimate of the result.  */           call gradG                            /*calculate the initial gradient.      */           delG= sqrt(z.0**2  +  z.1**2)         /*    "      "     "    norm.          */           b= alpha / delG                             do while delG>tolerance                             x.0= x.0   -   b * z.0;               x.1= x.1   -   b * z.1                             call gradG                             delG= sqrt(z.0**2  +  z.1**2);        if delG=0  then return                             b= alpha / delG                             g1= g()                                     /*find minimum.*/                             if g1>g0  then alpha= alpha * .5            /*adjust ALPHA.*/                                       else    g0= g1                    /*   "   G0.   */                             end   /*while*/           return/*──────────────────────────────────────────────────────────────────────────────────────*/exp:  procedure; parse arg x; ix= x%1;  if abs(x-ix)>.5  then ix= ix + sign(x);  x= x - ix      z=1;  _=1;  w=z;   do j=1;  _= _*x/j;  z= (z+_)/1;  if z==w  then leave;  w= z;  end      if z\==0  then z= z * e() ** ix;                                          return z/1/*──────────────────────────────────────────────────────────────────────────────────────*/sqrt: procedure; parse arg x; if x=0  then return 0; d= digits();  numeric digits;  h= d+6      numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .; g=g *.5'e'_ %2        do j=0  while h>9;      m.j=h;                h= h % 2 + 1;   end  /*j*/        do k=j+5  to 0  by -1;  numeric digits m.k;   g=(g+x/g)*.5;   end  /*k*/; return g`
output   when using the internal default inputs:
```═══════════════════ testing for the steepest descent method ═══════════════════
The minimum is at:     x[0]= 0.107626824      x[1]= -1.223259655
```

## Scala

Translation of: Go
`object GradientDescent {   /** Steepest descent method modifying input values*/  def steepestDescent(x : Array[Double], learningRate : Double, tolerance : Double) = {    val n = x.size    var h = tolerance    var alpha = learningRate    var g0 = g(x) // Initial estimate of result.     // Calculate initial gradient.    var fi = gradG(x,h)     // Calculate initial norm.    var delG = 0.0    for (i <- 0 until n by 1)  delG += fi(i) * fi(i)    delG = math.sqrt(delG)    var b = alpha / delG     // Iterate until value is <= tolerance.    while(delG > tolerance){      // Calculate next value.      for (i <- 0 until n by 1) x(i) -= b * fi(i)      h /= 2       // Calculate next gradient.      fi = gradG(x,h)       // Calculate next norm.      delG = 0.0      for (i <- 0 until n by 1) delG += fi(i) * fi(i)      delG = math.sqrt(delG)      b = alpha / delG       // Calculate next value.      var g1 = g(x)       // Adjust parameter.      if(g1 > g0) alpha = alpha / 2      else g0 = g1    }   }   /** Gradient of the input function given in the task*/  def gradG(x : Array[Double], h : Double) : Array[Double] = {    val n = x.size    val z : Array[Double] = Array.fill(n){0}    val y = x    val g0 = g(x)     for(i <- 0 until n by 1){      y(i) += h      z(i) = (g(y) - g0) / h    }     z   }   /** Bivariate function given in the task*/  def g( x : Array[Double]) : Double = {    ( (x(0)-1) * (x(0)-1) * math.exp( -x(1)*x(1) ) + x(1) * (x(1)+2) * math.exp( -2*x(0)*x(0) ) )  }   def main(args: Array[String]): Unit = {    val tolerance = 0.0000006    val learningRate = 0.1    val x =  Array(0.1, -1) // Initial guess of location of minimum.     steepestDescent(x, learningRate, tolerance)    println("Testing steepest descent method")    println("The minimum is at x : " + x(0) + ", y : " + x(1))  }} `
Output:
```Testing steepest descent method
The minimum is at x : 0.10756393294495799, y : -1.2234116852966237
```

## TypeScript

Translation of
•   [Numerical Methods, Algorithms and Tools in C# by Waldemar Dos Passos (18.2 Gradient Descent Method]

` // Using the steepest-descent method to search// for minimum values of a multi-variable functionexport const steepestDescent = (x: number[], alpha: number, tolerance: number) => {     let n: number = x.length; // size of input array    let h: number = 0.0000006; //Tolerance factor    let g0: number = g(x); //Initial estimate of result     //Calculate initial gradient    let fi: number[] = [n];     //Calculate initial norm    fi = GradG(x, h);    // console.log("fi:"+fi);     //Calculate initial norm    let DelG: number = 0.0;     for (let i: number = 0; i < n; ++i) {        DelG += fi[i] * fi[i];    }    DelG = Math.sqrt(DelG);    let b: number = alpha / DelG;     //Iterate until value is <= tolerance limit    while (DelG > tolerance) {        //Calculate next value        for (let i = 0; i < n; ++i) {            x[i] -= b * fi[i];        }        h /= 2;         //Calculate next gradient        fi = GradG(x, h);        //Calculate next norm        DelG = 0;        for (let i: number = 0; i < n; ++i) {            DelG += fi[i] * fi[i];        }         DelG = Math.sqrt(DelG);        b = alpha / DelG;         //Calculate next value        let g1: number = g(x);         //Adjust parameter        if (g1 > g0) alpha /= 2;        else g0 = g1;    }} // Provides a rough calculation of gradient g(x).export const GradG = (x: number[], h: number) => {     let n: number = x.length;    let z: number[] = [n];    let y: number[] = x;    let g0: number = g(x);     // console.log("y:" + y);     for (let i = 0; i < n; ++i) {        y[i] += h;        z[i] = (g(y) - g0) / h;    }    // console.log("z:"+z);    return z;} // Method to provide function g(x).export const g = (x: number[]) => {    return (x[0] - 1) * (x[0] - 1)        * Math.exp(-x[1] * x[1]) + x[1] * (x[1] + 2)        * Math.exp(-2 * x[0] * x[0]);} export const gradientDescentMain = () => {    let tolerance: number = 0.0000006;    let alpha: number = 0.1;    let x: number[] = [2];     //Initial guesses    x[0] = 0.1;    //of location of minimums     x[1] = -1;    steepestDescent(x, alpha, tolerance);     console.log("Testing steepest descent method");    console.log("The minimum is at x[0] = " + x[0]        + ", x[1] = " + x[1]);    // console.log("");} gradientDescentMain();  `
Output:
```Testing steepest descent method
The minimum is at x[0] = 0.10768224291553158, x[1] = -1.2233090211217854
```

### Linear Regression

Translation of
•   [Linear Regression using Gradient Descent by Adarsh Menon]

` let data: number[][] =    [[32.5023452694530, 31.70700584656990],    [53.4268040332750, 68.77759598163890],    [61.5303580256364, 62.56238229794580],    [47.4756396347860, 71.54663223356770],    [59.8132078695123, 87.23092513368730],    [55.1421884139438, 78.21151827079920],    [52.2117966922140, 79.64197304980870],    [39.2995666943170, 59.17148932186950],    [48.1050416917682, 75.33124229706300],    [52.5500144427338, 71.30087988685030],    [45.4197301449737, 55.16567714595910],    [54.3516348812289, 82.47884675749790],    [44.1640494967733, 62.00892324572580],    [58.1684707168577, 75.39287042599490],    [56.7272080570966, 81.43619215887860],    [48.9558885660937, 60.72360244067390],    [44.6871962314809, 82.89250373145370],    [60.2973268513334, 97.37989686216600],    [45.6186437729558, 48.84715331735500],    [38.8168175374456, 56.87721318626850],    [66.1898166067526, 83.87856466460270],    [65.4160517451340, 118.59121730252200],    [47.4812086078678, 57.25181946226890],    [41.5756426174870, 51.39174407983230],    [51.8451869056394, 75.38065166531230],    [59.3708220110895, 74.76556403215130],    [57.3100034383480, 95.45505292257470],    [63.6155612514533, 95.22936601755530],    [46.7376194079769, 79.05240616956550],    [50.5567601485477, 83.43207142132370],    [52.2239960855530, 63.35879031749780],    [35.5678300477466, 41.41288530370050],    [42.4364769440556, 76.61734128007400],    [58.1645401101928, 96.76956642610810],    [57.5044476153417, 74.08413011660250],    [45.4405307253199, 66.58814441422850],    [61.8962226802912, 77.76848241779300],    [33.0938317361639, 50.71958891231200],    [36.4360095113868, 62.12457081807170],    [37.6756548608507, 60.81024664990220],    [44.5556083832753, 52.68298336638770],    [43.3182826318657, 58.56982471769280],    [50.0731456322890, 82.90598148507050],    [43.8706126452183, 61.42470980433910],    [62.9974807475530, 115.24415280079500],    [32.6690437634671, 45.57058882337600],    [40.1668990087037, 54.08405479622360],    [53.5750775316736, 87.99445275811040],    [33.8642149717782, 52.72549437590040],    [64.7071386661212, 93.57611869265820],    [38.1198240268228, 80.16627544737090],    [44.5025380646451, 65.10171157056030],    [40.5995383845523, 65.56230126040030],    [41.7206763563412, 65.28088692082280],    [51.0886346783367, 73.43464154632430],    [55.0780959049232, 71.13972785861890],    [41.3777265348952, 79.10282968354980],    [62.4946974272697, 86.52053844034710],    [49.2038875408260, 84.74269780782620],    [41.1026851873496, 59.35885024862490],    [41.1820161051698, 61.68403752483360],    [50.1863894948806, 69.84760415824910],    [52.3784462192362, 86.09829120577410],    [50.1354854862861, 59.10883926769960],    [33.6447060061917, 69.89968164362760],    [39.5579012229068, 44.86249071116430],    [56.1303888168754, 85.49806777884020],    [57.3620521332382, 95.53668684646720],    [60.2692143939979, 70.25193441977150],    [35.6780938894107, 52.72173496477490],    [31.5881169981328, 50.39267013507980],    [53.6609322616730, 63.64239877565770],    [46.6822286494719, 72.24725106866230],    [43.1078202191024, 57.81251297618140],    [70.3460756150493, 104.25710158543800],    [44.4928558808540, 86.64202031882200],    [57.5045333032684, 91.48677800011010],    [36.9300766091918, 55.23166088621280],    [55.8057333579427, 79.55043667850760],    [38.9547690733770, 44.84712424246760],    [56.9012147022470, 80.20752313968270],    [56.8689006613840, 83.14274979204340],    [34.3331247042160, 55.72348926054390],    [59.0497412146668, 77.63418251167780],    [57.7882239932306, 99.05141484174820],    [54.2823287059674, 79.12064627468000],    [51.0887198989791, 69.58889785111840],    [50.2828363482307, 69.51050331149430],    [44.2117417520901, 73.68756431831720],    [38.0054880080606, 61.36690453724010],    [32.9404799426182, 67.17065576899510],    [53.6916395710700, 85.66820314500150],    [68.7657342696216, 114.85387123391300],    [46.2309664983102, 90.12357206996740],    [68.3193608182553, 97.91982103524280],    [50.0301743403121, 81.53699078301500],    [49.2397653427537, 72.11183246961560],    [50.0395759398759, 85.23200734232560],    [48.1498588910288, 66.22495788805460],    [25.1284846477723, 53.45439421485050]]; function lossFunction(arr0: number[], arr1: number[], arr2: number[]) {     let n: number = arr0.length; // Number of elements in X     //D_m = (-2/n) * sum(X * (Y - Y_pred))  # Derivative wrt m    let a: number = (-2 / n) * (arr0.map((a, i) => a * (arr1[i] - arr2[i]))).reduce((sum, current) => sum + current);    //D_c = (-2/n) * sum(Y - Y_pred)  # Derivative wrt c    let b: number = (-2 / n) * (arr1.map((a, i) => (a - arr2[i]))).reduce((sum, current) => sum + current);    return [a, b];} export const gradientDescentMain = () => {     // Building the model    let m: number = 0;    let c: number = 0;    let X_arr: number[];    let Y_arr: number[];    let Y_pred_arr: number[];    let D_m: number = 0;    let D_c: number = 0;     let L: number = 0.00000001;  // The learning Rate    let epochs: number = 10000000;  // The number of iterations to perform gradient descent     //Initial guesses    for (let i = 0; i < epochs; i++) {        X_arr = data.map(function (value, index) { return value[0]; });        Y_arr = data.map(function (value, index) { return value[1]; });         // The current predicted value of Y        Y_pred_arr = X_arr.map((a) => ((m * a) + c));         let all = lossFunction(X_arr, Y_arr, Y_pred_arr);        D_m = all[0];        D_c = all[1];         m = m - L * D_m;  // Update m        c = c - L * D_c;  // Update c    }     console.log("m: " + m + " c: " + c);} gradientDescentMain(); `

## Wren

Translation of: Go
Library: Wren-math
Library: Wren-fmt
`import "/math" for Mathimport "/fmt" for Fmt // Function for which minimum is to be found.var g = Fn.new { |x|    return (x[0]-1)*(x[0]-1)*        Math.exp(-x[1]*x[1]) + x[1]*(x[1]+2)*        Math.exp(-2*x[0]*x[0])} // Provides a rough calculation of gradient g(p).var gradG = Fn.new { |p|    var x = p[0]    var y = p[1]    return [2*(x-1)*Math.exp(-y*y) - 4*x*Math.exp(-2*x*x)*y*(y+2),            -2*(x-1)*(x-1)*y*Math.exp(-y*y) + Math.exp(-2*x*x)*(y+2) + Math.exp(-2*x*x)*y]} var steepestDescent = Fn.new { |x, alpha, tolerance|    var n = x.count    var g0 = g.call(x) // // Initial estimate of result.     // Calculate initial gradient.    var fi = gradG.call(x)     // Calculate initial norm.    var delG = 0    for (i in 0...n) delG = delG + fi[i]*fi[i]    delG = delG.sqrt    var b = alpha/delG     // Iterate until value is <= tolerance.    while (delG > tolerance) {        // Calculate next value.        for (i in 0...n) x[i] = x[i] - b*fi[i]         // Calculate next gradient.        fi = gradG.call(x)         // Calculate next norm.        delG = 0        for (i in 0...n) delG = delG + fi[i]*fi[i]        delG = delG.sqrt        b = alpha/delG         // Calculate next value.        var g1 = g.call(x)         // Adjust parameter.        if (g1 > g0) {            alpha = alpha / 2        } else {            g0 = g1        }    }} var tolerance = 0.0000006var alpha = 0.1var x = [0.1, -1] // Initial guess of location of minimum. steepestDescent.call(x, alpha, tolerance)System.print("Testing steepest descent method:")Fmt.print("The minimum is at x = \$f, y = \$f for which f(x, y) = \$f.", x[0], x[1], g.call(x))`
Output:
```Testing steepest descent method:
The minimum is at x = 0.107627, y = -1.223260 for which f(x, y) = -0.750063.
```

## zkl

Translation of: Go
`fcn steepestDescent(f, x,y, alpha, h){   g0:=f(x,y);	# Initial estimate of result.   fix,fiy := gradG(f,x,y,h);	# Calculate initial gradient    # Calculate initial norm.   b:=alpha / (delG := (fix*fix + fiy*fiy).sqrt());   while(delG > h){	# Iterate until value is <= tolerance.      x,y = x - b*fix, y - b*fiy;      # Calculate next gradient and next value      fix,fiy = gradG(f,x,y, h/=2);      b=alpha / (delG = (fix*fix + fiy*fiy).sqrt());	# Calculate next norm.      if((g1:=f(x,y)) > g0) alpha/=2 else g0 = g1;	# Adjust parameter.   }   return(x,y)} fcn gradG(f,x,y,h){	# gives a rough calculation of gradient f(x,y).   g0:=f(x,y);   return((f(x + h, y) - g0)/h, (f(x, y + h) - g0)/h)}`
`fcn f(x,y){	# Function for which minimum is to be found.   (x - 1).pow(2)*(-y.pow(2)).exp() +    y*(y + 2)*(-2.0*x.pow(2)).exp()} tolerance,alpha := 0.0000006, 0.1; x,y := 0.1, -1.0;	# Initial guess of location of minimum.x,y = steepestDescent(f,x,y,alpha,tolerance); println("Testing steepest descent method:");println("The minimum is at (x,y) = (%f,%f). f(x,y) = %f".fmt(x,y,f(x,y)));`
```Testing steepest descent method: