#sfb 25 oct 03

How to use the TI-89 to do a two dimensional steepest decent. An aid to
the big valley project. 

Strategy:
	Besides computing the next approximations we are going to
	want to store the answers in list variables.

The Data Editor:
	One way of handling the lists/data/matrices is to use the data
	editor. One gets to the data editor via the APPS key, it is #6
	on the list. It offers a choice of current/open/new. Use new to
	create new variables and open to find old ones. For new ones
	you will be given a choice of type Data/Matrix/List, we need 
	lists, and you need to name the variable. We will use px and py.
	We might have other uses for the data editor, but we will create
	px and py without it below.

Create our lists px and py
	Use newList, which is found in CATALOG (type n twice to jump to n's)
	scroll down to newList. Or alternately type newlist. Store into px
	and py.

	newList(0) sto> px
	newList(0) sto> py

Create local index variables
	1 sto> j1       # index into px and py
	1 sto> px[j1]   
	0 sto> py[j1]   # the zero^th point
	[Note: none of these are single letters -- this might be a good thing.]
	uninitialize variables
	p2, q2  # the current point obtained from px[j1], py[j1] at the top
	t1 # the t value of the next local minimum.

Create functions
	put f(x,y) into z1 but MODE Graphing 3D green diamond y=
	z1=x^2-x*y+y^2
	# The following need to be done only once
	d(z1(x,y),x) sto> fx(x,y)  #fx for f_x
	d(z1(x,y),y) sto> fy(x,y)  #fy for f_u
	p2-fx(p2,q2)*t sto> lx(t)
	q2-fy(p2,q2)*t sto> ly(t)  #<lx(t), ly(t)> is Pu line
	z1(lx(t),ly(t) sto> g(t)   #g is f restricted to Pu line

Repeat loop
	px[j1] sto> p2
	py[j1] sto> q2
	j1 + 1 sto> j1 # the location to fill

	zeros(diff(g(t),t), t) # zeros is `F2 alg #4' diff is `F3 calc #1'
	@@@ Here is the only point where there is need for human like
	choice. There could be more than one zero and you have to pick the
	correct zero. Use [index of correct zero]

	zeros(diff(g(t),t), t)[1] sto> t1
	lx(t1) sto> px[j1]
	ly(t1) sto> py[j1]

	Recall history to top of repeat

I did the loop 5 times (j1 = 6 at the end) and got
	px {1  2/7 3/28   3/98 9/784   9/2744}
	py {0 5/14    0 15/392     0 45/10976}
	Which agrees with the project page. WE HAVE SUCCESS.

More
	1. Get the z-values
	seq(z1(px[n],py[n]),n,1,6) sto> pz
	The values are
	pz {1 3/28 9/784 27/21952 81/614656 243/17210368}

	2. Draw a contour map (check your window) [DrwCtour in CATALOG]
	DrwCtour pz
	# it takes its time, default window too big, change to -1.5..1.5 for
	both x and y. But this changes the contours do the comand again
	DrwCtour pz

	3. Draw the zig-zag line. Change the graphing mode to function, set
	the window 0..1 0..0.5
	Line px[1],py[1],px[2],py[2]  # then home button to type the next line
	Line px[2],py[2],px[3],py[3]
	Line px[3],py[3],px[4],py[4]
	Line px[4],py[4],px[5],py[5]
	Line px[5],py[5],px[6],py[6]
	# one can zoom in, but you have to redraw the lines.

	4. We could put px and py into a data array and do and xyline plot
	might draw the lines faster.
	newData zig,px,py # We will use the data editor to recreate pz
	APPS 6 open variable=zig
	# define a statistics plot
	F2 F1 plot type -> xyline
	mark?
	x = c1
	y = c2
	enter enter
	green diamond y= # you can see plot1 appears and is checked
	check your window, 0..1 0..0.5 (MODE is function)
	look at graph, it is the zig-zag graph of #3, but you can zoom in on this 1
	green diamond y= then uparrow to highlight plot1 F4 to deselect

	#optional way to recreate pz list
	APPS 6 open variable=zig
	Move the cursor to the rectangle named c3 hit enter
	c3 = z1(c1,c2) # recreates the pz list

Notes:
	Apparently one can't use q1 or q3 as a variable name on ti-89 because
	they are used for the 1st and 3rd quartile for statistics.