HP HP-15C Stücklistenhandbuch

—

HEWLETT-PACKARD

HP-15C

ADVANCED

FUNCTIONS

HANDBOOK

NOTICE

Hewlett-PackardCompanymakes

express

implied

warranty

with

regard

tothe

keystrokeprocedures

and

program

material offered

theirmerchantability

their

fitness

forany

particularpurpose.

The

keystroke procedures

and

programmaterial

are

madeavailablesolely

onan"asis"

basis,

andthe

entirerisk

asto

theirquality

and

performance

with

the

user.Should

the

keystrokeprocedures

program

materialprovedefective,

the

user

(and

not

Hewlett-Packard

Company

norany

otherparty)shallbear

the

entirecost

ofall

necessary

correction

andall

incidental

consequential

damages.Hewlett-PackardCompany

shall

notbe

liable

for

any

incidental

consequentialdamages

connection

with

arising

outofthe

furnishing,use,

performance

ofthe

keystroke

procedures

programmaterial.

HEWLETT

PACKARD

HP-15C

August

1982

00015-90011

Printed

U.S.A.

Hewlett-PackardCompany

1982

Contents

Introduction

Section

Using

[SOLVE

[Effectively

Finding

Roots

How

SOLVE]Samples

HandlingTroublesomeSituations

Easy

VersusHardEquations

Inaccurate

Equations

With

SeveralRoots

Using

SOLVE

With

Polynomials

Solving

System

Equations

FindingLocalExtremes

ofa

Function

Using

the

Derivative

Using

ApproximateSlope

UsingRepeatedEstimation

Applications

Annuities

and

Compound

Amounts

Discounted

CashFlowAnalysis

Section

Working

with

(TH

Numerical

IntegrationUsing

|7T|

Accuracy

ofthe

Function

tobe

Integrated

FunctionsRelated

PhysicalSituations

Round-Off

Error

InternalCalculations

ShorteningCalculationTime

Subdividing

the

Interval

Integration

Transformation

Variables

Calculating

DifficultIntegrals

Application

Section

Calculating

Complex

Mode

UsingComplex

Mode

TrignomometricModes

Definitions

Math

Functions

ArithmeticOperations

Single-ValuedFunctions

Contents

MultivaluedFunctions

Using

SOLVE

and

|7T|

Complex

Mode

Acuracy

Complex

Mode

Applications

Storing

and

Recalling

ComplexNumbersUsing

Matrix

Calculating

the

/?th

Roots

ofa

ComplexNumber

Solving

Equation

forIts

ComplexRoots

ContourIntegrals

ComplexPotentials

Section

Using

Matrix

Operations

Understanding

theLU

Decomposition

Ill-Conditioned

Matrices

and

the

ConditionNumber

The

Accuracy

NumericalSolutions

LinearSystems

103

MakingDifficultEquations

Easier

104

Scaling

104

Preconditioning

1 07

Least-SquaresCalculations

110

NormalEquations

110

OrthogonalFactorization

113

Singular

and

NearlySingularMatrices

117

Applications

119

Constructing

IdentityMatrix

119

One-Step

ResidualCorrection

119

Solving

System

NonlinearEquations

122

Solving

LargeSystem

ComplexEquations

128

Least-Squares

UsingNormalEquations

131

Least-Squares

UsingSuccessiveRows

140

Eigenvalues

ofa

Symmetric

Real

Matrix

148

Eigenvectors

ofa

Symmetric

Real

Matrix

154

Optimization

1 60

Appendix:

Accuracy

NumericalCalculations

172

Misconceptions

About

Errors

172

Hierarchy

Errors

178

Level

0:No

Error

178

Level

oo:

Overflow/Underflow

179

Level

CorrectlyRounded,

Nearly

179

Level

1 C:

ComplexLevel

183

Level

CorrectlyRounded

for

PossiblyPerturbed

Input

184

TrigonometricFunctions

Real

Radian

Angles

184

BackwardErrorAnalysis

187

Contents

BackwardErrorAnalysisVersusSingularities

192

Summary

Here

194

BackwardErrorAnalysis

MatrixInversion

200

Backward

Error

Analysis

Good

Idea?

204

Index

......................................................

212

Introduction

The

HP-15Cprovidesseveraladvancedcapabilitiesnever

before

combined

conveniently

ina

handheldcalculator:

•

Finding

the

roots

equations.

•

Evaluatingdefiniteintegrals.

•

Calculatingwithcomplexnumbers.

•

Calculatingwithmatrices.

The

HP-15C

Owner's Handbook gives

the

basicinformationabout

performing

theseadvancedoperations.

alsoincludesnumerous

examples

that

show

howtouse

these

features.

The

owner'shand-

book

yourprimaryreference

for

informationabout

the

advanced

functions.

This

HP-15C

Advanced

Functions

Handbook continueswhere

the

owner's

handbookleaves

off.

this

handbook

you

will

find

information

about

howthe

HP-15Cperforms

the

advancedcomputa-

tions

and

information

that

explains

howto

interpret

the

results

that

you

get.

This

handbookalsocontainsnumerousprograms,

applications.

Theseprogramsserve

two

purposes.

First,

theysuggestways

using

the

advancedfunctions,

that

you

might

use

thesecapa-

bilitiesmore

effectively

your

own

applications.Second,

the

programscover

widerange

applications—they

maybe

useful

youinthe

form

presented

this

handbook.

Note:

The

discussions

mosttopics

thishandbook

presume

that

you

already

understand

the

basic

information

about

using

the

advanced

functions

and

that

youare

generally

familiar

with

the

subject

matter

being

discussed.

Section

Using

SOLVE

Effectively

The

[SOLVE|

algorithmprovides

effective

method

for

finding

root

ofan

equation.

This

sectiondescribes

the

numericalmethod

used

[SOLVE

| and

givespracticalinformationaboutusing

SOLVE

varioussituations.

FindingRoots

general,

numericaltechnique

canbe

guaranteed

find

root

everyequation

that

has

one.

Because

finitenumber

digits

are

used,

the

calculated

function

may

differ

from

the

theoretical

function

certainintervals

ofx,itmaynotbe

possible

represent

the

rootsexactly,

oritmaybe

impossible

distinguish

between

zeros

and

discontinuities

ofthe

function

beingused.

Because

the

function

canbe

sampled

only

finitenumber

places,

it's

alsopossible

concludefalsely

that

the

equation

has

roots.

Despite

theseinherentlimitations

onany

numericalmethod

for

finding

roots,

effective

method—like

that

used

by[

SOLVE

—

should

strive

meet

each

ofthe

followingobjectives:

•Ifa

realrootexists

andcanbe

exactlyrepresented

bythe

calculator,

should

returned.Note

that

the

calculated

function

may

underflow

(and

besetto

zero)

for

somevalues

other

than

the

trueroots.

•Ifa

realrootexists,

butit

can't

exactlyrepresented

bythe

calculator,

the

valuereturnedshould

differ

from

the

trueroot

only

inthe

last

significantdigit.

•Ifno

realrootexists,

errormessageshould

displayed.

The

SOLVE

algorithm

was

designedwith

these

objectives

mind.

Itis

also

easy

touseand

requireslittle

ofthe

calculator'smemory.

And

because

SOLVE

|ina

program

can

detect

the

situation

ofnot

finding

root,yourprograms

can

remainentirelyautomatic

regardless

whether

SOLVE

finds

root.

How

SOLVESamples

The

[SOLVE

routineusesonly

five

registers

allocatablememory

inthe

HP-15C.

The

five

registersholdthreesamplevalues

(a,b,

andc)andtwo

previousfunctionvalues

(/(a)

and

f(b))

whileyour

function

subroutine

calculates/(c).

Thekeytothe

effectiveness

of|

SOLVE]

ishowthe

nextsamplevalue

found.

Normally,

[SOLVE

uses

the

secantmethod

select

the

nextvalue.

This

methoduses

the

values

ofa,

b,f(a),

andf(b)

predict

value

where

/(c)

might

zero.

f(x)

isn't

root,

but

/(c)

closer

zero

than

f(b), then

b is

relabeled

asa,c is

relabeled

asb,andthe

predictionprocess

repeated.Provided

the

graph

f(x)

smooth

and

provided

the

initialvalues

ofa andb are

toa

simpleroot,

the

secant

method

rapidlyconverges

toa

root.

However,

undercertainconditions

the

secantmethoddoesn't

suggest

nextvalue

that

willbound

the

move

the

closer

toa

root,such

finding

signchange

ora

smallerfunction

magnitude.

suchcases,

[SOLVE[uses

different

approach.

the

calculated

secant

nearlyhorizontal,

SOLVE

modifies

the

secantmethod

ensure

that

—

1001

—

61.

This

especially

importantbecause

alsoreduces

the

tendency

forthe

secant

method

togo

astray

whenroundingerrorbecomessignificantnear

root.

Section

Using

[SOLVE]

Effectively

f(x)

[SOLVE

[has

already

found

values

a andb

such

that/(a)

haveoppositesigns,

modifies

the

secantmethod

ensure

that

alwayslieswithin

the

intervalcontaining

the

signchange.

This

guarantees

that

the

searchintervaldecreaseswitheachiteration,

eventually

finding

root.

f(x)

SOLVE

hasn't

found

signchange

anda

samplevalue

doesn't

yield

functionvaluewithdiminishedmagnitude,then

SOLVE

fits

parabolathrough

the

function

values

ata,b,andc.\E |

finds

the

value

d at

which

the

parabola

hasits

maximum

minimum,

relabels

d asa,and

thencontinues

the

searchusing

the

secant

method.

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