; NEWTON method for equations

; solves f(x)=0
; function is programmed at label f(x), it receives x on the stack
; it can use the whole stack and registers 4-9 and b-d

; set precision into register "e" (0.01 for example)
; in the X register, put a first guess of the solution and RUN.

; it returns the solution.
; you can start again with new parameters (precision or starting x)

; usage: store error in register e (example 0.01 STO e)
; go to first step ( B/0)
; key in the first x value and RUN.

#c3 1		; auto save to C3 file

#reg e error
#reg a x
#reg 0 delta
#reg 1 h
#reg 2 fxh
#reg 3 fx

DO
	=x
	$1e-4 =delta
	REPEAT
		@x @delta *
		IF(x=0)THEN
			@delta
		ENDIF
		=h			; h=0.0001x or 0.0001 if x=0
		@x + GOSUB f(x) =fxh	; calculates f(x+h)
		@x GOSUB f(x) =fx	; calculates f(x)
		@h *
		@fx @ fxh - /		; calculates dx = hf(x)/(f(x)-f(x+h)) = -f(x)/f'(x)
		ENTER
		sum x			; new x = x + dx
		x<>y abs
		@error -		; abs(dx) - error
	UNTIL(x<0)			; stop if <0 (if abs(dx) < error)
	@x				; display x
	STOP
LOOP

f(x):
RTN

  Step | Code | Keys
------------------------
  000  |  4A  | STO a
  001  |  01  | 1
  002  |  0C  | EE
  003  |  04  | 4
  004  |  0B  | /-/
  005  |  40  | STO 0
  006  |  6A  | RCL a
  007  |  60  | RCL 0
  008  |  12  | * 
  009  |  5E  | F X=0 
  010  |  12  | 1 2
  011  |  60  | RCL 0
  012  |  41  | STO 1
  013  |  6A  | RCL a
  014  |  10  | + 
  015  |  53  | GOSUB 
  016  |  42  | 4 2
  017  |  42  | STO 2
  018  |  6A  | RCL a
  019  |  53  | GOSUB 
  020  |  42  | 4 2
  021  |  43  | STO 3
  022  |  61  | RCL 1
  023  |  12  | * 
  024  |  63  | RCL 3
  025  |  62  | RCL 2
  026  |  11  | - 
  027  |  13  | / 
  028  |  0E  | ENTER 
  029  |  6A  | RCL a
  030  |  10  | + 
  031  |  4A  | STO a
  032  |  14  | X<>Y 
  033  |  31  | K ABS 
  034  |  6E  | RCL e
  035  |  11  | - 
  036  |  5C  | F X<0 
  037  |  06  | 0 6
  038  |  6A  | RCL a
  039  |  50  | STOP 
  040  |  51  | GOTO 
  041  |  00  | 0 0
  042  |  52  | RTN 
------------------------

  Addr |  Name
------------------------
  042  |  F(X)
------------------------

  Reg  |  Name
------------------------
   0   |  DELTA
   1   |  H
   2   |  FXH
   3   |  FX
   a   |  X
   e   |  ERROR
------------------------

#reg 0 n

DO
  =n
  1
  FOR n
    @n *
  NEXT
  STOP
LOOP

  Step | Code | Keys
------------------------
  000  |  40  | STO 0
  001  |  01  | 1 
  002  |  60  | RCL 0
  003  |  12  | * 
  004  |  5D  | F L0 
  005  |  02  | 0 2
  006  |  50  | STOP 
  007  |  51  | GOTO 
  008  |  00  | 0 0
------------------------

  Reg  |  Name
------------------------
   0   |  N
------------------------

STEP   KEY
 n     +
 n+1   GOTO
 n+2   n

Prog i               Prog reference
STEP   KEY           STEP   KEY
 00    RCL i          00    NOP
 01    STO i          01    NOP
 02    L3             02    L3
 03    00             03    00
 04    STOP           04    STOP