"File Newton"

"AUXILIARY FUNCTIONS"

EVAL_(y,a):=ITERATE(y,x,a,1)

E_(z,i):=ELEMENT(z,i)

V_LINE(y,p):=[p,t*EVAL_(y,p)]

N_LINE(y,p,q):=[(1-t)*p+t*q,(1-t)*EVAL_(y,p)]

T_LINE(y,p,q):=[V_LINE(y,p),N_LINE(y,p,q)]

A_LINE(y,z,k):=VECTOR(T_LINE(y,E_(z,i),E_(z,i+1)),i,1,k)

NEW(n,x,w):=[n+1,EVAL_(w,x)]

N_(y,a,n):=ITERATES(x-y/DIF(y,x),x,a,n)

NS_(y,a,n):=ITERATE(x-y/DIF(y,x),x,a,n)

"########################################################"

"USER FUNCTIONS"

NEWTON(y,a,n):=ITERATES(NEW(E_(z,1),E_(z,2),x-y/DIF(y,x)),z,[0,a],n)

PICTURE(y,a,n):=[y,A_LINE(y,N_(y,a,n),n-1)]

START(y,n,a,b,s):=VECTOR([k,NS_(y,k,n)],k,a,b,s)

"######################################"

"INSTRUCTIONS"

"NEWTON(Y,A,N) produces n iterations of Newton's method"

".....applied to y=f(x) starting at x=a."

"PICTURE(Y,A,N) produces a picture of n iterations of "

".....Newton's method applied to y=f(x) starting at x=a."

".....THE PLOT IS PARAMETRIC. "

"PARAMETER DOMAIN MUST BE SET TO  MIN:0  MAX:1"

".....Use <Ctrl Enter> to accept parameter domain."

"START(Y,N,A,B,S) returns the ordered pairs [k,z] where z"

".....is the result of applying N iterations of Newoton's method"

".....to Y starting at k.  k ranges from A to B in steps of S."

"###################"

"Example 1"

"approX the following to see 10 steps of Newton's method"

"... starting at a=1.1 applied to y=cos x + sin x"

NEWTON(SIN(x)+COS(x),1.1,10)

"Example 2"

"approX the following and the Plot (see graphics settings above)"

"... to see a picture of Example 1"

PICTURE(SIN(x)+COS(x),1.1,10)