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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.roots.bad_guess"></a><a class="link" href="bad_guess.html" title="The Effect of a Poor Initial Guess">The Effect of a Poor Initial
      Guess</a>
</h3></div></div></div>
<p>
        It's instructive to take our "toy" example algorithms, and use
        deliberately bad initial guesses to see how the various root finding algorithms
        fair. We'll start with the cubed root, and using the cube root of 500 as
        the test case:
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  Initial Guess=
                </p>
              </th>
<th>
                <p>
                  -500% (&#8776;1.323)
                </p>
              </th>
<th>
                <p>
                  -100% (&#8776;3.97)
                </p>
              </th>
<th>
                <p>
                  -50% (&#8776;3.96)
                </p>
              </th>
<th>
                <p>
                  -20% (&#8776;6.35)
                </p>
              </th>
<th>
                <p>
                  -10% (&#8776;7.14)
                </p>
              </th>
<th>
                <p>
                  -5% (&#8776;7.54)
                </p>
              </th>
<th>
                <p>
                  5% (&#8776;8.33)
                </p>
              </th>
<th>
                <p>
                  10% (&#8776;8.73)
                </p>
              </th>
<th>
                <p>
                  20% (&#8776;9.52)
                </p>
              </th>
<th>
                <p>
                  50% (&#8776;11.91)
                </p>
              </th>
<th>
                <p>
                  100% (&#8776;15.87)
                </p>
              </th>
<th>
                <p>
                  500 (&#8776;47.6)
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  bracket_and_solve_root
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  10
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  13
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  newton_iterate
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  9
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  halley_iterate
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  schroder_iterate
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        As you can see <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
        is relatively insensitive to starting location - as long as you don't start
        many orders of magnitude away from the root it will take roughly the same
        number of steps to bracket the root and solve it. On the other hand the derivative-based
        methods are slow to start, but once they have some digits correct they increase
        precision exceptionally fast: they are therefore quite sensitive to the initial
        starting location.
      </p>
<p>
        The next table shows the number of iterations required to find the second
        radius of an ellipse with first radius 50 and arc-length 500:
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  Initial Guess=
                </p>
              </th>
<th>
                <p>
                  -500% (&#8776;20.6)
                </p>
              </th>
<th>
                <p>
                  -100% (&#8776;61.81)
                </p>
              </th>
<th>
                <p>
                  -50% (&#8776;61.81)
                </p>
              </th>
<th>
                <p>
                  -20% (&#8776;98.9)
                </p>
              </th>
<th>
                <p>
                  -10% (&#8776;111.3)
                </p>
              </th>
<th>
                <p>
                  -5% (&#8776;117.4)
                </p>
              </th>
<th>
                <p>
                  5% (&#8776;129.8)
                </p>
              </th>
<th>
                <p>
                  10% (&#8776;136)
                </p>
              </th>
<th>
                <p>
                  20% (&#8776;148.3)
                </p>
              </th>
<th>
                <p>
                  50% (&#8776;185.4)
                </p>
              </th>
<th>
                <p>
                  100% (&#8776;247.2)
                </p>
              </th>
<th>
                <p>
                  500 (&#8776;741.7)
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  bracket_and_solve_root
                </p>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  9
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  10
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  newton_iterate
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  halley_iterate
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  schroder_iterate
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        Interestingly this function is much more resistant to a poor initial guess
        when using derivatives.
      </p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
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      Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
      Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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	6	<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
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	24	</div>
	25	<div class="section">
	26	<div class="titlepage"><div><div><h3 class="title">
	27	<a name="math_toolkit.roots.bad_guess"></a><a class="link" href="bad_guess.html" title="The Effect of a Poor Initial Guess">The Effect of a Poor Initial
	28	Guess</a>
	29	</h3></div></div></div>
	30	<p>
	31	It's instructive to take our "toy" example algorithms, and use
	32	deliberately bad initial guesses to see how the various root finding algorithms
	33	fair. We'll start with the cubed root, and using the cube root of 500 as
	34	the test case:
	35	</p>
	36	<div class="informaltable"><table class="table">
	37	<colgroup>
	38	<col>
	39	<col>
	40	<col>
	41	<col>
	42	<col>
	43	<col>
	44	<col>
	45	<col>
	46	<col>
	47	<col>
	48	<col>
	49	<col>
	50	<col>
	51	</colgroup>
	52	<thead><tr>
	53	<th>
	54	<p>
	55	Initial Guess=
	56	</p>
	57	</th>
	58	<th>
	59	<p>
	60	-500% (≈1.323)
	61	</p>
	62	</th>
	63	<th>
	64	<p>
65	-100% (≈3.97)
66	</p>
67	</th>
68	<th>
69	<p>
70	-50% (≈3.96)
71	</p>
72	</th>
73	<th>
74	<p>
75	-20% (≈6.35)
76	</p>
77	</th>
78	<th>
79	<p>
80	-10% (≈7.14)
81	</p>
82	</th>
83	<th>
84	<p>
85	-5% (≈7.54)
86	</p>
87	</th>
88	<th>
89	<p>
90	5% (≈8.33)
91	</p>
92	</th>
93	<th>
94	<p>
95	10% (≈8.73)
96	</p>
97	</th>
98	<th>
99	<p>
100	20% (≈9.52)
101	</p>
102	</th>
103	<th>
104	<p>
105	50% (≈11.91)
106	</p>
107	</th>
108	<th>
109	<p>
110	100% (≈15.87)
111	</p>
112	</th>
113	<th>
114	<p>
115	500 (≈47.6)
116	</p>
117	</th>
118	</tr></thead>
119	<tbody>
120	<tr>
121	<td>
122	<p>
123	bracket_and_solve_root
124	</p>
125	</td>
126	<td>
127	<p>
128	12
129	</p>
130	</td>
131	<td>
132	<p>
133	8
134	</p>
135	</td>
136	<td>
137	<p>
138	8
139	</p>
140	</td>
141	<td>
142	<p>
143	10
144	</p>
145	</td>
146	<td>
147	<p>
148	11
149	</p>
150	</td>
151	<td>
152	<p>
153	11
154	</p>
155	</td>
156	<td>
157	<p>
158	11
159	</p>
160	</td>
161	<td>
162	<p>
163	11
164	</p>
165	</td>
166	<td>
167	<p>
168	11
169	</p>
170	</td>
171	<td>
172	<p>
173	11
174	</p>
175	</td>
176	<td>
177	<p>
178	7
179	</p>
180	</td>
181	<td>
182	<p>
183	13
184	</p>
185	</td>
186	</tr>
187	<tr>
188	<td>
189	<p>
190	newton_iterate
191	</p>
192	</td>
193	<td>
194	<p>
195	12
196	</p>
197	</td>
198	<td>
199	<p>
200	7
201	</p>
202	</td>
203	<td>
204	<p>
205	7
206	</p>
207	</td>
208	<td>
209	<p>
210	5
211	</p>
212	</td>
213	<td>
214	<p>
215	5
216	</p>
217	</td>
218	<td>
219	<p>
220	4
221	</p>
222	</td>
223	<td>
224	<p>
225	4
226	</p>
227	</td>
228	<td>
229	<p>
230	5
231	</p>
232	</td>
233	<td>
234	<p>
235	5
236	</p>
237	</td>
238	<td>
239	<p>
240	6
241	</p>
242	</td>
243	<td>
244	<p>
245	7
246	</p>
247	</td>
248	<td>
249	<p>
250	9
251	</p>
252	</td>
253	</tr>
254	<tr>
255	<td>
256	<p>
257	halley_iterate
258	</p>
259	</td>
260	<td>
261	<p>
262	7
263	</p>
264	</td>
265	<td>
266	<p>
267	4
268	</p>
269	</td>
270	<td>
271	<p>
272	4
273	</p>
274	</td>
275	<td>
276	<p>
277	3
278	</p>
279	</td>
280	<td>
281	<p>
282	3
283	</p>
284	</td>
285	<td>
286	<p>
287	3
288	</p>
289	</td>
290	<td>
291	<p>
292	3
293	</p>
294	</td>
295	<td>
296	<p>
297	3
298	</p>
299	</td>
300	<td>
301	<p>
302	3
303	</p>
304	</td>
305	<td>
306	<p>
307	4
308	</p>
309	</td>
310	<td>
311	<p>
312	4
313	</p>
314	</td>
315	<td>
316	<p>
317	6
318	</p>
319	</td>
320	</tr>
321	<tr>
322	<td>
323	<p>
324	schroder_iterate
325	</p>
326	</td>
327	<td>
328	<p>
329	11
330	</p>
331	</td>
332	<td>
333	<p>
334	6
335	</p>
336	</td>
337	<td>
338	<p>
339	6
340	</p>
341	</td>
342	<td>
343	<p>
344	4
345	</p>
346	</td>
347	<td>
348	<p>
349	3
350	</p>
351	</td>
352	<td>
353	<p>
354	3
355	</p>
356	</td>
357	<td>
358	<p>
359	3
360	</p>
361	</td>
362	<td>
363	<p>
364	3
365	</p>
366	</td>
367	<td>
368	<p>
369	4
370	</p>
371	</td>
372	<td>
373	<p>
374	5
375	</p>
376	</td>
377	<td>
378	<p>
379	5
380	</p>
381	</td>
382	<td>
383	<p>
384	8
385	</p>
386	</td>
387	</tr>
388	</tbody>
389	</table></div>
390	<p>
391	As you can see <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
392	is relatively insensitive to starting location - as long as you don't start
393	many orders of magnitude away from the root it will take roughly the same
394	number of steps to bracket the root and solve it. On the other hand the derivative-based
395	methods are slow to start, but once they have some digits correct they increase
396	precision exceptionally fast: they are therefore quite sensitive to the initial
397	starting location.
398	</p>
399	<p>
400	The next table shows the number of iterations required to find the second
401	radius of an ellipse with first radius 50 and arc-length 500:
402	</p>
403	<div class="informaltable"><table class="table">
404	<colgroup>
405	<col>
406	<col>
407	<col>
408	<col>
409	<col>
410	<col>
411	<col>
412	<col>
413	<col>
414	<col>
415	<col>
416	<col>
417	<col>
418	</colgroup>
419	<thead><tr>
420	<th>
421	<p>
422	Initial Guess=
423	</p>
424	</th>
425	<th>
426	<p>
427	-500% (≈20.6)
428	</p>
429	</th>
430	<th>
431	<p>
432	-100% (≈61.81)
433	</p>
434	</th>
435	<th>
436	<p>
437	-50% (≈61.81)
438	</p>
439	</th>
440	<th>
441	<p>
442	-20% (≈98.9)
443	</p>
444	</th>
445	<th>
446	<p>
447	-10% (≈111.3)
448	</p>
449	</th>
450	<th>
451	<p>
452	-5% (≈117.4)
453	</p>
454	</th>
455	<th>
456	<p>
457	5% (≈129.8)
458	</p>
459	</th>
460	<th>
461	<p>
462	10% (≈136)
463	</p>
464	</th>
465	<th>
466	<p>
467	20% (≈148.3)
468	</p>
469	</th>
470	<th>
471	<p>
472	50% (≈185.4)
473	</p>
474	</th>
475	<th>
476	<p>
477	100% (≈247.2)
478	</p>
479	</th>
480	<th>
481	<p>
482	500 (≈741.7)
483	</p>
484	</th>
485	</tr></thead>
486	<tbody>
487	<tr>
488	<td>
489	<p>
490	bracket_and_solve_root
491	</p>
492	</td>
493	<td>
494	<p>
495	11
496	</p>
497	</td>
498	<td>
499	<p>
500	5
501	</p>
502	</td>
503	<td>
504	<p>
505	5
506	</p>
507	</td>
508	<td>
509	<p>
510	8
511	</p>
512	</td>
513	<td>
514	<p>
515	8
516	</p>
517	</td>
518	<td>
519	<p>
520	7
521	</p>
522	</td>
523	<td>
524	<p>
525	7
526	</p>
527	</td>
528	<td>
529	<p>
530	8
531	</p>
532	</td>
533	<td>
534	<p>
535	9
536	</p>
537	</td>
538	<td>
539	<p>
540	8
541	</p>
542	</td>
543	<td>
544	<p>
545	6
546	</p>
547	</td>
548	<td>
549	<p>
550	10
551	</p>
552	</td>
553	</tr>
554	<tr>
555	<td>
556	<p>
557	newton_iterate
558	</p>
559	</td>
560	<td>
561	<p>
562	4
563	</p>
564	</td>
565	<td>
566	<p>
567	4
568	</p>
569	</td>
570	<td>
571	<p>
572	4
573	</p>
574	</td>
575	<td>
576	<p>
577	3
578	</p>
579	</td>
580	<td>
581	<p>
582	3
583	</p>
584	</td>
585	<td>
586	<p>
587	3
588	</p>
589	</td>
590	<td>
591	<p>
592	3
593	</p>
594	</td>
595	<td>
596	<p>
597	3
598	</p>
599	</td>
600	<td>
601	<p>
602	3
603	</p>
604	</td>
605	<td>
606	<p>
607	4
608	</p>
609	</td>
610	<td>
611	<p>
612	4
613	</p>
614	</td>
615	<td>
616	<p>
617	4
618	</p>
619	</td>
620	</tr>
621	<tr>
622	<td>
623	<p>
624	halley_iterate
625	</p>
626	</td>
627	<td>
628	<p>
629	4
630	</p>
631	</td>
632	<td>
633	<p>
634	3
635	</p>
636	</td>
637	<td>
638	<p>
639	3
640	</p>
641	</td>
642	<td>
643	<p>
644	3
645	</p>
646	</td>
647	<td>
648	<p>
649	3
650	</p>
651	</td>
652	<td>
653	<p>
654	2
655	</p>
656	</td>
657	<td>
658	<p>
659	2
660	</p>
661	</td>
662	<td>
663	<p>
664	3
665	</p>
666	</td>
667	<td>
668	<p>
669	3
670	</p>
671	</td>
672	<td>
673	<p>
674	3
675	</p>
676	</td>
677	<td>
678	<p>
679	3
680	</p>
681	</td>
682	<td>
683	<p>
684	3
685	</p>
686	</td>
687	</tr>
688	<tr>
689	<td>
690	<p>
691	schroder_iterate
692	</p>
693	</td>
694	<td>
695	<p>
696	4
697	</p>
698	</td>
699	<td>
700	<p>
701	3
702	</p>
703	</td>
704	<td>
705	<p>
706	3
707	</p>
708	</td>
709	<td>
710	<p>
711	3
712	</p>
713	</td>
714	<td>
715	<p>
716	3
717	</p>
718	</td>
719	<td>
720	<p>
721	2
722	</p>
723	</td>
724	<td>
725	<p>
726	2
727	</p>
728	</td>
729	<td>
730	<p>
731	3
732	</p>
733	</td>
734	<td>
735	<p>
736	3
737	</p>
738	</td>
739	<td>
740	<p>
741	3
742	</p>
743	</td>
744	<td>
745	<p>
746	3
747	</p>
748	</td>
749	<td>
750	<p>
751	3
752	</p>
753	</td>
754	</tr>
755	</tbody>
756	</table></div>
757	<p>
758	Interestingly this function is much more resistant to a poor initial guess
759	when using derivatives.
760	</p>
761	</div>
762	<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
763	<td align="left"></td>
764	<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
765	Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
766	Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
767	Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
768	Distributed under the Boost Software License, Version 1.0. (See accompanying
769	file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
770	</p>
771	</div></td>
772	</tr></table>
773	<hr>
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