HP Mathematics II User manual

USER MANUAL
Mathematics II

CALCULUS
MATHEMATICS II
VER1.0
A mathematical program for calculators
Odd Bringslid
tsv
Postbox 10143601Kongsberg
NORWAY
.

Copyright© 1991 Odd Bringslid ISV
All rights reserved
The authorshould notbe liable forany errors or conse-
quential or incidentialdamagesconnectingwith the furnis-
hing performance oruseofthe applicationcard.
JS5
First editionAugust1992

Contents
Generaly 6
Hardware requirements 6
Starting up7
User interface 7
The input editor 9
Echoingfrom the stack 10
Calculation finished 10
Moving in the menu 10
STAT and MATR menu 11
LeavingCALCULUS 11
Intermediateresults 12
Flagstatus andCST menu 12
Linear algebra 13
Linear equations(Gaussalgo-
rithm) 13
Matrix calculations 16
Addition 16
Multiplication 16
Inverting 16
Determinant 16
Rank 17
Trace 17
Orthogonal matrix 17
Transposedmatrix 17
Symmetric 17

Linear transformations 18
2D transformations (two dimen-
sions) 18
Rotation 18
Translation 19
Scaling 20
Concatinating 21
3D transformations (3 dimen-
sions) 23
Translation 23
Scaling 23
Rotation 23
Concatinating 24
Eigenvalueproblems 25
Eigenvalues 25
Eigenvectors 26
Diagonalization 27
Diffequations 28
Vectorspaces 30
Basis? 30
Norm 31
Norming 31
Scalar product 31
Orthogonalization 31
Orthogonal? 32
Orthonorming 32
Vector innew basis 32
Transformationmatrix innew basis 34
rr
-~ L

Laplacetransforms 36
Laplace transform 37
Invers Laplacetransform 38
Inverse L Partialfractions 39
Diffequations 41
Probability 42
Withoutreplacement 43
Combinationsunordered 43
Combinationsordered 44
Hypergeometric distribution 44
Hypergeometric distr.function 44
Withreplacement 46
Combinationsunordered 46
Combinationsordered 47
Binomial distribution 48
Binomial dsitr. function 49
Negative binomial distribution 49
Negativebinomial distribution func-
tion 50
Pascal distribution 51
Pascaldistribution function 52
Normal distribution 53
Poisson distribution 54
Poisson distributionfunction 55
Info 55
Binomialcoefficients 55
Statistics 57
Distributions 58

Normal distribution 58
Inverse normaldistribution 58
Kjisquare distribution 59
Inverse Kjisquare 60
Studen-t distribution 60
Inversestudent-t 61
Confidence intervals 62
Mean, known o- 62
Mean, uknown a
Variance uknown ^ 63
Samplemean, stdev, median 65
Fitting 66
Normal ditsribution, "best fit" 66
Hypothesis normaldistribution 67
Hypothesis binomialdistribution 68
Hypothesis Poisson distribution 69 L_ —
Class table 69 ^^T
Mean, stdev 70 —
Discrete table 70 it
Description of samples 71
Diskrete table £DAT 71
Classes KSDAT 71
Cummulativ table 72
72
^
2DAT mean and st.deviation 72 [gr-
Histogram KXDAT 73
Frequency polygon KXDAT 74 §2*j
Linear regression and correlation 74 j

Fourierseries 75
Fourier series, symbolic form 76
Fourier series numericform 78
Half rangeexpansions 79
Linearprogramming 81

1
Generaly
This is part IIof CALCULUSmathematics.Togetherwith
part HI thiswillrepresent a completemath pacfor higher
technicaleducation.
As in part I a pedagogicalinterface is stressed.CALCULUS
mathematics isa pedagogicaltool in addition toa package for
gettingthingscalculated.
Hardware requirements
CALCULUSMath II runsunder the calculator HP 48SX.
The programcard maybe inserted into either of thetwo ports
and Math I could beinthe otherport.
1. Generaly

Starting up
The LIBRARYmenuwill show up MAILPushing the MAJI
key willlead you into the main, menu and then you simply
push the STARTkey.
User interface
The CALCULUS meny system is easy to use.Using thear-
row keys allows youto move the dark barand select by pus-
hing ENTER.
Inthe following example you willenter the submenu for
LINEARALGEBRA and selectMatrices/Multiply.
RAD
{HOME }
PRG
LAPLACETRANSFORMS
FOURIER SERIES
LINEARPROGRAMMING
\r^
-
1. Generaly

RAD
{HOME }
PRG
Linearequations
+ J&~ -k t
Transformations
Eigenvalueproblems
*m*r-
UnderMatrices you willchooseMultiply andyoumay mul-
tiply two symbolicmatrices. The matrices areput into the
SYMBOLICMATRIX WRITER.
RAD
{HOME}
PRG
Sum
Powers
Inverting
»*-<-»—»>«^ p«-v~«
1- 2*-
Inthe MatrixWriter youmay delete,add, and echo from the
stack (seemanual for 48SX).
1. Generaly

The inputEditor
If you selectLinearequationsunderLINEARALGEBRA
you willenter the editor for input(inputscreen).
RAD
{HOME }
:PartAns
w -"'• sp"*r
, , ,JV. »
PRG
Y/N:J
.}: {123}
}: {xyz}
Here the inputdata can be modified and deleted and youcan
move around by using the arrowkeys.
The cursor is placedrightbehind :PartAns Y/N: and here you
enter Y ifyou wantintermediateresults. The arrow keys are
used toget rightbehind :B {Bl...}: and here you enter the
right sidevector ofthe system.
If you havedone a mistake youmay alter yourinput by using
the delete keys onthe calculatorkeyboard. You will notbe
able to continuebefore the data are correctly putin.
1. Generaly

Inthe inputscreenthere is often informationabout the pro-
blem youare going to solve (formulaes etc). Remember the
'' in algebraics and separation of severaldata onthe same
input line by usingblanks (space).
Echoing fromthe stack
If an expression ora valuelaying onthe stack is going tobe
used,then the EDrT/tSTK/ECHO/ENTER sequencewill
load datainto the inputscreen. Be sure to place the cursor
correctly
Calculation finished
When a calculation isfinished CALCULUSwill either show
up intermediateresults by using the VIEW routine(intrinsic
MAII) or returndirectly tothe menu. Inthe lastcase you will
need touse the ->STKkey toseethe resultlaying onthe stack.
Moving upand down inthe menu
You can movedownwards inthe menusystem by scrolling
the dark barand pressing ENTER. Ifyou need to move up-
wards the UPDIR key will help. Atany time youmay HALT
CALCULUS andusethe calculatorindependent of CAL-
CULUS by pressing the -^STK key. CONT will getyou back
to the menusystem.
1. Generaly 10

STAT and MATR menues
On the menu line at the bottom the choices STAT and MATR
are possible. Here you willhaveaccess to someroutines re-
gardless of yourcurrentmenuposition.
STAT:
MATR:
• NORMNormaldistribution
• INVN Inversenormaldistribution
• USD AT Samplemean, st. dev., median
• KSDAT Classtable
• SDAT Discretetable, two columns
•ADDAdd symbolicmatrices
• MULTMultiply
• INV Invert
• TRN Transpose
•DET Determinant
Leaving CALCULUS
Pushing the EXIT key willleave CALCULUS.
1. Generaly 11

Intermediate results
Inthe inputscreen youmay choose PartAns Y/N. Choosing
N the result will be laying onthe stack andyou have touse
-^STK toseethe answer.Choosing Y, different pages ofin-
termediateresultswill show upor morethan one result is lay-
ing onthe stack.
The degree of details inthe partialanswers is somewhatdif-
ferent, but some ofthe resultscovers"thewholeanswer". In
E:'"^
"~ ~-~""
every casethiswillgive the user a goodhelp.
Different parts ofan answer maybe found on different pages
and the pagenumber canbee seen(usearrowup/down).
WhenPartAnsY(es) is chosen all numberswillshow up with
two figures behindcomma. Ifa moreaccurateanswer isne-
cessary, you willhave to look onthe stack and perhaps use
theN FIX option. s=^-r
Flag status andCST menu
Theflag status andCST menu you havebeforegoinginto
CALCULUSwill be restoredwhen you leave by pushing
EXIT.
3
5
s=*
1. Generaly 12

Linearalgebra
The subjectlinearalgebracovers linear eqautions withsolu-
tionalso for singularsystems,matrixmanipulation(symbo-
lic),eigenvalueproblemsincludedsystems of linear
differential equations, linear transformations intwoand
threedimensions and vectorspaces.
Linearequations(Gaussmethod)
Linearequationswithsymbolicparameters are handled. The
equationshave to be ordered to reckognize the coefficient
matrix andthe right side. The equations are given inthe form:
E^-sr
r~
Br-g A isthe coefficient matrix , X a columnvector fortheun-
I j knowns and B the right sidecolumnvector.Symbolic coeffi-
| cients are possible.
2. Linearalgebra _ 13

If Det(A) = 0 (determinant) the systemwill be singular(self
contradictory or indefinite). This is stated as "Self contradic-
tory" orthe solutionwillgiven in terms ofoneore more of
the unknowns(indefinite).Example:
{x,y,z} = {x,2*x-l,x-4}
The value ofx is arbitrary so there isan infinitenumber of
solutions.
If the system is underdetermined (too few equations), theso- JS^g
lution will b e given inthe idefinite form. Ifthe system is over- [__
determined(toomanyequations) the solutionwill be given
inthe indefinite form ifthe equations are linearydependent
oras "Self contradictory" if they are linearyindependent.
L
The solutionalgorithm isthe Gausselimination. If PartAns
Y(es) is selected, the differentstages inthe processwill be
given as matrices onthe stackwhich maybe viewed by using
the MATW option(LIBRARY). The coefficient matrix and
theright sidevector are assembled inone matrix (Bisthe ^;_^
rightmost column). ^- L
2. Linearalgebra 14

Interface:
RAD
{HOME}
PRG
A*X =B
: PartAns Y/N: Y
: B{B1...}: {7 6 0}
...}: {xlx2x3x4x5}
V, f, m j,,.>™» „,,«
' .- H tt
The symbolicmatrixwriterwill now appear. The following
matrix isput into it:
' 2
i i
1 1
5
-1
2
-4
3
1
-1
2
-1
3
-1
1
-1
The examplesolves the system:
2xi-X2 +3x3 + 2x4-xs = 7
xi + 2x2+X3-X4+xs =6
xi - 4x2 -xs + 3x4-xs =0
The system is indefinite(too few eqautions)) andthe solu-
tion is given in terms of xs and X4.
2. Linearalgebra 15

Matrix calculations
Some operations on symbolicmatrices are done(notcove- gjr
red bythe 48SXintrinsic functions). The matrices are put into
the SYMBOLICMATRIX WRITER andthe matrix isput *"**
on the stack by pushing ENTER.
Addition
Both matices areput into the matrixwriter and added. Aner-
ror message is given for wrongdimension.
Multiplication
Both matrices areput into the matrixwriter and multiplied.
An errormessage is given for wrongdimensions. The first
matrix hasto have the samenumber of columns asthe sec-
ond has rows. Be aware ofthe order ofthe matrices.
Inverting
The mark isput into the matrixwriter and inverted. Aner- ^
ror message is given ifitsnot quadratic.
Determinant ^T
The determinant ofa symbolicmatrix is calculated. Thema- ~~
trix isput into the matrixwriter. An errormessage is given if
its not quadratic.
2. Linearalgebra 16

Rank ofa matrix
The rank ofa mark is calculated.This routine maybe used
for testing linear independency of rowvectors. The routine
makes the matrixupper traingular and PartAns Y gives the
different stages ofthe process.
Trace
The trace ofa squarematrix is calculated. An errormessage
is given ifthe matrix isnot quadratic.
Orthogonalmatrix
Thisroutine is testingwhether the matrix is orthogonal i.e.
the inverse is equal tothe transpose. An errormessage isgi-
venfor wrongdimension(must be quadratic). The answer is
logic 0or1.Maybe used to investigate if rowvectors are ort-
hogonal i.e. isan orthogonalbasis ofa vectorspace.
Transpose matrix
The transpose ofa symbolicmatrix is calculated.
Symmetric
Investigates whether a matrix is symmetric or not.Logic 0or
1.
2. Linearalgebra 17