Halette C. Burgess

Calc 3

A gradient search is used to locate critical points in a localized area. The partials - that form the gradient - approach zero when they approach a critical point. We will be using this program on Maple to find local minimums.

Exercise one:

> restart:with(plottools):with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> func:=(x,y)->(x+1)^4 +(y-1)^4+1/(x^2*y^2+1);

> with(plottools):plot3d((x+1)^4 +(y-1)^4+1/(x^2*y^2+1),x=-10..10,y=-10..10);

I would guess the local minimum would be at x=0 and y=0

> a := 0: b := 0: del := 1: evalf(func(a, b));

> plot3d(func(x,y),x=a-del..a+del, y=b-del..b+del,axes=boxed, style=patchcontour);

> plots[contourplot](func(x,y),x=a-del..a+del, y=b-del..b+del);

> funcx := diff(func(x,y),x);

> funcy := diff(func(x,y),y);

> fxpoint := evalf(subs({x=a, y=b}, funcx));

> fypoint := evalf(subs({x=a, y=b}, funcy));

> plot(func(a+fxpoint*t, b+fypoint*t), t=-1..0);

> pathdiff := diff(func(a+fxpoint*t, b+fypoint*t),t);

> tmin := fsolve(pathdiff,t=0);

> a1 := a + tmin*fxpoint; b1 := b + tmin*fypoint; func(a1, b1);

> a := a1; b := b1; currentval := func(a, b);

> print(`currrent a `=a, `current b `=b, `current val `= currentval);

> fxpoint := evalf(subs({x=a, y=b}, funcx));

> fypoint := evalf(subs({x=a, y=b}, funcy));

> print(`current gradient `=[fxpoint,fypoint]);

> pathdiff := diff(func(a+fxpoint*t, b+fypoint*t),t):

> tmin := fsolve(pathdiff,t=0);

> a1 := a + tmin*fxpoint; b1 := b + tmin*fypoint; func(a1, b1);

The "very accurate" approximation of the local minimum is at x= -1.391040050 and y=1.391040050.

> plot3d(func(x,y),x=a1-del..a1+del, y=b1-del..b1+del,axes=boxed, style=patchcontour);

> plots[contourplot](func(x,y),x=a1-del..a1+del, y=b1-del..b1+del);

Exercise two:

> restart:with(plottools):with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> func := (x,y) -> .15*x+3*(.8*y)^(-.87)+(.8*y)*(1+3/x);

> with(plottools):plot3d(.15*x+3*(.8*y)^(-.87)+(.8*y)*(1+3/x),x=0..10,y=0..10);

>

I would guess the local minimum in this window to be at about x=1 and y=1

> a := 1: b := 1: del := 1: evalf(func(a, b));

> plot3d(func(x,y),x=a-del..a+del, y=b-del..b+del,axes=boxed, style=patchcontour);

> plots[contourplot](func(x,y),x=a-del..a+del, y=b-del..b+del);

>

func := (x,y) -> .15*x+3*(.8*y)^(-.87)+(.8*y)*(1+3/x);

initx := 10; inity := 10;

funcx := diff(func(x,y),x);

funcy := diff(func(x,y),y);

a1 := initx: b1 := inity:

for i from 1 to 10 do

a := a1; b := b1; currentval := evalf(func(a, b));

print(`currrent a `=a, `current b `=b, `current val `= currentval);

fxpoint := evalf(subs({x=a, y=b}, funcx));

fypoint := evalf(subs({x=a, y=b}, funcy));

print(`current gradient `=[fxpoint,fypoint]);

pathdiff := diff(func(a+fxpoint*t, b+fypoint*t),t):

tmin := fsolve(pathdiff,t=0);

a1 := a + tmin*fxpoint; b1 := b + tmin*fypoint; func(a1, b1);

od:

Local minimum is at x=5.108160844 and y=1.630830381

> a1=5.108160844,b1=1.63083081;

> plot3d(func(x,y),x=a1-del..a1+del, y=b1-del..b1+del,axes=boxed, style=patchcontour);

> plots[contourplot](func(x,y),x=a1-del..a1+del, y=b1-del..b1+del);