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+;;; "limit.scm" Scheme Implementation of one-side limit algorithm.
+;Copyright 2005 Aubrey Jaffer
+;
+;Permission to copy this software, to modify it, to redistribute it,
+;to distribute modified versions, and to use it for any purpose is
+;granted, subject to the following restrictions and understandings.
+;
+;1.  Any copy made of this software must include this copyright notice
+;in full.
+;
+;2.  I have made no warranty or representation that the operation of
+;this software will be error-free, and I am under no obligation to
+;provide any services, by way of maintenance, update, or otherwise.
+;
+;3.  In conjunction with products arising from the use of this
+;material, there shall be no use of my name in any advertising,
+;promotional, or sales literature without prior written consent in
+;each case.
+
+;;@code{(require 'limit)}
+
+(define (inv-root f1 f2 f3 prec)
+  (define f1^2 (* f1 f1))
+  (define f2^2 (* f2 f2))
+  (define f3^2 (expt f3 2))
+  (require 'root)			; SLIB
+  (newton:find-root (lambda (f0)
+		      (+ (- (* (expt f0 2) f1))
+			 (* f0 f1^2)
+			 (* (- (* 2 (expt f0 2)) (* 3 f1^2)) f2)
+			 (* (+ (- (* 2 f0)) (* 3 f1)) f2^2)
+			 (* (- (+ (- (expt f0 2)) (* 2 f1^2)) f2^2)
+			    f3)
+			 (* (+ (- f0 (* 2 f1)) f2) f3^2)))
+		    (lambda (f0)
+		      (+ (- (+ (* -2 f0 f1) f1^2 (* 4 f0 f2))
+			    (* 2 f2^2)
+			    (* 2 f0 f3))
+			 f3^2))
+		    f1
+		    prec))
+
+(define (invintp f1 f2 f3)
+  (define f1^2 (* f1 f1))
+  (define f2^2 (* f2 f2))
+  (define f3^2 (expt f3 2))
+  (let ((c (+ (* -3 f1^2 f2)
+	      (* 3 f1 f2^2)
+	      (* (- (* 2 f1^2) f2^2) f3)
+	      (* (- f2 (* 2 f1)) f3^2)))
+	(b (+ (- f1^2 (* 2 f2^2)) f3^2))
+	(a (- (* 2 f2) f1 f3)))
+    (define disc (- (* b b) (* 4 a c)))
+    ;;(printf "discriminant: %g\n" disc)
+        (if (negative? (real-part disc))
+	(/ b -2 a)
+	(let ((sqrt-disc (sqrt disc)))
+	  (define root+ (/ (- sqrt-disc b) 2 a))
+	  (define root- (/ (+ sqrt-disc b) -2 a))
+	  (if (< (magnitude (- root+ f1)) (magnitude (- root- f1)))
+	      root+
+	      root-)))))
+
+(define (extrapolate-0 fs)
+  (define n (length fs))
+  (define (choose n k)
+    (do ((kdx 1 (+ 1 kdx))
+	 (prd 1 (/ (* (- n kdx -1) prd) kdx)))
+	((> kdx k) prd)))
+  (do ((k 1 (+ 1 k))
+       (lst fs (cdr lst))
+       (L 0 (+ (* -1 (expt -1 k) (choose n k) (car lst)) L)))
+      ((null? lst) L)))
+
+(define (sequence->limit proc sequence)
+  (define lval (proc (car sequence)))
+  (if (finite? lval)
+      (let ((val (proc (cadr sequence))))
+	(define h_n*nsamps (* (length sequence) (magnitude (- val lval))))
+	(if (finite? val)
+	    (let loop ((sequence (cddr sequence))
+		       (fxs (list val lval))
+		       (trend #f)
+		       (ldelta (- val lval))
+		       (jdx (+ -1 (length sequence))))
+	      (cond ((null? sequence)
+		     (case trend
+		       ((diverging) (and (real? val) (/ ldelta 0.0)))
+		       ((bounded) (invintp val lval (caddr fxs)))
+		       (else (cond ((zero? ldelta) val)
+				   ((not (real? val)) #f)
+				   (else (extrapolate-0 fxs))))))
+		    (else
+		     (set! lval val)
+		     (set! val (proc (car sequence)))
+		     ;;(printf "f(%12g)=%12g; delta=%12g hyp=%12g j=%3d %s\n" (car sequence) val (- val lval) (/ h_n*nsamps jdx) jdx (or trend ""))
+		     (if (finite? val)
+			 (let ((delta (- val lval)))
+			   (define h_j (/ h_n*nsamps jdx))
+			   (cond ((case trend
+				    ((converging) (<= (magnitude delta) h_j))
+				    ((bounded)    (<= (magnitude ldelta) (magnitude delta)))
+				    ((diverging)  (>= (magnitude delta) h_j))
+				    (else #f))
+				  (loop (cdr sequence) (cons val fxs) trend delta (+ -1 jdx)))
+				 (trend #f)
+				 (else
+				  (loop (cdr sequence) (cons val fxs)
+					(cond ((> (magnitude delta) h_j) 'diverging)
+					      ((< (magnitude ldelta) (magnitude delta)) 'bounded)
+					      (else 'converging))
+					delta (+ -1 jdx)))))
+			 (and (eq? trend 'diverging) val)))))
+	    (and (real? val) val)))
+      (and (real? lval) lval)))
+
+(define (limit proc x1 x2 . k)
+  (set! k (if (null? k) 8 (car k)))
+  (cond ((not (finite? x2)) (slib:error 'limit 'infinite 'x2 x2))
+	((not (finite? x1))
+	 (or (positive? (* x1 x2)) (slib:error 'limit 'start 'mismatch x1 x2))
+	 (limit (lambda (x) (proc (/ x))) 0.0 (/ x2) k))
+	((= x1 (+ x1 x2)) (slib:error 'limit 'null 'range x1 (+ x1 x2)))
+	(else (let ((dec (/ x2 k)))
+		(do ((x (+ x1 x2 0.0) (- x dec))
+		     (cnt (+ -1 k) (+ -1 cnt))
+		     (lst '() (cons x lst)))
+		    ((negative? cnt)
+		     (sequence->limit proc (reverse lst))))))))