This file is indexed.

/usr/share/libctl/base/math-utils.scm is in libctl5 3.2.2-2.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
; libctl: flexible Guile-based control files for scientific software 
; Copyright (C) 1998-2014 Massachusetts Institute of Technology and Steven G. Johnson
;
; This library is free software; you can redistribute it and/or
; modify it under the terms of the GNU Lesser General Public
; License as published by the Free Software Foundation; either
; version 2 of the License, or (at your option) any later version.
;
; This library is distributed in the hope that it will be useful,
; but WITHOUT ANY WARRANTY; without even the implied warranty of
; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
; Lesser General Public License for more details.
; 
; You should have received a copy of the GNU Lesser General Public
; License along with this library; if not, write to the
; Free Software Foundation, Inc., 59 Temple Place - Suite 330,
; Boston, MA  02111-1307, USA.
;
; Steven G. Johnson can be contacted at stevenj@alum.mit.edu.

; ****************************************************************
; Miscellaneous math utilities

; Return the arithmetic sequence (list): start start+step ... (n values)
(define (arith-sequence start step n)
  (define (s x n L) ; tail-recursive helper function
    (if (= n 0)
      L
      (s (binary+ x step) (- n 1) (cons x L))))
  (reverse (s start n '())))

; Given a list of numbers, linearly interpolates n values between
; each pair of numbers.
(define (interpolate n nums)
  (map 
   unary->inexact
   (cons 
    (car nums)
    (fold-right
     append '()
     (map
      (lambda (x y)
	(reverse (arith-sequence y (binary/ (binary- x y) (+ n 1)) (+ n 1))))
      (reverse (cdr (reverse nums))) ; nums w/o last value
      (cdr nums)))))) ; nums w/o first value

; Like interpolate, except only interpolates n values *on average*
; between each pair of numbers.  The actual number of interpolated
; points varies for each pair to try to keep the density of points
; uniform.
(define (interpolate-uniform n nums)
  (define meandiff
    (/ (fold-left + 0 (map unary-abs (map binary- (cdr nums)
					  (reverse (cdr (reverse nums))))))
       (length (cdr nums))))
  (map unary->inexact
       (if (zero? n)
	   nums
	   (cons
	    (car nums)
	    (fold-right
	     append '()
	     (map
	      (lambda (x y)
		(let ((m (inexact->exact (round
			  (+ -0.5 (* (+ n 1) (/ (unary-abs (binary- x y))
						meandiff)))))))
		  (reverse (arith-sequence y (binary/ (binary- x y) (+ m 1)) 
					   (+ m 1)))))
	      (reverse (cdr (reverse nums))) ; nums w/o last value
	      (cdr nums))))))) ; nums w/o first value

; ****************************************************************
; Minimization and root-finding utilities (useful in ctl scripts)

; The routines are:
;    minimize: minimize a function of one argument
;    minimize-multiple: minimize a function of multiple arguments
;    maximize, maximize-multiple : as above, but maximize
;    find-root: find the root of a function of one argument
; All routines use quadratically convergent methods.

; ****************************************************************

(define min-arg car)
(define min-val cdr)
(define max-arg min-arg)
(define max-val min-val)

; One-dimensional minimization (using Brent's method):

; (minimize f tol) : minimize (f x) with fractional tolerance tol
; (minimize f tol guess) : as above, but gives starting guess
; (minimize f tol x-min x-max) : as above, but gives range to optimize in
;                                (this is preferred)
; All variants return a result that contains both the argument and the
; value of the function at its minimum.
;      (min-arg result) : the argument of the function at its minimum
;      (min-val result) : the value of the function at its minimum

(define (minimize f tol . min-max)
  (define (midpoint a b) (* 0.5 (+ a b)))

  (define (quadratic-min-denom x a b fx fa fb)
    (magnitude (* 2.0 (- (* (- x a) (- fx fb)) (* (- x b) (- fx fa))))))
  (define (quadratic-min-num x a b fx fa fb)
    (let ((den (* 2.0 (- (* (- x a) (- fx fb)) (* (- x b) (- fx fa)))))
	  (num (- (* (- x a) (- x a) (- fx fb))
		  (* (- x b) (- x b) (- fx fa)))))
      (if (> den 0) (- num) num)))

  (define (tol-scale x) (* tol (+ (magnitude x) 1e-6)))
  (define (converged? x a b)
    (<= (magnitude (- x (midpoint a b))) (- (* 2 (tol-scale x)) (* 0.5 (- b a)))))
  
  (define golden-ratio (* 0.5 (- 3 (sqrt 5))))
  (define (golden-interpolate x a b)
    (* golden-ratio (if (>= x (midpoint a b)) (- a x) (- b x))))

  (define (sign x) (if (< x 0) -1 1))

  (define (brent-minimize x a b v w fx fv fw prev-step prev-prev-step)
    (define (guess-step proposed-step)
      (let ((step (if (> (magnitude proposed-step) (tol-scale x))
		      proposed-step
		      (* (tol-scale x) (sign proposed-step)))))
	(let ((u (+ x step)))
	  (let ((fu (f u)))
	    (if (<= fu fx)
		(if (> u x)
		    (brent-minimize u x b w x fu fw fx step prev-step)
		    (brent-minimize u a x w x fu fw fx step prev-step))
		(let ((new-a (if (< u x) u a))
		      (new-b (if (< u x) b u)))
		  (if (or (<= fu fw) (= w x))
		      (brent-minimize x new-a new-b w u fx fw fu
				      step prev-step)
		      (if (or (<= fu fv) (= v x) (= v w))
			  (brent-minimize x new-a new-b u w fx fu fw
					  step prev-step)
			  (brent-minimize x new-a new-b v w fx fv fw
					  step prev-step)))))))))
	      
    (if (converged? x a b)
	(cons x fx)
	(if (> (magnitude prev-prev-step) (tol-scale x))
	    (let ((p (quadratic-min-num x v w fx fv fw))
		  (q (quadratic-min-denom x v w fx fv fw)))
	      (if (or (>= (magnitude p) (magnitude (* 0.5 q prev-prev-step)))
		      (< p (* q (- a x))) (> p (* q (- b x))))
		  (guess-step (golden-interpolate x a b))
		  (guess-step (/ p q))))
	    (guess-step (golden-interpolate x a b)))))

  (define (bracket-minimum a b c fa fb fc)
    (if (< fb fc)
	(list a b c fa fb fc)
	(let ((u (/ (quadratic-min-num b a c fb fa fc)
		    (max (quadratic-min-denom b a c fb fa fc) 1e-20)))
	      (u-max (+ b (* 100 (- c b)))))
	  (cond
	   ((positive? (* (- b u) (- u c)))
	    (let ((fu (f u)))
	      (if (< fu fc)
		  (bracket-minimum b u c fb fu fc)
		  (if (> fu fb)
		      (bracket-minimum a b u fa fb fu)
		      (bracket-minimum b c (+ c (* 1.6 (- c b)))
				       fb fc (f (+ c (* 1.6 (- c b)))))))))
	   ((positive? (* (- c u) (- u u-max)))
	    (let ((fu (f u)))
	      (if (< fu fc)
		  (bracket-minimum c u (+ c (* 1.6 (- c b)))
				   fc fu (f (+ c (* 1.6 (- c b)))))
		  (bracket-minimum b c u fb fc fu))))
	   ((>= (* (- u u-max) (- u-max c)) 0)
	    (bracket-minimum b c u-max fb fc (f u-max)))
	   (else
	    (bracket-minimum b c (+ c (* 1.6 (- c b)))
			     fb fc (f (+ c (* 1.6 (- c b))))))))))

   (if (= (length min-max) 2)
       (let ((x-min (first min-max))
	     (x-max (second min-max)))
	 (let ((xm (midpoint x-min x-max)))
	   (let ((fm (f xm)))
	     (brent-minimize xm x-min x-max xm xm fm fm fm 0 0))))
       (let ((a (if (= (length min-max) 1) (first min-max) 1.0)))
	 (let ((b (if (= a 0) 1.0 0)))
	   (let ((fa (f a)) (fb (f b)))
	     (let ((aa (if (> fb fa) b a))
		   (bb (if (> fb fa) a b))
		   (faa (max fa fb))
		   (fbb (max fa fb)))
	       (let ((bracket
		      (bracket-minimum aa bb (+ bb (* 1.6 (- bb aa)))
				       faa fbb (f (+ bb (* 1.6 (- bb aa)))))))
		 (brent-minimize
		  (second bracket)
		  (min (first bracket) (third bracket))
		  (max (first bracket) (third bracket))
		  (first bracket)
		  (third bracket)
		  (fifth bracket)
		  (fourth bracket)
		  (sixth bracket)
		  0 0))))))))

; ****************************************************************

; (minimize-multiple f tol arg1 arg2 ... argN) :
;      Minimize a function f of N arguments, given the fractional tolerance
; desired and initial guesses for the arguments.
;
; (min-arg result) : list of argument values at the minimum
; (min-val result) : list of function values at the minimum

(define (minimize-multiple-expert f tol max-iters fmin guess-args arg-scales)
  (let ((best-val 1e20) (best-args '()))
    (subplex
     (lambda (args)
       (let ((val (apply f args)))
	 (if (or (null? best-args) (< val best-val))
	     (begin
	       (print "extremization: best so far is " 
			     val " at " args "\n")
	       (set! best-val val)
	       (set! best-args args)))
	 val))
     guess-args tol max-iters
     (if fmin fmin 0.0) (if fmin true false)
     arg-scales)))

(define (minimize-multiple f tol . guess-args)
  (minimize-multiple-expert f tol 999999999 false guess-args '(0.1)))

; Yet another alternate multi-dimensional minimization (Simplex algorithm).
(define (simplex-minimize-multiple f tol . guess-args)
  (let ((simplex-result (simplex-minimize f guess-args tol)))
    (cons (simplex-point-x simplex-result)
	  (simplex-point-val simplex-result))))

; Alternate multi-dimensional minimization (using Powell's method):
; (not the default since it seems to have convergence problems sometimes)

(define (powell-minimize-multiple f tol . guess-args)
  (define (create-unit-vector i n)
    (let ((v (make-vector n 0)))
      (vector-set! v i 1)
      v))
  (define (initial-directions n)
    (make-initialized-list n (lambda (i) (create-unit-vector i n))))

  (define (v- v1 v2) (vector-map - v1 v2))
  (define (v+ v1 v2) (vector-map + v1 v2))
  (define (v* s v) (vector-map (lambda (x) (* s x)) v))
  (define (v-dot v1 v2) (vector-fold-right + 0 (vector-map * v1 v2)))
  (define (v-norm v) (sqrt (v-dot v v)))
  (define (unit-v v) (v* (/ (v-norm v)) v))

  (define (fv v) (apply f (vector->list v)))
  (define guess-vector (list->vector guess-args))
  (define (f-dir p0 dir) (lambda (x) (fv (v+ p0 (v* x dir)))))

  (define (minimize-dir p0 dir)
    (let ((min-result (minimize (f-dir p0 dir) tol)))
      (cons
       (v+ p0 (v* (min-arg min-result) dir))
       (min-val min-result))))

  (define (minimize-dirs p0 dirs)
    (if (null? dirs)
	(cons p0 '())
	(let ((min-result (minimize-dir p0 (car dirs))))
	  (let ((min-results (minimize-dirs (min-arg min-result) (cdr dirs))))
	    (cons (min-arg min-results)
		  (cons (min-val min-result) (min-val min-results)))))))

  (define (replace= val vals els el)
    (if (null? els) '()
	(if (= (car vals) val)
	    (cons el (cdr els))
	    (cons (car els) (replace= val (cdr vals) (cdr els) el)))))
  
  ; replace direction where largest decrease occurred:
  (define (update-dirs decreases dirs p0 p)
    (replace= (apply max decreases) decreases dirs (v- p p0)))

  (define (minimize-aux p0 fp0 dirs)
    (let ((min-results (minimize-dirs p0 dirs)))
      (let ((decreases (map (lambda (val) (- fp0 val)) (min-val min-results)))
	    (p (min-arg min-results))
	    (fp (first (reverse (min-val min-results)))))
	(if (<= (v-norm (v- p p0))
		(* tol 0.5 (+ (v-norm p) (v-norm p0) 1e-20)))
	    (cons (vector->list p) fp)
	    (let ((min-result (minimize-dir p (v- p p0))))
	      (minimize-aux (min-arg min-result) (min-val min-result)
			    (update-dirs decreases dirs p0 p)))))))

  (minimize-aux guess-vector (fv guess-vector)
		(initial-directions (length guess-args))))

; Maximization variants of the minimize functions:

(define (maximize f tol . min-max)
  (let ((result (apply minimize (append (list (compose - f) tol) min-max))))
    (cons (min-arg result) (- (min-val result)))))

(define (maximize-multiple f tol . guess-args)
  (let ((result (apply minimize-multiple
		       (append (list (compose - f) tol) guess-args))))
    (cons (min-arg result) (- (min-val result)))))

; ****************************************************************
; Find a root of a function of one argument using Ridder's method.

; (find-root f tol x-min x-max) : returns the root of the function (f x),
; within a fractional tolerance tol.  x-min and x-max must bracket the
; root; that is, (f x-min) must have a different sign than (f x-max).

(define (find-root f tol x-min x-max)
  (define (midpoint a b) (* 0.5 (+ a b)))
  (define (sign x) (if (< x 0) -1 1))
  
  (define (best-bracket a b x1 x2 fa fb f1 f2)
    (if (positive? (* f1 f2))
	(if (positive? (* fa f1))
	    (list (max x1 x2) b (if (> x1 x2) f1 f2) fb)
	    (list a (min x1 x2) fa (if (< x1 x2) f1 f2)))
	(if (< x1 x2)
	    (list x1 x2 f1 f2)
	    (list x2 x1 f2 f1))))

  (define (converged? a b x) (< (min (magnitude (- x a)) (magnitude (- x b))) 
				(* tol (magnitude x))))
  
  ; find the root by Ridder's method:
  (define (ridder a b fa fb)
    (if (or (= fa 0) (= fb 0))
	(if (= fa 0) a b)
	(begin
	  (if (> (* fa fb) 0)
	      (error "x-min and x-max in find-root must bracket the root!"))
	  (let ((m (midpoint a b)))
	    (let ((fm (f m)))
	      (let ((x (+ m (/ (* (- m a) (sign (- fa fb)) fm)
			       (sqrt (- (* fm fm) (* fa fb)))))))
		(if (or (= fm 0) (converged? a b x))
		    (if (= fm 0) m x)
		    (let ((fx (f x)))
		      (apply ridder (best-bracket a b x m fa fb fx fm))))))))))
	
  (ridder x-min x-max (f x-min) (f x-max)))

; ****************************************************************
; Find a root by Newton's method with bounds and bisection,
; given a function f that returns a pair of (value . derivative)

(define (find-root-deriv f tol x-min x-max . x-guess)
  ; Some trickiness: we only need to evaluate the function at x-min and
  ; x-max if a Newton step fails, and even then only if we haven't already
  ; bracketed the root, so do this via lazy evaluation.
  (define f-memo (memoize f))
  (define (lazy x) (if (number? x) x (x)))
  (define (pick-bound which?)
    (lambda ()
      (let ((fmin-pair (f-memo x-min)) (fmax-pair (f-memo x-max)))
	(let ((fmin (car fmin-pair)) (fmax (car fmax-pair)))
	  (if (which? fmin) x-min
	      (if (which? fmax) x-max
		  (error "failed to bracket the root in find-root-deriv")))))))

  (define (in-bounds? x f df a b)
    (negative? (* (- f (* df (- x a)))
		  (- f (* df (- x b))))))
	  
  (define (newton x a b dx)
    (if (< (abs dx) (abs (* tol x)))
	x
	(let ((fx-pair (f-memo x)))
	  (let ((f (car fx-pair)) (df (cdr fx-pair)))
	    (if (= f 0)
		x
		(let ((a' (if (< f 0) x a)) (b' (if (> f 0) x b)))
		  (if (and (not (= dx (- x-max x-min)))
			   (negative? (* dx (/ f df)))
			   (positive? (* (car (f-memo (lazy a')))
					 (car (f-memo (lazy b'))))))
		      (error "failed to bracket the root in find-root-deriv"))
		  (if (and (if (and (number? a) (number? b))
			       (in-bounds? x f df a b)
			       (in-bounds? x f df x-min x-max))
;			   (> (abs (* 0.5 dx df)) (abs f))
			   )
		      (newton (- x (/ f df)) a' b' (/ f df))
		      (let ((av (lazy a)) (bv (lazy b)))
			(let ((dx' (* 0.5 (- bv av)))
			      (a'' (if (eq? a a') av a'))
			      (b'' (if (eq? b b') bv b')))
			  (newton (* (+ av bv) 0.5) a'' b'' dx'))))))))))

  (newton (if (null? x-guess) (* (+ x-min x-max) 0.5) (car x-guess))
	  (pick-bound negative?)
	  (pick-bound positive?)
	  (- x-max x-min)))

; ****************************************************************

; Numerical differentiation:
;   Compute the numerical derivative of a function f at x, using
; Ridder's method of polynomial extrapolation, described e.g. in
; Numerical Recipes in C (section 5.7).

; This is the basic routine, but we wrap it in another interface below
; so that dx and tol can be optional arguments.
(define (do-derivative f x dx tol)

  ; Using Neville's algorithm, compute successively higher-order
  ; extrapolations of the derivative (the "Neville tableau"):
  (define (deriv-a a0 prev-a fac fac0)
    (if (null? prev-a)
	(list a0)
	(cons a0 (deriv-a (binary/
			   (binary- (binary* a0 fac) (car prev-a)) 
			   (- fac 1))
			  (cdr prev-a) (* fac fac0) fac0))))
  
  (define (deriv dx df0 err0 prev-a fac0)
    (let ((a (deriv-a (binary/ (binary- (f (+ x dx)) (f (- x dx))) (* 2 dx))
		      prev-a fac0 fac0)))
      (if (null? prev-a)
	  (deriv (/ dx (sqrt fac0)) (car a) err0 a fac0)
	  (let* ((errs
		  (map max
		       (map unary-abs (map binary- (cdr a) (reverse (cdr (reverse a)))))
		       (map unary-abs (map binary- (cdr a) prev-a))))
		 (errmin (apply min errs))
		 (err (min errmin err0))
		 (df (if (> err err0)
			 df0
			 (cdr (assoc errmin (map cons errs (cdr a)))))))
	    (if (or (<= err (* tol (unary-abs df)) )
		    (> (unary-abs (binary- (car (reverse a)) (car (reverse prev-a))))
		       (* 2 err)))
		(list df err)
		(deriv (/ dx (sqrt fac0)) df err a fac0))))))

  (deriv dx 0 1e30 '() 2))
      
(define (do-derivative-wrap do-deriv f x dx-and-tol)
  (let ((dx (if (> (length dx-and-tol) 0)
		(car dx-and-tol)
		(max (magnitude (* x 0.01)) 0.01)))
	(tol (if (> (length dx-and-tol) 1)
		 (cadr dx-and-tol)
		 0)))
    (do-deriv f x dx tol)))

(define derivative-df car)
(define derivative-df-err cadr)
(define derivative-d2f caddr)
(define derivative-d2f-err cadddr)

(define (derivative f x . dx-and-tol)
  (do-derivative-wrap do-derivative f x dx-and-tol))
(define (deriv f x . dx-and-tol)
  (derivative-df (do-derivative-wrap do-derivative f x dx-and-tol)))

; Compute both the first and second derivatives at the same time
; (using minimal extra function evaluations).
(define (derivative2 f x . dx-and-tol)
  (define f-memo (memoize f))
  (define (f-deriv y)
    (binary* (binary- (f-memo y) (f-memo x)) (/ 2 (- y x))))
  (append
   (do-derivative-wrap do-derivative f-memo x dx-and-tol)
   (do-derivative-wrap do-derivative f-deriv x dx-and-tol)))

(define (deriv2 f x . dx-and-tol)
  (derivative-d2f (apply derivative2 (cons f (cons x dx-and-tol)))))


; Below, we have variants of the above routine which only compute the
; *one-sided* derivative df/dx for dx > 0.  (Adapted from Ridder's
; algorithm by SGJ.  Note that these are generally less accurate
; than the ordinary two-sided derivative, above.)

(define (do-derivative+ f x dx tol)

  ; Using Neville's algorithm, compute successively higher-order
  ; extrapolations of the derivative (the "Neville tableau"):
  (define (deriv-a a0 prev-a fac fac0)
    (if (null? prev-a)
	(list a0)
	(cons a0 (deriv-a (binary/
			   (binary- (binary* a0 fac) (car prev-a)) 
			   (- fac 1))
			  (cdr prev-a) (* fac fac0) fac0))))
  (define fx (f x))
  
  (define (deriv dx df0 err0 prev-a fac0)
    (let ((a (deriv-a (binary/ (binary- (f (+ x dx)) fx) dx)
		      prev-a fac0 fac0)))
      (if (null? prev-a)
	  (deriv (/ dx fac0) (car a) err0 a fac0)
	  (let* ((errs
		  (map max
		       (map unary-abs (map binary- (cdr a) (reverse (cdr (reverse a)))))
		       (map unary-abs (map binary- (cdr a) prev-a))))
		 (errmin (apply min errs))
		 (err (min errmin err0))
		 (df (if (> err err0)
			 df0
			 (cdr (assoc errmin (map cons errs (cdr a)))))))
	    (if (or (< err (* tol (unary-abs df)) )
		    (> (unary-abs (binary- (car (reverse a)) (car (reverse prev-a))))
		       (* 2 err)))
		(list df err)
		(deriv (/ dx fac0) df err a fac0))))))

  (deriv dx 0 1e30 '() (sqrt 2)))
      
; Compute both the first and second derivatives at the same time
; (using minimal extra function evaluations).
(define (do-derivative-wrap2+ do-deriv only2? f x dx-and-tol)
  (define f-memo (memoize f))
  (define (f-deriv y)
    (if (= y x) 
	0.0
	(binary* (binary+ (f-memo y) 
			  (binary- (f-memo x)
				   (binary* 2.0 (f-memo (* 0.5 (+ x y))))))
		 (/ 4 (- y x)))))
  (append
   (if only2? 
       (list 0 0)
       (do-derivative-wrap do-deriv f-memo x dx-and-tol))
   (do-derivative-wrap do-deriv f-deriv x dx-and-tol)))

(define (derivative+ f x . dx-and-tol)
  (do-derivative-wrap do-derivative+ f x dx-and-tol))
(define (deriv+ f x . dx-and-tol)
  (derivative-df (do-derivative-wrap do-derivative+ f x dx-and-tol)))
(define (derivative2+ f x . dx-and-tol)
  (do-derivative-wrap2+ do-derivative+ false f x dx-and-tol))
(define (deriv2+ f x . dx-and-tol)
  (derivative-d2f (do-derivative-wrap2+ do-derivative+ true f x dx-and-tol)))

; as do-derivative+, but taking derivative from left
(define (do-derivative- f x dx tol)
  (do-derivative+ f x (- dx) tol))
(define (derivative- f x . dx-and-tol)
  (do-derivative-wrap do-derivative- f x dx-and-tol))
(define (deriv- f x . dx-and-tol)
  (derivative-df (do-derivative-wrap do-derivative- f x dx-and-tol)))
(define (derivative2- f x . dx-and-tol)
  (do-derivative-wrap2+ do-derivative- false f x dx-and-tol))
(define (deriv2- f x . dx-and-tol)
  (derivative-d2f (do-derivative-wrap2+ do-derivative- true f x dx-and-tol)))

; ****************************************************************

; Some simple integration routines using an adaptive trapezoidal rule
; (see e.g. Numerical Recipes, Sec. 4.2).  It might be nice to have
; Gaussian quadratures and what-not, but on the other hand the
; functions we are integrating may well be the result of a computation
; on a finite grid (somehow interpolated), and so will not be smooth.
; Also, implementing thse simple algorithms in Scheme lets us use our
; polymorphic arithmetic functions so that we can easily integrate
; real, complex, and vector-valued functions.
;
; UPDATE: quadrature/cubature rules are now implemented via C

; Integrate the 1d function (f x) from x=a..b to within the specified
; fractional tolerance.
(define (integrate-1d f a b tol)
  (define (pow2 n) (if (<= n 0) 1 (* 2 (pow2 (- n 1))))) ; 2^n
  (define (trap0 n sum)
    (binary*
     0.5
     (binary+
      sum
      (if (<= n 1)
	  (binary* (- b a) (binary+ (f a) (f b)))
	  (let ((steps (pow2 (- n 2))))
	    (let ((dx (/ (- b a) steps)))
	      (binary* 
	       dx
	       (do ((cur-sum 0) (i 0 (+ i 1)) (x (+ a dx) (+ x dx)))
		   ((>= i steps) cur-sum)
		 (set! cur-sum (binary+ cur-sum (f x)))))))))))
  (define (trap n sum)
    (let ((newsum (trap0 n sum)))
      (if (and (> n 5)
	       (or (> n 20) 
		   (binary= newsum sum)
		    (< (unary-abs (binary- newsum sum))
		       (* tol (unary-abs newsum)))))
	  newsum
	  (trap (+ n 1) newsum))))
  (trap 1 0.0))
	  
; Integrate the multi-dimensional function f from a..b, within the
; specified tolerance.  a and b are either numbers (for 1d integrals),
; or vectors/lists of the same length giving the bounds in each dimension.
;  NOTE: this is our *old* routine that uses the trapezoidal rule
(define (integrate-old f a b tol)
  (define (int f a b)
    (if (null? a)
	(f)
	(integrate-1d
	 (lambda (x) (int (lambda (. y) (apply f (cons x y))) (cdr a) (cdr b)))
	 (car a) (car b) tol)))
  (cond
   ((and (vector? a) (vector? b))
    (integrate-old f (vector->list a) (vector->list b) tol))
   ((and (number? a) (number? b))
    (integrate-old f (list a) (list b) tol))
   (else (int f a b))))

; As above, but use adaptive cubature rules in integrator.c
; Optionally, can take absolute tolerance and max # function evals as args.
(define (integrate f a b reltol . abstol-and-maxnfe)
  (define (to-list x)
    (cond ((number? x) (list x))
	  ((vector? x) (vector->list x))
	  (else x)))
  ((if (defined? 'cadaptive-integration) 
       cadaptive-integration ; only compiled when complex nums are available
       adaptive-integration)
   (lambda (x) (apply f x)) 
   (to-list a) (to-list b) 
   (if (null? abstol-and-maxnfe) 0.0 (car abstol-and-maxnfe))
   reltol
   (if (< (length abstol-and-maxnfe) 2) 0 (cadr abstol-and-maxnfe))))

; ****************************************************************