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Gradient descent (view source)

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[https://books.google.co.uk/books?id=dFHvBQAAQBAJ&pg=PA543&lpg=PA543&dq=c%23+steepest+descent+method+to+find+minima+of+two+variable+function&source=bl&ots=TCyD-ts9ui&sig=ACfU3U306Og2fOhTjRv2Ms-BW00IhomoBg&hl=en&sa=X&ved=2ahUKEwitzrmL3aXjAhWwVRUIHSEYCU8Q6AEwCXoECAgQAQ#v=onepage&q=c%23%20steepest%20descent%20method%20to%20find%20minima%20of%20two%20variable%20function&f=false This book excerpt] shows sample C# code for solving this task.
<br><br>
 
=={{header|ALGOL 68}}==
{{works with|ALGOL 68G|Any - tested with release 2.8.3.win32}}
{{Trans|Go}} modified to use the actual gradient function -
{{Trans|Fortran}}
THe results agree with the Fortran sample and the Julia sample to 6 places.
<syntaxhighlight lang="algol68">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
OD
END # 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;
z
END # 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</syntaxhighlight>
{{out}}
<pre>
Testing steepest descent method:
The minimum is at x[0] = 0.107627, x[1] = -1.223260
</pre>
 
=={{header|ALGOL W}}==
{{Trans|ALGOL 68}} which is a {{Trans|Go}} with the gradient function from {{Trans|Fortran}}
The results agree (to 6 places) with the Fortran and Julia samples.
<syntaxhighlight lang="algolw">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 ) )
end
end.</syntaxhighlight>
{{out}}
<pre>
Testing steepest descent method:
The minimum is at x(0) = 0.1076268, x(1) = -1.2232596
</pre>
 
 
 
 
=={{header|Common Lisp}}==
{{works with|SBCL}}
 
<syntaxhighlight lang="Lisp">
(defparameter *epsilon* (sqrt 0.0000001d0))
 
(defun myfun (x y)
(+ (* (- x 1)
(- x 1)
(exp (- (expt y 2))))
(* y (+ 2 y)
(exp (- (* 2 (expt x 2)))))))
 
(defun xd (x y)
(/ (- (myfun (* x (+ 1 *epsilon*)) y)
(myfun (* x (- 1 *epsilon*)) y))
(* 2 x *epsilon*)))
 
(defun yd (x y)
(/ (- (myfun x (* y (+ 1 *epsilon*)))
(myfun x (* y (- 1 *epsilon*))))
(* 2 y *epsilon*)))
 
(defun gd ()
(let ((x 0.1d0)
(y -1.0d0))
(dotimes (i 31)
(setf x (- x (* 0.3d0 (xd x y))))
(setf y (- y (* 0.3d0 (yd x y )))))
(list x y)))
 
 
 
</syntaxhighlight>
{{out}}
<pre>
(0.10762684380416455 -1.2232596637047382)
</pre>
 
 
=={{header|Forth}}==
{{works with|SwiftForth}}
{{libheader|fpmath}}
 
 
Approach with partial differential equations GD1, GD2
 
<syntaxhighlight lang="forth">
 
0e fvalue px
0e fvalue py
fvariable x 0.1e x F!
fvariable y -1e y F!
 
: ^2 FDUP F* ;
 
\ gradient for x, analytical function
 
: GD1 ( F: x y -- dfxy)
to py to px
2e px 1e F- F* py ^2 FNEGATE FEXP F*
-4e px F* py F* py 2e F+ F* px ^2 -2e F* FEXP F*
F+
;
 
\ gradient for y, analytical function
 
: GD2 ( F: x y -- dfxy )
to py to px
px ^2 -2e F* FEXP FDUP
py F* FSWAP
py 2e F+ F* F+
py ^2 FNEGATE FEXP py F*
px 1e F- ^2 F*
-2e F*
F+
;
 
\ gradient descent
: GD ( x y n -- ) \ gradient descent, initial guess and number of iterations
FOVER FOVER y F! x F!
0 do
FOVER FOVER
GD1 0.3e F* x F@ FSWAP F- x F!
GD2 0.3e F* y F@ FSWAP F- y F!
x F@ y F@ FOVER FOVER FSWAP F. F. CR
loop
;
 
</syntaxhighlight>
 
Approach with main function and numerical gradients xd xy:
 
<syntaxhighlight lang="forth">
 
0e fvalue px
0e fvalue py
fvariable x 0.1e x F!
fvariable y -1e y F!
 
1e-6 fvalue EPSILON
 
: fep EPSILON FSQRT ;
 
 
: MYFUN ( x y -- fxy )
FDUP FROT FDUP ( y y x x )
( x ) 1e F- FDUP F* ( y y x a )
FROT ( y x a y )
( y ) FDUP F* FNEGATE FEXP F* ( y x b )
FROT ( x b y )
( y ) FDUP 2e F+ F* ( x b c )
FROT ( b c x )
( x ) FDUP F* -2e F* FEXP F*
F+
;
 
 
: xd ( x y )
to py to px
px 1e fep F+ F* py myfun px 1e fep F- F* py myfun F- 2e px F* fep F* F/
;
 
: xy ( x y )
to py to px
py 1e fep F+ F* px FSWAP myfun py 1e fep F- F* px FSWAP myfun F- 2e py F* fep F* F/
;
 
 
: GDB ( x y n -- ) \ gradient descent, initial guess and number of iterations
CR FOVER FOVER y F! x F!
0 do
FOVER FOVER
xd 0.3e F* x F@ FSWAP F- x F!
xy 0.3e F* y F@ FSWAP F- y F!
x F@ y F@ FOVER FOVER FSWAP F. F. CR
loop
;
</syntaxhighlight>
 
{{out}}
<pre>
 
0.1e -1e 20 GD
0.1810310 -1.1787894
0.1065262 -1.1965306
0.1144636 -1.2181779
0.1075002 -1.2206140
0.1082816 -1.2227594
0.1076166 -1.2230041
0.1076894 -1.2232108
0.1076261 -1.2232350
0.1076328 -1.2232549
0.1076267 -1.2232573
0.1076274 -1.2232592
0.1076268 -1.2232594
0.1076269 -1.2232596
0.1076268 -1.2232596
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
 
0.1e -1e 20 GDB
0.1810310 -1.1787893
0.1065262 -1.1965305
0.1144636 -1.2181779
0.1075002 -1.2206140
0.1082816 -1.2227594
0.1076166 -1.2230041
0.1076894 -1.2232108
0.1076261 -1.2232350
0.1076328 -1.2232549
0.1076268 -1.2232573
0.1076274 -1.2232592
0.1076268 -1.2232594
0.1076269 -1.2232596
0.1076268 -1.2232596
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
0.1076268 -1.2232597
</pre>
 
 
 
 
=={{header|Fortran}}==
'''Compiler:''' [[gfortran 8.3.0]]<br>
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.
<syntaxhighlight lang="fortran"> 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
</syntaxhighlight>
{{out}}
<pre>After 31 steps, found minimum at x= 0.107626843548372 y= -1.223259663839920 of f= -0.750063420551493
STOP program complete
</pre>
 
=={{header|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.
For some unknown reason the results differ from the other solutions after the first 4 decimal places but are near enough for an approximate method such as this.
<langsyntaxhighlight lang="go">package main
 
import (
Line 25 ⟶ 489:
func steepestDescent(x []float64, alpha, tolerance float64) {
n := len(x)
h := tolerance
g0 := g(x) // Initial estimate of result.
 
// Calculate initial gradient.
fi := gradG(x, h)
 
// Calculate initial norm.
Line 45 ⟶ 508:
x[i] -= b * fi[i]
}
h /= 2
 
// Calculate next gradient.
fi = gradG(x, h)
 
// Calculate next norm.
Line 70 ⟶ 532:
}
 
// Provides a rough calculation of gradient g(xp).
func gradG(xp []float64, h float64) []float64 {
nz := make([]float64, len(xp))
zx := make(p[0]float64, n)
y := make(p[1]float64, n)
z[0] = 2*(x-1)*math.Exp(-y*y) - 4*x*math.Exp(-2*x*x)*y*(y+2)
copy(y, x)
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
g0 := g(x)
 
for i := 0; i < n; i++ {
y[i] += h
z[i] = (g(y) - g0) / h
}
return z
}
Line 99 ⟶ 556:
steepestDescent(x, alpha, tolerance)
fmt.Println("Testing steepest descent method:")
fmt.PrintlnPrintf("The minimum is at x[0] =" %f, y = %f for which f(x[0], "y) = %f.\bn", x[10] =", x[1], g(x))
}</syntaxhighlight>
}
</lang>
 
{{out}}
<pre>
Testing steepest descent method:
The minimum is at x[0] = 0.10764302056464771107627, x[1]y = -1.223351901171944223260 for which f(x, y) = -0.750063.
</pre>
 
=={{header|J}}==
{{works with|Jsoftware|>= 9.0.1}}
{{libheader|math/calculus}}
 
In earlier versions without calculus library, D. can be used instead of pderiv
 
<syntaxhighlight lang="J">
 
 
load 'math/calculus'
coinsert 'jcalculus'
 
NB. initial guess
 
go=: 0.1 _1
 
NB. function
 
func=: monad define
'xp yp' =. y
((1-xp)*(1-xp) * ] ^-(yp)^2) + yp*(yp+2)* ] ^ _2 * xp^2 NB. evaluates from right to left without precedence
)
 
NB. gradient descent
 
shortygd =: monad define
go=.y
go=. go - 0.03 * func pderiv 1 ] go NB. use D. instead of pderiv for earlier versions
)
 
NB. use:
 
shortygd ^:_ go
 
</syntaxhighlight>
{{out}}
<pre>
0.107627 _1.22326
</pre>
 
 
 
 
=={{header|jq}}==
'''Adapted from [[#Go|Go]]'''
 
'''Works with jq and gojq, the C and Go implementations of jq'''
 
The function, g, and its gradient, gradG, are defined as 0-arity filters so that they
can be passed as formal parameters to the `steepestDescent` function.
<syntaxhighlight lang=jq>
def sq: .*.;
 
def ss(stream): reduce stream as $x (0; . + ($x|sq));
 
# Function for which minimum is to be found: input is a 2-array
def g:
. as $x
| (($x[0]-1)|sq) *
(( - ($x[1]|sq))|exp) + $x[1]*($x[1]+2) *
((-2*($x[0]|sq))|exp);
 
# Provide a rough calculation of gradient g(p):
def gradG:
. as [$x, $y]
| ($x*$x) as $x2
| [2*($x-1)*(-($y|sq)|exp) - 4 * $x * (-2*($x|sq)|exp) * $y*($y+2),
- 2*($x-1)*($x-1)*$y*((-$y*$y)|exp)
+ ((-2*$x2)|exp)*($y+2) + ((-2*$x2)|exp)*$y] ;
 
def steepestDescent(g; gradG; $x; $alpha; $tolerance):
($x|length) as $n
| {g0 : (x|g), # Initial estimate of result
alpha: $alpha,
fi: ($x|gradG), # Calculate initial gradient
x: $x,
iterations: 0
}
 
# Calculate initial norm:
| .delG = (ss( .fi[] ) | sqrt )
| .b = .alpha/.delG
 
# Iterate until value is <= $tolerance:
| until (.delG <= $tolerance;
.iterations += 1
# Calculate next value:
| reduce range(0; $n) as $i(.;
.x[$i] += (- .b * .fi[$i]) )
 
# Calculate next gradient:
| .fi = (.x|gradG)
 
# Calculate next norm:
| .delG = (ss( .fi[] ) | sqrt)
| .b = .alpha/.delG
 
# Calculate next value:
| .g1 = (.x|g)
 
# Adjust parameter:
| if .g1 > .g0
then .alpha /= 2
else .g0 = .g1
end
) ;
 
def tolerance: 0.0000006;
def alpha: 0.1;
def x: [0.1, -1]; # Initial guess of location of minimum.
 
def task:
steepestDescent(g; gradG; x; alpha; tolerance)
| "Testing the steepest descent method:",
"After \(.iterations) iterations,",
"the minimum is at x = \(.x[0]), y = \(.x[1]) for which f(x, y) = \(.x|g)." ;
 
task
</syntaxhighlight>
{{output}}
<pre>
Testing the steepest descent method:
After 24 iterations,
the minimum is at x = 0.10762682432948058, y = -1.2232596548816101 for which f(x, y) = -0.7500634205514924.
</pre>
 
=={{header|Julia}}==
<langsyntaxhighlight lang="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()))
</langsyntaxhighlight>{{out}}
<pre>
Results of Optimization Algorithm
Line 136 ⟶ 718:
</pre>
 
=={{header|Perl 6Nim}}==
{{trans|Go}}
The gradient function has been rewritten to remove a term in the partial derivative with respect to y (two terms instead of three). This doesn’t change the result which agrees with those of Go, Fortran, Julia, etc.
<lang perl6>#!/usr/bin/env perl6
 
<syntaxhighlight lang="nim">import math, strformat
use v6.d;
 
 
sub steepestDescent(@x, $alpha is copy, $tolerance) {
func steepDescent(g: proc(x: openArray[float]): float;
my \N = +@x ; my $h = $tolerance ;
gradG: proc(p: openArray[float]): seq[float];
x: var openArray[float]; alpha, tolerance: float) =
let n = x.len
var alpha = alpha
var g0 = g(x) # Initial estimate of result.
 
# Calculate initial gradient.
var fi = gradG(x)
 
# Calculate initial norm.
var delG = 0.0
for i in 0..<n:
delG += fi[i] * fi[i]
delG = sqrt(delG)
var b = alpha / delG
 
# Iterate until value is <= tolerance.
while delG > tolerance:
# Calculate next value.
for i in 0..<n:
x[i] -= b * fi[i]
 
# Calculate next gradient.
fi = gradG(x)
 
# Calculate next norm.
delG = 0
for i in 0..<n:
delG += fi[i] * fi[i]
delG = sqrt(delG)
b = alpha / delG
 
# Calculate next value.
let g1 = g(x)
 
# Adjust parameter.
if g1 > g0: alpha *= 0.5
else: g0 = g1
 
 
when isMainModule:
 
func g(x: openArray[float]): float =
## Function for which minimum is to be found.
(x[0]-1) * (x[0]-1) * exp(-x[1]*x[1]) + x[1] * (x[1]+2) * exp(-2*x[0]*x[0])
 
func gradG(p: openArray[float]): seq[float] =
## Provides a rough calculation of gradient g(p).
result = newSeq[float](p.len)
let x = p[0]
let y = p[1]
result[0] = 2 * (x-1) * exp(-y*y) - 4 * x * exp(-2*x*x) * y * (y+2)
result[1] = -2 * (x-1) * (x-1) * y * exp(-y*y) + 2 * (y+1) * exp(-2*x*x)
 
const
Tolerance = 0.0000006
Alpha = 0.1
 
var x = [0.1, -1.0] # Initial guess of location of minimum.
 
steepDescent(g, gradG, x, Alpha, Tolerance)
echo "Testing steepest descent method:"
echo &"The minimum is at x = {x[0]:.12f}, y = {x[1]:.12f} for which f(x, y) = {g(x):.12f}"</syntaxhighlight>
 
{{out}}
<pre>Testing steepest descent method:
The minimum is at x = 0.107626824329, y = -1.223259654882 for which f(x, y) = -0.750063420551</pre>
 
=={{header|Perl}}==
Calculate with <code>bignum</code> for numerical stability.
{{trans|Raku}}
<syntaxhighlight lang="perl">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(@x, $h, @x) ; # Calculate initial gradient
 
# Calculate initial norm.
my $delG = 0;
for ^(0..$N-1) { $delG += @$fi[$_]²**2 }
my $b = $alpha / sqrt($delG.sqrt);
 
while ( $delG > $tolerance ) { # Iterate until value is <= tolerance.
# Calculate next value.
for ^(0..$N-1) { @$x[$_] -= $b * @$fi[$_] }
$h /= 2;
 
@fi = gradG(@x, $h, @x); # Calculate next gradient.
 
# Calculate next norm.
$delG = 0;
for ^(0..$N-1) { $delG += @$fi[$_]²**2 }
$b = $alpha / sqrt($delG.sqrt);
 
my $g1 = g(@x); # Calculate next value.
 
$g1 > $g0 ?? ( $alpha /= 2 ) !!: ( $g0 = $g1 ) ; # Adjust parameter.
}
@x
}
 
sub gradG(@x, $h) { # Provides a rough calculation of gradient g(x).
sub gradG {
my \N = +@x ; my ( @y , @z );
my($h, @yx) = @x_;
my $N = @x;
my @y = @x;
my $g0 = g(@x);
my @z;
for ^N { @y[$_] += $h ; @z[$_] = (g(@y) - $g0) / $h }
for (0..$N-1) { $y[$_] += $h ; $z[$_] = (g(@y) - $g0) / $h }
return @z
}
 
# Function for which minimum is to be found.
sub g { my(\@x) {= @_; ($x[0]-1)²**2 * exp(-$x[1]²**2).exp + $x[1]*($x[1]+2) * exp(-2*$x[0]²**2).exp };
 
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);</syntaxhighlight>
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];
</lang>
{{out}}
<pre>The minimum is at x[0] = 0.107653, x[1] = -1.223370</pre>
<pre>Testing steepest descent method:
The minimum is at x[0] = 0.10743450794656964, x[1] = -1.2233956711774543
</pre>
 
=={{header|Phix}}==
{{trans|Go}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
... and just like Go, the results don't quite match anything else.
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<lang Phix>-- Function for which minimum is to be found.
<span style="color: #000080;font-style:italic;">-- Function for which minimum is to be found.</span>
function g(sequence x)
<span style="color: #008080;">function</span> <span style="color: #000000;">g</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
atom {x0,x1} = x
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x</span>
return (x0-1)*(x0-1)*exp(-x1*x1) +
<span style="color: #008080;">return</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*(</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">exp</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span>
x1*(x1+2)*exp(-2*x0*x0)
<span style="color: #000000;">x1</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">exp</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">)</span>
end function
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
-- Provides a rough calculation of gradient g(x).
<span style="color: #000080;font-style:italic;">-- Provides a rough calculation of gradient g(x).</span>
function gradG(sequence x, atom h)
<span style="color: #008080;">function</span> <span style="color: #000000;">gradG</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
integer n = length(x)
<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;">p</span><span style="color: #0000FF;">,</span>
sequence z = repeat(0, n)
<span style="color: #000000;">xm1</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: #0000FF;">,</span>
atom g0 := g(x)
<span style="color: #000000;">emyy</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">exp</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>
for i=1 to n do
<span style="color: #000000;">em2xx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">exp</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;">x</span><span style="color: #0000FF;">),</span>
x[i] += h
<span style="color: #000000;">xm12emyy</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">xm1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">emyy</span>
z[i] = (g(x) - g0) / h
<span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">xm12emyy</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">em2xx</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;">2</span><span style="color: #0000FF;">),</span>
end for
<span style="color: #0000FF;">-</span><span style="color: #000000;">xm12emyy</span><span style="color: #0000FF;">*</span><span style="color: #000000;">xm1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">em2xx</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;">2</span><span style="color: #0000FF;">)}</span>
return z
<span style="color: #008080;">return</span> <span style="color: #000000;">p</span>
end function
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
function steepestDescent(sequence x, atom alpha, tolerance)
<span style="color: #008080;">function</span> <span style="color: #000000;">steepestDescent</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">alpha</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">tolerance</span><span style="color: #0000FF;">)</span>
integer n = length(x)
<span style="color: #004080;">sequence</span> <span style="color: #000000;">fi</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gradG</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- Calculate initial gradient</span>
atom h = tolerance,
<span style="color: #004080;">atom</span> <span style="color: #000000;">g0</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">g</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span> <span style="color: #000080;font-style:italic;">-- Initial estimate of result</span>
g0 = g(x) -- Initial estimate of result.
<span style="color: #000000;">delG</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fi</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fi</span><span style="color: #0000FF;">))),</span> <span style="color: #000080;font-style:italic;">-- & norm</span>
 
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alpha</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">delG</span>
-- Calculate initial gradient.
<span style="color: #004080;">integer</span> <span style="color: #000000;">icount</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> <span style="color: #000080;font-style:italic;">-- iteration limitor/sanity</span>
sequence fi = gradG(x, h)
<span style="color: #008080;">while</span> <span style="color: #000000;">delG</span><span style="color: #0000FF;">></span><span style="color: #000000;">tolerance</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">-- Iterate until &lt;= tolerance</span>
 
<span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fi</span><span style="color: #0000FF;">))</span> <span style="color: #000080;font-style:italic;">-- Calculate next value</span>
-- Calculate initial norm.
<span style="color: #000000;">fi</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">gradG</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- next gradient</span>
atom delG = sqrt(sum(sq_mul(fi,fi))),
<span style="color: #000000;">delG</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fi</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fi</span><span style="color: #0000FF;">)))</span> <span style="color: #000080;font-style:italic;">-- next norm</span>
b = alpha / delG
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alpha</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">delG</span>
 
<span style="color: #004080;">atom</span> <span style="color: #000000;">g1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">g</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- next value</span>
-- Iterate until value is <= tolerance.
<span style="color: #008080;">if</span> <span style="color: #000000;">g1</span><span style="color: #0000FF;">></span><span style="color: #000000;">g0</span> <span style="color: #008080;">then</span>
while delG>tolerance do
<span style="color: #000000;">alpha</span> <span style="color: #0000FF;">/=</span> <span style="color: #000000;">2</span> <span style="color: #000080;font-style:italic;">-- Adjust parameter</span>
-- Calculate next value.
x <span style="color: sq_sub(x,sq_mul(b,fi))#008080;">else</span>
<span style="color: #000000;">g0</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">g1</span>
h /= 2
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">icount</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
-- Calculate next gradient.
<span style="color: #7060A8;">assert</span><span style="color: #0000FF;">(</span><span style="color: #000000;">icount</span><span style="color: #0000FF;"><</span><span style="color: #000000;">100</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (increase if/when necessary)</span>
fi = gradG(x, h)
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">x</span>
-- Calculate next norm.
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
delG = sqrt(sum(sq_mul(fi,fi)))
b = alpha / delG
<span style="color: #008080;">constant</span> <span style="color: #000000;">alpha</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">tolerance</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.00000000000001</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">steepestDescent</span><span style="color: #0000FF;">({</span><span style="color: #000000;">0.1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span> <span style="color: #000000;">alpha</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">tolerance</span><span style="color: #0000FF;">)</span>
-- Calculate next value.
<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;">"Testing steepest descent method:\n"</span><span style="color: #0000FF;">)</span>
atom g1 = g(x)
<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;">"The minimum is at x = %.13f, y = %.13f for which f(x, y) = %.15f\n"</span><span style="color: #0000FF;">,</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: #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;">g</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<!--</syntaxhighlight>-->
-- Adjust parameter.
if g1>g0 then
alpha /= 2
else
g0 = g1
end if
end while
return x
end function
constant tolerance = 0.0000006, alpha = 0.1
sequence x = steepestDescent({0.1,-1}, alpha, tolerance)
printf(1,"Testing steepest descent method:\n")
printf(1,"The minimum is at x[1] = %.16f, x[1] = %.16f\n", x)</lang>
{{out}}
Results match Fortran, most others to 6 or 7dp<br>
Some slightly unexpected/unusual (but I think acceptable) variance was noted when playing with different tolerances, be warned.<br>
Results on 32/64 bit Phix agree to 13dp, which I therefore choose to show in full here (but otherwise would not really trust).
<pre>
Testing steepest descent method:
The minimum is at x[1] = 0.10765720809349961076268435484, x[1]y = -1.22329760804758902232596638399 for --which f(64x, bity) = -0.750063420551493
The minimum is at x[1] = 0.1073980565405569, x[1] = -1.2233251778997771 -- (32 bit)
</pre>
 
Line 276 ⟶ 922:
I could have used ∇ and Δ in the variable names, but it looked too confusing, so I've gone with <var>grad-</var> and <var>del-</var>
 
<langsyntaxhighlight lang="racket">#lang racket
 
(define (apply-vector f v)
Line 323 ⟶ 969:
(module+ main
(Gradient-descent))
</syntaxhighlight>
</lang>
 
{{out}}
<pre>'#(0.10760797905122492 -1.2232993981966753)</pre>
 
=={{header|Raku}}==
(formerly Perl 6)
{{trans|Go}}
<syntaxhighlight lang="raku" line># 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];
</syntaxhighlight>
{{out}}
<pre>Testing steepest descent method:
The minimum is at x[0] = 0.10762682432947938, x[1] = -1.2232596548816097
</pre>
 
=={{header|REXX}}==
The &nbsp; ''tolerance'' &nbsp; can be much smaller; &nbsp; a tolerance of &nbsp; '''1e-200''' &nbsp; was tested. &nbsp; It works, but causes the program to execute a bit slower, but still sub-second execution time.
<syntaxhighlight lang="rexx">/*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= -1
say 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 2.718281828459045235360287471352662497757247093699959574966967627724
g: 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</syntaxhighlight>
{{out|output|text=&nbsp; when using the internal default inputs:}}
<pre>
═══════════════════ testing for the steepest descent method ═══════════════════
The minimum is at: x[0]= 0.107626824 x[1]= -1.223259655
</pre>
 
=={{header|Scala}}==
{{trans|Go}}
<syntaxhighlight lang="scala">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))
}
}
</syntaxhighlight>
{{out}}
<pre>
Testing steepest descent method
The minimum is at x : 0.10756393294495799, y : -1.2234116852966237
</pre>
 
=={{header|TypeScript}}==
Line 333 ⟶ 1,160:
<br><br>
 
<syntaxhighlight lang="typescript">
<lang Typescript>
// Using the steepest-descent method to search
// for minimum values of a multi-variable function
Line 430 ⟶ 1,257:
gradientDescentMain();
 
</syntaxhighlight>
</lang>
{{out}}
<pre>
Line 441 ⟶ 1,268:
* &nbsp; [Linear Regression using Gradient Descent by Adarsh Menon]
<br><br>
<syntaxhighlight lang="typescript">
<lang Typescript>
let data: number[][] =
[[32.5023452694530, 31.70700584656990],
Line 589 ⟶ 1,416:
 
gradientDescentMain();
</syntaxhighlight>
</lang>
 
=={{header|Wren}}==
{{trans|Go}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./fmt" for Fmt
 
// Function for which minimum is to be found.
var g = Fn.new { |x|
return (x[0]-1)*(x[0]-1)*
(-x[1]*x[1]).exp + x[1]*(x[1]+2)*
(-2*x[0]*x[0]).exp
}
 
// 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)*(-y*y).exp - 4*x*(-2*x*x).exp*y*(y+2),
-2*(x-1)*(x-1)*y*(-y*y).exp + (-2*x*x).exp*(y+2) + (-2*x*x).exp*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 = 0.0000006
var alpha = 0.1
var 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))</syntaxhighlight>
 
{{out}}
<pre>
Testing steepest descent method:
The minimum is at x = 0.107627, y = -1.223260 for which f(x, y) = -0.750063.
</pre>
 
=={{header|zkl}}==
{{trans|Go}} with tweaked gradG
<syntaxhighlight lang="zkl">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)
}</syntaxhighlight>
<syntaxhighlight lang="zkl">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)));</syntaxhighlight>
{{out}}
<pre>
Testing steepest descent method:
The minimum is at (x,y) = (0.107608,-1.223299). f(x,y) = -0.750063
</pre>
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