1

Homework Set #3, Math 475A

2

Part

3

I:

4

The-

5

ory

6

An-

7

swer

8

the

9

fol-

10

low-

11

ing

12

ques-

13

tions

14

as

15

rig-

16

or-

17

ously

18

as

19

pos-

20

si-

21

ble.

22

The

23

first

24

3

25

ques-

26

tions

27

come

28

from

29

Kin-

30

caid

31

&

32

Ch-

33

eney’s

34

book.

35

1.

36

Let

37

cn =

38

12(an+

39

bn),

40

r =

41

limn→∞ cn,

42

and

43

en =

44

r−

45

cn.

46

Here

47

[an, bn],

48

with

49

n ≥

50

0,

51

de-

52

notes

53

the

54

suc-

55

ces-

56

sive

57

in-

58

ter-

59

vals

60

that

61

arise

62

in

63

the

64

bi-

65

sec-

66

tion

67

method

68

when

69

it

70

is

71

ap-

72

plied

73

to

74

a

75

con-

76

tin-

77

u-

78

ous

79

func-

80

tion

81

f .

82

(a)

83

Show

84

that

85

|en| ≤

86

2−n(b1−

87

a1).

88

(b)

89

Show

90

that

91

en =

92

O(2−n

93

as

94

n→

95

∞.

96

(c)

97

Is

98

it

99

true

100

that

101

|e0 ≥

102

|e1| ≥

103

· · ·?

104

Ex-

105

plain.

106

(d)

107

Show

108

that

109

|cn−

110

cn+1| =

111

2−n−2(b0−

112

a0)

113

(f)

114

Show

115

that

116

for

117

all

118

n

119

and

120

m,

121

am ≤

122

bn.

123

(g)

124

Show

125

that

126

for

127

all

128

n,

129

[an+1, bn+1] ⊆

130

[an, bn].

131

2.

132

Give

133

an

134

ex-

135

am-

136

ple

137

in

138

which

139

a0 =

140

a1 <

141

a2 =

142

a3 <

143

a4 =

144

a5 <

145

a6 =

146

· · ·

147

3.

148

If

149

the

150

bi-

151

sec-

152

tion

153

method

154

is

155

used

156

in

157

IEEE

158

32

159

bit

160

sin-

161

gle

162

pre-

163

ci-

164

sion,

165

start-

166

ing

167

with

168

the

169

in-

170

ter-

171

val

172

[128, 129],

173

can

174

we

175

com-

176

pute

177

the

178

root

179

with

180

ab-

181

so-

182

lute

183

pre-

184

ci-

185

sion

186

<

187

10−6?

188

An-

189

swer

190

the

191

same

192

ques-

193

tion

194

for

195

the

196

rel-

197

a-

198

tive

199

pre-

200

ci-

201

sion.

202

4.

203

For

204

the

205

Con-

206

trac-

207

tive

208

Map-

209

ping

210

The-

211

o-

212

rem

213

(also

214

called

215

the

216

Fixed

217

Point

218

The-

219

o-

220

rem)

221

to

222

be

223

valid,

224

three

225

con-

226

di-

227

tions

228

are

229

es-

230

sen-

231

tial:

232

(a)

233

The

234

set

235

C

236

is

237

a

238

closed

239

sub-

240

set

241

of

242

R;

243

(b)

244

The

245

func-

246

tion

247

F

248

maps

249

C

250

into

251

C;

252

(c)

253

|F (x)−

254

F (y)| ≤

255

λ|x−

256

y|

257

for

258

all

259

x, y ∈

260

C

261

with

262

λ <

263

1.

264

Find

265

counter-

266

examples

267

to

268

the

269

con-

270

clu-

271

sion

272

of

273

the

274

the-

275

o-

276

rem

277

when

278

only

279

two

280

of

281

the

282

three

283

con-

284

di-

285

tions

286

are

287

re-

288

quired.

289

i.e.,

290

one

291

coun-

292

terex-

293

am-

294

ple

295

when

296

con-

297

di-

298

tions

299

(a)

300

and

301

(b)

302

but

303

not

304

(c)

305

are

306

true,

307

one

308

where

309

(a)

310

and

311

(c),

312

but

313

not

314

(b),

315

and

316

one

317

where

318

(b)

319

and

320

(c)

321

but

322

not

323

(a).

324

Part

325

II:

326

This

327

part

328

should

329

be

330

done

331

us-

332

ing

333

MAT-

334

LAB.

335

You

336

have

337

learned

338

some

339

ba-

340

sics

341

about

342

MAT-

343

LAB

344

in

345

the

346

pre-

347

vi-

348

ous

349

home-

350

work.

351

Here

352

you

353

will

354

learn

355

more

356

about

357

how

358

to

359

write

360

M-

361

files.

362

There

363

are

364

two

365

types

366

of

367

M-

368

files:

369

script

370

files

371

and

372

func-

373

tion

374

files.

375

A

376

script

377

file

378

con-

379

sists

380

of

381

a

382

se-

383

quence

384

of

385

nor-

386

mal

387

MAT-

388

LAB

389

state-

390

ments

391

as

392

you

393

have

394

seen

395

in

396

home-

397

work

398

set

399

#1.

400

A

401

func-

402

tion

403

file

404

is

405

a

406

spe-

407

cial-

408

ized

409

form

410

of

411

M-

412

file

413

which

414

cre-

415

ates

416

a

417

new

418

func-

419

tion

420

spe-

421

cific

422

to

423

your

424

prob-

425

lem.

426

(a)

427

Open

428

two

429

win-

430

dows,

431

one

432

for

433

edit-

434

ing

435

files,

436

the

437

other

438

for

439

run-

440

ning

441

MAT-

442

LAB.

443

The

444

two

445

win-

446

dows

447

should

448

work

449

in

450

the

451

same

452

di-

453

rec-

454

tory

455

or

456

folder.

457

In-

458

voke

459

MAT-

460

LAB

461

in

462

one

463

of

464

the

465

win-

466

dows.

467

You

468

don’t

469

need

470

to

471

use

472

di-

473

ary

474

this

475

time

476

to

477

record

478

your

479

ses-

480

sion.

481

(b)

482

You

483

can

484

write

485

a

486

func-

487

tion

488

file

489

for

490

any

491

func-

492

tion.

493

Put

494

the

495

fol-

496

low-

497

ing

498

in

499

the

500

file

501

fun1.m.

502

503

function y = fun1(x)

504

y = exp(x) - sin(x);

505

506

Save

507

the

508

file

509

and

510

in

511

the

512

MAT-

513

LAB

514

win-

515

dow,

516

type

517

fun1(-

518

1.1)

519

It

520

will

521

re-

522

turn

523

the

524

value

525

of

526

e−1.1−

527

sin(−1.1).

528

So

529

fun1(x)

530

works

531

just

532

like

533

any

534

built-

535

in

536

MAT-

537

LAB

538

func-

539

tion

540

such

541

as

542

sin(x).

543

No-

544

tice

545

that

546

for

547

a

548

func-

549

tion

550

file,

551

the

552

file

553

name

554

and

555

the

556

func-

557

tion

558

name

559

need

560

to

561

be

562

the

563

same!

564

The

565

fol-

566

low-

567

ing

568

is

569

a

570

MAT-

571

LAB

572

code

573

for

574

the

575

bi-

576

sec-

577

tion

578

al-

579

go-

580

rithm.

581

Put

582

it

583

in

584

the

585

script

586

file

587

bi-

588

sec-

589

tion.m.

590

591

% bisection.m

592

% MATLAB code for the bisection algorithm.

593

594

clear; % Clear the MATLAB environment

595

596

a = input(’The left value of the interval [a,b]: ’);

597

b = input(’The right value of the interval [a,b]: ’);

598

M = input(’Maximum number of iterations: ’);

599

d = input(’The smallest interval size: ’);

600

ep = input(’The smallest absolute function value: ’);

601

602

u = fun1(a); v = fun1(b);

603

e = b - a;

604

disp([a,b,u,v]);

605

if sign(u) ~= sign(v)

606

for k = 1:M

607

e = e/2; c = a + e; w = fun1(c);

608

disp([k,c,w,e]);

609

if ( abs(e) < d ) | ( abs(w) < ep )

610

break;

611

end

612

if sign(w) ~= sign(u)

613

b = c; v = w;

614

else

615

a = c; u = w;

616

end

617

end

618

end

619

620

Compare

621

the

622

above

623

with

624

the

625

pseudo-

626

code

627

in

628

the

629

notes

630

line-

631

by-

632

line

633

and

634

also

635

use

636

MAT-

637

LAB

638

on-

639

line

640

help

641

to

642

un-

643

der-

644

stand

645

the

646

above

647

M-

648

file.

649

Now

650

in

651

MAT-

652

LAB,

653

type:

654

bisection

655

It

656

then

657

will

658

ask

659

you

660

for

661

the

662

val-

663

ues

664

of

665

a, b,M, d, ep.

666

Use

667

a =

668

1; b =

669

2,M =

670

20, d =

671

1.e−

672

12, ep =

673

1.e−

674

14.

675

Can

676

you

677

tell

678

what

679

the

680

best

681

ap-

682

prox-

683

i-

684

ma-

685

tion

686

to

687

the

688

root

689

is?

690

Uti-

691

lize

692

the

693

above

694

m-

695

file

696

but

697

with

698

other

699

func-

700

tion

701

m-

702

files

703

to

704

solve

705

the

706

fol-

707

low-

708

ing

709

root-

710

finding

711

prob-

712

lems:

713

•

714

x−1−

715

tanx,

716

on

717

[0, π/2].

718

•

719

2−x+

720

ex+

721

2 cosx−

722

6,

723

on

724

[1, 3].

725

Hand

726

in

727

both

728

the

729

codes

730

(script

731

files

732

and

733

func-

734

tion

735

files)

736

and

737

the

738

out-

739

put

740

of

741

20

742

it-

743

er-

744

a-

745

tions.

746

Part

747

III:

748

5.

749

In

750

this

751

ex-

752

er-

753

cise

754

we

755

com-

756

pare

757

the

758

Bi-

759

sec-

760

tion,

761

Se-

762

cant,

763

False

764

Po-

765

si-

766

tion

767

and

768

New-

769

ton

770

meth-

771

ods

772

for

773

the

774

fol-

775

low-

776

ing

777

prob-

778

lems:

779

(a)

780

x3−

781

2x2−

782

5 =

783

0 , [1, 4]

784

785

(b)

786

x =

787

cosx , [0, π/2]

788

(c)

789

ex+

790

2−x+

791

2 cosx−

792

6 =

793

0 , [1, 2]

794

(d)

795

(10−6x−

796

2)2−

797

lnx+

798

6 ln 10 =

799

0 , [106, 2·

800

106]

801

(e)

802

erf(0.5+

803

x) =

804

0.520508, , [10−6, 10−3]

805

For

806

New-

807

ton’s

808

method

809

use

810

the

811

mid-

812

point

813

of

814

the

815

pro-

816

vided

817

in-

818

ter-

819

val

820

as

821

ini-

822

tial

823

guess.

824

You

825

should

826

find

827

the

828

so-

829

lu-

830

tions

831

within

832

10−14

833

(rel-

834

a-

835

tive)

836

ac-

837

cu-

838

racy

839

(but

840

start

841

with

842

lower

843

ac-

844

cu-

845

racy

846

while

847

de-

848

bug-

849

ging

850

your

851

code).

852

What

853

you

854

should

855

hand

856

in:

857

(a)

858

Your

859

pro-

860

grams

861

for

862

the

863

four

864

meth-

865

ods

866

for

867

case

868

(5a).

869

Make

870

your

871

pro-

872

grams

873

easy

874

to

875

un-

876

der-

877

stand

878

by

879

us-

880

ing

881

mean-

882

ing-

883

ful

884

names

885

for

886

vari-

887

ables,

888

adding

889

com-

890

ments,

891

writ-

892

ing

893

in

894

a

895

struc-

896

tured

897

for-

898

mat,

899

etc.

900

(b)

901

For

902

each

903

case,

904

give

905

the

906

fi-

907

nal

908

so-

909

lu-

910

tion,

911

the

912

num-

913

ber

914

of

915

it-

916

er-

917

a-

918

tions

919

and

920

the

921

cpu

922

run-

923

time

924

for

925

each

926

of

927

the

928

three

929

meth-

930

ods.

931

(c)

932

Two

933

ta-

934

bles

935

to

936

com-

937

pare

938

these

939

meth-

940

ods

941

in

942

the

943

for-

944

mat:

945

946

Bisection False Position Secant Newton

a

b

c

d

e

947

using:

948

i.

949

number

950

of

951

it-

952

er-

953

a-

954

tions

955

and

956

ii.

957

cpu

958

run-

959

time

960

(for

961

cpu

962

run-

963

time

964

you

965

may

966

need

967

to

968

av-

969

er-

970

age

971

over

972

sev-

973

eral

974

runs).

975

(d)

976

Do

977

these

978

ta-

979

bles

980

al-

981

ways

982

agree?

983

If

984

not,

985

why?

986

(e)

987

Which

988

ta-

989

ble

990

should

991

we

992

con-

993

sult

994

when

995

judg-

996

ing

997

which

998

method

999

is

1000

faster?

1001

1002

1003