HP HP-15C Parts list manual

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HEWLETT-PACKARD
HP-15C
ADVANCED FUNCTIONS
HANDBOOK
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NOTICE
Hewlett-PackardCompanymakes no express or implied
warranty with regard tothe keystrokeprocedures and
program material offered or theirmerchantability or their
fitness forany particularpurpose. The keystroke procedures
and programmaterial are madeavailablesolely onan"asis"
basis, andthe entirerisk asto theirquality and performance is
with the user.Should the keystrokeprocedures or program
materialprovedefective, the user (and not Hewlett-Packard
Company norany otherparty)shallbear the entirecost ofall
necessary correction andall incidental or consequential
damages.Hewlett-PackardCompany shall notbe liable for
any incidental or consequentialdamages in connection with
or arising outofthe furnishing,use, or performance ofthe
keystroke procedures or programmaterial.

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HEWLETT
PACKARD
HP-15C
August 1982
00015-90011
Printed in U.S.A. © Hewlett-PackardCompany 1982

Contents
Introduction 5
Section 1: Using [SOLVE [Effectively 6
Finding Roots 6
How | SOLVE]Samples 7
HandlingTroublesomeSituations 9
Easy VersusHardEquations 9
Inaccurate Equations 10
Equations With SeveralRoots 10
Using | SOLVE | With Polynomials 10
Solving a System of Equations 15
FindingLocalExtremes ofa Function 17
Using the Derivative 17
Using an ApproximateSlope 20
UsingRepeatedEstimation 23
Applications 26
Annuities and Compound Amounts 26
Discounted CashFlowAnalysis 39
Section 2: Working with (TH 45
Numerical IntegrationUsing |7T| 45
Accuracy ofthe Function tobe Integrated 47
FunctionsRelated to PhysicalSituations 47
Round-Off Error in InternalCalculations 49
ShorteningCalculationTime 49
Subdividing the Interval of Integration 50
Transformation of Variables 54
Calculating DifficultIntegrals 55
Application 60
Section 3: Calculating in Complex Mode 65
UsingComplex Mode 65
TrignomometricModes 68
Definitions of Math Functions 68
ArithmeticOperations 69
Single-ValuedFunctions 69
2

Contents 3
MultivaluedFunctions 69
Using | SOLVE | and |7T| in Complex Mode 73
Acuracy in Complex Mode 73
Applications 76
Storing and Recalling ComplexNumbersUsing a Matrix 76
Calculating the /?th Roots ofa ComplexNumber 78
Solving an Equation forIts ComplexRoots 80
ContourIntegrals 85
ComplexPotentials 89
Section 4: Using Matrix Operations 96
Understanding theLU Decomposition 96
Ill-Conditioned Matrices and the ConditionNumber 98
The Accuracy of NumericalSolutions to LinearSystems 103
MakingDifficultEquations Easier 104
Scaling 104
Preconditioning 1 07
Least-SquaresCalculations 110
NormalEquations 110
OrthogonalFactorization 113
Singular and NearlySingularMatrices 117
Applications 119
Constructing an IdentityMatrix 119
One-Step ResidualCorrection 119
Solving a System of NonlinearEquations 122
Solving a LargeSystem of 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 of NumericalCalculations 172
Misconceptions About Errors 172
A Hierarchy of Errors 178
Level 0:No Error 178
Level oo: Overflow/Underflow 179
Level 1: CorrectlyRounded, or Nearly So 179
Level 1 C: ComplexLevel 1 183
Level 2: CorrectlyRounded for PossiblyPerturbed Input 184
TrigonometricFunctions of Real Radian Angles 184
BackwardErrorAnalysis 187

4 Contents
BackwardErrorAnalysisVersusSingularities 192
Summary to Here 194
BackwardErrorAnalysis of MatrixInversion 200
Is Backward Error Analysis a Good Idea? 204
Index ...................................................... 212

Introduction
The HP-15Cprovidesseveraladvancedcapabilitiesnever before
combined so conveniently ina handheldcalculator:
• Finding the roots of equations.
• Evaluatingdefiniteintegrals.
• Calculatingwithcomplexnumbers.
• Calculatingwithmatrices.
The HP-15C Owner's Handbook gives the basicinformationabout
performing theseadvancedoperations. It alsoincludesnumerous
examples that show howtouse these features. The owner'shand-
book is yourprimaryreference for informationabout the advanced
functions.
This HP-15C Advanced Functions Handbook continueswhere the
owner's handbookleaves off. In 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, or applications.
Theseprogramsserve two purposes. First, theysuggestways of
using the advancedfunctions, so that you might use thesecapa-
bilitiesmore effectively in your own applications.Second, the
programscover a widerange of applications—they maybe useful
to youinthe form presented in this handbook.
Note: The discussions of mosttopics in 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 1
Using | SOLVE | Effectively
The [SOLVE| algorithmprovides an effective method for finding a
root ofan equation. This sectiondescribes the numericalmethod
used by [SOLVE | and givespracticalinformationaboutusing
| SOLVE | in varioussituations.
FindingRoots
In general, no numericaltechnique canbe guaranteed to find a root
of everyequation that has one. Because a finitenumber of digits
are used, the calculated function may differ from the theoretical
function in certainintervals ofx,itmaynotbe possible to
represent the rootsexactly, oritmaybe impossible to distinguish
between zeros and discontinuities ofthe function beingused.
Because the function canbe sampled at only a finitenumber of
places, it's alsopossible to concludefalsely that the equation has
no roots.
Despite theseinherentlimitations onany numericalmethod for
finding roots, an effective method—like that used by[ SOLVE | —
should strive to meet each ofthe followingobjectives:
•Ifa realrootexists andcanbe exactlyrepresented bythe
calculator, it should be returned.Note that the calculated
function may underflow (and besetto zero) for somevalues of
x other than the trueroots.
•Ifa realrootexists, butit can't be exactlyrepresented bythe
calculator, the valuereturnedshould differ from the trueroot
only inthe last significantdigit.
•Ifno realrootexists, an errormessageshould be displayed.
The | SOLVE | algorithm was designedwith these objectives in mind.
Itis also easy touseand requireslittle ofthe calculator'smemory.
And because I SOLVE |ina program can detect the situation ofnot
finding a root,yourprograms can remainentirelyautomatic
regardless of whether | SOLVE | finds a root.
6

How SOLVESamples
The [SOLVE I routineusesonly five registers of 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
c is found.
Normally, [SOLVE | uses the secantmethod to select the nextvalue.
This methoduses the values ofa, b,f(a), andf(b) to predict a value
c where /(c) might be close to zero.
f(x)
If c isn't a root, but /(c) is closer to zero than f(b), then b is
relabeled asa,c is relabeled asb,andthe predictionprocess is
repeated.Provided the graph of f(x) is smooth and provided the
initialvalues ofa andb are close toa simpleroot, the secant
method rapidlyconverges toa root.
However, undercertainconditions the secantmethoddoesn't
suggest a nextvalue that willbound the search or move the search
closer toa root,such as finding a signchange ora smallerfunction
magnitude. In suchcases, [SOLVE[uses a different approach.
If the calculated secant is nearlyhorizontal, | SOLVE | modifies the
secantmethod to ensure that |c — 61 ^ 1001 a — 61. This is especially
importantbecause it alsoreduces the tendency forthe secant
method togo astray whenroundingerrorbecomessignificantnear
a root.

Section 1: Using [SOLVE] Effectively
f(x)
If [SOLVE [has already found values a andb such that/(a) an
haveoppositesigns, it modifies the secantmethod to ensure that c
alwayslieswithin the intervalcontaining the signchange. This
guarantees that the searchintervaldecreaseswitheachiteration,
eventually finding a root.
f(x)
b /
If | SOLVE | hasn't found a signchange anda samplevalue c doesn't
yield a functionvaluewithdiminishedmagnitude,then | SOLVE | fits
a parabolathrough the function values ata,b,andc.\E | finds
the value d at which the parabola hasits maximum or minimum,
relabels d asa,and thencontinues the searchusing the secant
method.

Section 1: Using | SOLVE] Effectively 9
I SOLVE | abandons the search fora rootonlywhenthreesuccessive
parabolic fits yield no decrease inthe functionmagnitude or when
d — b. Under these conditions, the calculatordisplays Error 8.
Because b represents the pointwith the smallestsampledfunction
magnitude, b and f(b) are returned intheX-and Z-registers,
respectively. The Y-registercontains the value ofa orc. With this
information, youcan decidewhat todo next. You mightresume the
search where it left off, or direct the searchelsewhere, or decide
that f(b) is negligible so that x = b isa root, or transform the
equationintoanotherequationeasier to solve, or conclude that no
root exists.
HandlingTroublesomeSituations
The followinginformation is useful for working with problems that
could yieldmisleadingresults.Inaccurateroots are caused by
calculatedfunctionvalues that differ from the intendedfunction
values. Youcan frequently avoidtrouble by knowing howto
diagnoseinaccuracy and reduce it.
Easy Versus Hard Equations
Thetwo equationsf(x) = 0 and e^(x) — 1=0 have the same real
roots, yetoneis almostalwaysmucheasier to solvenumerically
than the other. For instance,whenf(x) = 6x — x4— 1,the first
equation is easier.Whenf(x) = ln(6x — x4), the second is easier. The
difference lies inhowthe function's graph behaves,particularly in
the vicinity ofa root.
/(x) = 6x-x -1
4-- 60--
-60--

10 Section 1: Using | SOLVE | Effectively
In general,everyequation isoneofan infinitefamily of equivalent
equations with the samerealroots. And some of thoseequations
must be easier to solve than others.While | SOLVE |may fail to find a
root foroneof thoseequations, itmay succeedwithanother.
Inaccurate Equations
[SOLVE| can't calculate an equation'srootincorrectlyunless the
function is incorrectly calculated. The accuracy of yourfunction
subroutine affects the accuracy ofthe root that you find.
You should be aware of conditions that mightcauseyour
calculated functionvalue to differ from the theoreticalvalue you
want itto have. [SOLVE| can't inferintendedvalues of your
function. Frequently, youcan minimizecalculationerror by
carefully writingyourfunctionsubroutine.
Equations With Several Roots
The task of finding all roots ofan equationbecomesmore difficult
asthe number of rootsincreases. Andany roots that clusterclosely
will usually defy attempts at accurateresolution. Youcanuse
deflation to eliminateroots, as described inthe HP-15C Owner's
Handbook.
An equationwith a multipleroot is characterized bythe function
andits first few higher-orderderivativesbeingzero atthe multiple
root. When | SOLVE | finds a doubleroot, the last half of its digits may
be inaccurate. Fora tripleroot,two-thirds ofthe root's digits tend
tobe obscured. A quadrupleroottends to loseaboutthree-fourths of
its digits.
Using | SOLVE | With Polynomials
Polynomials are among the easiestfunctions to evaluate. That is
why they are traditionallyused to approximatefunctions that
model physicalprocesses or morecomplexmathematical
functions.
A polynomial of degree n canbe represented as
anxn + an _ ixn ~l + ... + a±x + a0 .
This functionequals zero atno more than n realvalues ofx, called
zeros ofthe polynomial. A limit tothe number of positive zeros of
this function canbe determined by counting the number of times

Section 1: Using ["SOLVE | Effectively 11
the signs ofthe coefficients change asyou scan the polynomial
from left to right.Similarly, a limit tothe number of negative zeros
canbe determined by scanning a new functionobtained by
substituting — x in place ofx inthe originalpolynomial. Ifthe
actualnumber of realpositive or negativezeros is less than its
limit, it will differ byan evennumber.(These relationships are
known as Descartes' Rule of Signs.)
Asan example,consider the third-degreepolynomialfunction
Itcan have no more than threerealzeros. Ithasat most two
positiverealzeros(observe the signchanges from the first to
second and third to fourth terms) andat most one negative real
zero(obtained from /( ~x) = — x3 — 3x'2 +6x +8).
Polynomial functions are usuallyevaluatedmostcompactlyusing
nestedmultiplication.(This is sometimesreferred toas Homer's
method.) Asan illustration, the function from the previous
example canbe rewritten as
f(x) = [(x-3)x-6]x + 8.
Thisrepresentation is moreeasilyprogrammed and more
efficiently executed than the original form, especiallysince | SOLVE)
fills the stackwith the value ofx.
Example: During the winter of '78, ArcticexplorerJean-Claude
Coulerre, isolated athis frozen camp inthefar north,began
scanning the southernhorizon in anticipation ofthe sun's
reappearance.Coulerreknew that thesun would notbe visible to
him untilearlyMarch,when it reached a declination of 5°18'S. On
what dayand time in March wasthe chillyexplorer'svigil
rewarded?
The time in Marchwhen thesun reached 5°18'S declination canbe
computed by solving the followingequation for t:
D = a4t4 + a3t3 + a2t2 + alt + a0
where D isthe declination in degrees, t isthe time in days from the
beginning ofthe month, and

12 Section 1: Using | SOLVE | Effectively
a4 = 4.2725 X 1(T8
a3 = -1.9931X10~5
a2= 1.0229 X1CT3
al= 3.7680 X 10"1
a0 = -8.1806 .
Thisequation is valid for1 ^ t < 32, representingMarch,1978.
First convert 5°18'S to decimaldegrees (press 5.18 [CHS][g][*H]),
obtaining —5.3000 (using |FIX| 4 display mode). (Southernlatitudes
are expressed as negativenumbers for calculationpurposes.)
The solution to Coulerre'sproblems isthe value oft satisfying
—5.3000 = a4t + a%t + a2t + a^ + a0.
Expressed inthe form required by [SOLVE |,the equation is
0 = a4t4 + a3t3 + a2t2 + a^t - 2.8806
where the last,constantterm now incorporates the value ofthe
declination.
Using Homer's method, the function tobeset equal to zero is
f(t) = (((a4t + a3)t + a2)t + ajt - 2.8806 .
To shorten the subroutine,store and recall the constantsusing the
registerscorresponding tothe exponent oft.
Keystrokes Display
|ON |/ PI Pr Error Clearscalculator's
memory.*
H o.oooo
rifllP/R I 000- Program mode.
"This step is included here only to ensure that sufficient memory isavailableforthe
examples that follow in this handbook.

Section 1: Using [SOLVE | Effectively
13
Keystrokes
mrTBLifAi
|RCL|4
E
|RCL|3
s
s
|RCL|2
E
E
|RCL|1
0
S
IRCLIO
Display
001-42
002-
003-
004-
005-
006-
007-
008-
009-
010-
011-
012-
013-
014-
015-
,21,11
45 4
20
45 3
40
20
45 2
40
20
45 1
40
20
45 0
40
4332
InRun mode, keyinthe fivecoefficients:
Keystrokes Display
4. 2725 fx] 8 [CHS]
4.2725
4.2725
1.9931
1.0229
0.0010
3.7680
0.3768
2.8806
-08
-08
-05
-03
-01
Run mode.
Coefficient oft4.
Coefficient of t\t of £•.
Coefficient oft.
Constantterm.
1.9931 |CHS||EEX|
5|CHS|[STOl3
1.0229rEEXl3|CHS|
|STO|2
3.7680|EEX|1 |CHS|
|STO|1
2.8806rCHSl|STO|0
Because the desiredsolutionshould be between 1 and32,keyin
these two values for initial estimates.Then use [SOLVE |to find the
roots.
KeystrokesDisplay
1 [ENTER! 1.0000
32 32 Initial estimates.
[Til SOLVE | [A] 7.5137Root found.
7.5137Samepreviousestimate.

14 Section 1: Using [SOLVE | Effectively
Keystrokes Display
FRTI
HOOD
0.0000
7.5137
Functionvalue.
Restoresstack.
Thedaywas March 7th. Convert the fractionalportion ofthe
number to decimalhours and then to hours,minutes, and seconds.
Keystrokes
240
[T1| -HH. MS |
Display
0.5137
12.3293
12.1945
Fractionalportion of day.
Decimal hours.
Hours, minutes,seconds.
Explorer Coulerreshouldexpect toseethesunon March 7that
12h 19m 45s (CoordinatedUniversalTime).
By examiningCoulerre'sfunctionf(t), you realize that itcan have
as many as four real roots—three positive andone negative. Tryto
find additionalpositiveroots by using | SOLVE | withlargerpositive
estimates.
Two larger,positive
estimates.
No rootfound.
Last estimatetried.
A previousestimate.
Nonzero value of function.
Restoresstack to original
state.
Again, no root found.
Approximately same
estimate.
A previousestimate.
Samefunctionvalue.
Keystrokes
1 000 1 ENTER | 1 1 00
rri i SOLVE i [A]
s
[MI
EB
LUtEBELRl]
Display
1,100
Error 8
278.4497
276.7942
7.8948
278.4497
LKB
LIB
Error 8
278.4398
278.4497
7.8948
You have found a positivelocalminimum rather than a root. Now
tryto find the negativeroot.

Section 1: Using [SOLVE I Effectively
1 i~
Keystrokes Display
-1,000.0000
-1,100
-108.9441
-108.9441
Two larger, negative
estimates.
Negativeroot.
Samepreviousestimate.
1.6000 -08 Functionvalue.
There isno need to search further—you have found all possible
roots. The negativeroot hasno meaningsince itis outside ofthe
range for which the declinationapproximation is valid. The graph
of the function confirms the results you have found.
-20--
Solving a System of Equations
[SOLVE| is designed to find a singlevariablevalue that satisfies a
single equation. Ifa problem involves a system of equationswith
severalvariables, youmay still be able to| SOLVE |to find a solution.
For somesystems of equations,expressed as
., xn) =0
itis possiblethroughalgebraicmanipulation to eliminate allbut
one variable. That is,youcanusethe equations to derive

16 Section 1: Using | SOLVE | Effectively
expressions forallbutone variable in terms ofthe remaining
variable. By usingtheseexpressions, youcan reduce the problem to
using [SOLVE |to find the root ofa singleequation. The values of the
othervariables atthe solution can then be calculatedusing the
derived expressions.
This is oftenuseful for solving a complexequation fora complex
root. For such a problem, the complexequation canbe expressed as
two real-valued equations—one forthe realcomponent andonefor
the imaginary component—with two real variables—representing
the real and imaginary parts ofthe complexroot.
For example, the complexequation z + 9+ 8e~z = 0hasno real roots
z,butithas infinitelymanycomplexroots z — x + iy. This equation
canbe expressed astwo realequations
x +9+ 8e~xcos y = 0
y — Seisin y = 0 .
The following manipulations canbe used to eliminate y from the
equations.Because the sign ofy doesn'tmatter inthe equations,
assume y > 0,so that any solution(x,y) givesanothersolution
(x,-y). Rewrite the secondequation as
x = ln(8(siny)/y),
which requires that siny > 0,so that 2mr < y < (2n + l)rr for
integer n = 0,1, ....
From the firstequation
y = cos~1(-ex(x + 9)/8) + 2mr
= (2n + !)TT - cos~1(ex(x + 9)/8)
for n — 0,1,... Substitute this expressioninto the secondequation,
, (2n + !)TT - cos-l(ex(x + 9)/8) \
+ lnl —^^^^^^ 1 = 0.
7 64 - (ex(x + 9))2 /

Section 1: Using ("SOLVE | Effectively 17
You can then use| SOLVE |to find the root x of this equation(for any
givenvalue of n, the number ofthe root).Knowing x,youcan
calculate the correspondingvalue of y.
A finalconsideration for this example isto choose the initial
estimates that would be appropriate.Because the argument ofthe
inversecosinemust be between -1and1,x must be morenegative
than about -0.1059 (found by trial and error orby using | SOLVE |).
The initialguessesmight be near but morenegative than this
value, -0.11 and -0.2 for example.
(The complexequationused in this example is solvedusing an
iterativeprocedure inthe example on page 81. Anothermethod for
solving a system of nonlinearequations is described on page122.)
FindingLocalExtremes ofa Function
Using the Derivative
The traditional wayto find localmaximums and minimums ofa
function's graphuses the derivative ofthe function. The derivative
isa function that describes the slope ofthe graph.Values ofx at
which the derivative is zerorepresentpotentiallocalextremes of
the function. (Althoughlesscommon for well-behavedfunctions,
values ofx where the derivative is infinite or undefined are also
possibleextremes.) Ifyoucan express the derivative ofa function
in closed form, youcanuse| SOLVE | to find where the derivative is
zero—showing where the function maybe maximum or minimum.
Example: Forthe design ofa verticalbroadcastingtower,radio
engineer Ann Tenorwants to find the angle from the tower at
which the relative field intensity is mostnegative. The relative
intensitycreated bythe tower is given by
cos(2?r/icos d)— cos(2?r/i)
E = -
-cos(27r/i)]sin0
where E isthe relative field intensity, h isthe antenna height in
wavelengths, and6 isthe angle from vertical in radians. The
height is0.6 wavelengths forher design.
The desiredangle isoneat which the derivative ofthe intensity
with respect to6is zero.

18
Section 1: Using | SOLVE | Effectively
To saveprogrammemoryspace and executiontime,store the
following constants in registers and recallthem as needed:
andis stored in register R0,
andis stored in register Rl5
andis stored in register R2.
The derivative ofthe intensity E withrespect tothe angle 0is given
by
rl = cos(2irh)
-cos(27r/i)]
dE
-
d6
\
— i~9 rnsi
"
cos(r0cosd) —
sine tan 6
Key ina subroutine to calculate the derivative.
Keystrokes
[TIlP/Rl
rflCLEARfPRGMl
[cos!
|RCL|0
E
[COS |
[RCL|1
E
fSJNl
H
I TAN I
E
Display
IRCLIO
000-
001-42,
002-
003-
004-
005-
006-
007-
008-
009-
010-
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016-
017-
,21, 0
24
45 0
20
24
451
30
34
23
10
34
25
10
16
34
24
45 0
Programmode.