Gradient descent
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.
- Task
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
modified to use the actual gradient function -
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
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- Output:
Testing steepest descent method: The minimum is at x[0] = 0.107627, x[1] = -1.223260
ALGOL W
which is a
with the gradient function from
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 ) )
end
end.- Output:
Testing steepest descent method: The minimum is at x(0) = 0.1076268, x(1) = -1.2232596
Common 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)))
- Output:
(0.10762684380416455 -1.2232596637047382)
Forth
Approach with partial differential equations GD1, GD2
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
;
Approach with main function and numerical gradients xd xy:
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
;
- Output:
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
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.
J
In earlier versions without calculus library, D. can be used instead of pderiv
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
- Output:
0.107627 _1.22326
jq
Adapted from 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.
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- Output:
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.
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
Nim
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.
import math, strformat
func steepDescent(g: proc(x: openArray[float]): float;
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}"
- Output:
Testing steepest descent method: The minimum is at x = 0.107626824329, y = -1.223259654882 for which f(x, y) = -0.750063420551
Perl
Calculate with bignum for numerical stability.
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
with javascript_semantics -- 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, xm1 = x-1, emyy = exp(-y*y), em2xx = exp(-2*x*x), xm12emyy = xm1*2*emyy p = {xm12emyy - 4*x*em2xx*y*(y+2), -xm12emyy*xm1*y + em2xx*(2*y+2)} return p end function function steepestDescent(sequence x, atom alpha, tolerance) sequence fi = gradG(x) -- Calculate initial gradient atom g0 = g(x), -- Initial estimate of result delG = sqrt(sum(sq_mul(fi,fi))), -- & norm b = alpha / delG integer icount = 0 -- iteration limitor/sanity while delG>tolerance do -- Iterate until <= tolerance x = sq_sub(x,sq_mul(b,fi)) -- Calculate next value fi = gradG(x) -- next gradient delG = sqrt(sum(sq_mul(fi,fi))) -- next norm b = alpha / delG atom g1 = g(x) -- next value if g1>g0 then alpha /= 2 -- Adjust parameter else g0 = g1 end if icount += 1 assert(icount<100) -- (increase if/when necessary) end while return x end function constant alpha = 0.1, tolerance = 0.00000000000001 sequence 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) = %.15f\n", {x[1], x[2], g(x)})
- Output:
Results match Fortran, most others to 6 or 7dp
Some slightly unexpected/unusual (but I think acceptable) variance was noted when playing with different tolerances, be warned.
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 = 0.1076268435484, y = -1.2232596638399 for which f(x, y) = -0.750063420551493
Racket
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)
# 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= -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
- 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
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 function
export 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
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))
- Output:
Testing steepest descent method: The minimum is at x = 0.107627, y = -1.223260 for which f(x, y) = -0.750063.
zkl
with tweaked gradG
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)));- Output:
Testing steepest descent method: The minimum is at (x,y) = (0.107608,-1.223299). f(x,y) = -0.750063