4. The following algorithm which generates the gamma function is defined in “gamma.kws”…
double gamma(float n){
/* Local Variables */
double zold=0.000001,znew,Aold=0.00,Anew;
/* Make calculations using the given specifications and report the results. */
/* We use the Trapezoidal Rule.*/
while(zold<=30) {
znew=zold+0.0010;
Anew=Aold+0.00050*(exp((n-1)*log(zold)-zold)+exp((n-1)*log(znew)-znew));
Aold=Anew; zold=znew;
}
return Aold;
}
5. This algorithm was used in a program called “BVN.cpp”...
Index=Refinement+25;
Pold=0.0; /* Initialize the probability . */
for (RowCount=-Index;RowCount<=Index;RowCount++) {
Aold=0.0; /* Reset the marginal accumulation . */
for (ColumnCount=-Index;ColumnCount<=Index;ColumnCount++) {
/* Independent variables. */
x = 5*ColumnCount*pow(Index,-1.0);
y = 5*RowCount*pow(Index,-1.0);
dx=5*pow(Index,-1.0);
dy=5*pow(Index,-1.0);
/* Depenedent variable: BVN((m1,m2),(s1,s2;r)) . */
BVN = exp(-((x-m1)*(x-m1)*pow(s1,-1.0)*pow(s1,-1.0)-
2*rho*(x-m1)*(y-m2)*pow(s1,-1.0)*pow(s2,-1.0)+
(y-m2)*(y-m2)*pow(s2,-1.0)*pow(s2,-1.0))*
pow(2*(1-rho*rho),-1.0))*
pow(2*3.14159265*s1*s2*sqrt(1-rho*rho),-1.0);
/* Record BVN in the matrix. */
fprintf(Destination1,"%0.6f ",BVN);
/* Integrate out x . */
Anew=Aold+BVN*dx;
Aold=Anew;
} /* Go to the next value of x. */
Pnew=Pold+Aold*dy; /* Accumulate the probability . */
fprintf(Destination1,"n"); /* Continue to the next row of BVN . */
/* Record the Marginal distribution and the probability . */
fprintf(Destination2,"%0.6f%12.6f%12.6fn",y,Aold,Pold);
Pold=Pnew; /* Update the probability . */
} /* Go to the next value of y and resume computing the marginal */
/* distribution . */
} /* WriteData */
6. The following algorithm, which generates solutions to ordinary differential
equations, is defined in “BSSLCALC.cpp” and “CALC2010.cpp”…
/* RK4 */
f1 = pow(a,-1.0)*(e-b*zold-c*xold-d);
g1 = zold;
f2 = pow(a,-1.0)*(e-b*(zold+0.5*refinement*f1)-c*(xold+0.5*refinement*g1)-d);
g2 = zold + 0.5*refinement*f1;
f3 = pow(a,-1.0)*(e-b*(zold+0.5*refinement*f2)-c*(xold+0.5*refinement*g2)-d);
g3 = zold + 0.5*refinement*f2;
f4 = pow(a,-1.0)*(e-b*(zold+refinement*f1)-c*(xold+refinement*g1)-d);
g4 = zold + refinement*f3;
znew = zold + refinement*pow(6.00,-1.00)*(f1 + 2*f2 + 2*f3 + f4);
xnew = xold + refinement*pow(6.00,-1.00)*(g1 + 2*g2 + 2*g3 + g4);
tnew = told+refinement;
7. This algorithm is used in “WAVECALC.cpp”…
xold=xLower;
while (xold<xUpper) { /* next row */
xnew=xold+RefineMent; yold=yLower; Aold=0.00 /* compute the next point of the marginal distribution */;
while (yold<yUpper) { /* nex column */
ynew=yold+RefineMent;
RR=xold*xold+yold*yold;
R=exp(0.5*log(RR));
if (menuchoice=='A')
z=exp(-attenuation*exp(viscosity*log(R)))*cos(frequency*exp(viscosity*log(R)));
if (menuchoice=='B')
z=cos(xold)*cos(yold);
if (menuchoice=='C')
z=cos(xold)*sin(yold);
if (menuchoice=='D')
z=cos(xold)+sin(yold);
if (menuchoice=='E')
z=cos(xold)+cos(yold);
if (menuchoice=='F')
/* Depenedent variable: BVN((m1,m2),(s1,s2;r)) . */
z=exp(-((xold-m1)*(xold-m1)*pow(s1,-1.0)*pow(s1,-1.0)-
2*rho*(xold-m1)*(yold-m2)*pow(s1,-1.0)*pow(s2,-1.0)+
(yold-m2)*(yold-m2)*pow(s2,-1.0)*pow(s2,-1.0))*
pow(2*(1-rho*rho),-1.0))*
pow(2*3.14159265*s1*s2*sqrt(1-rho*rho),-1.0);
fprintf(Destination1," %0.6f",z);
pointcount++;
yold=ynew;
/* integrate out y if BVN is the choice */
if (menuchoice=='F') {
Anew=Aold+z*RefineMent;
Aold=Anew; }
} /* next column */
fprintf(Destination1,"n");
xold=xnew;
/* compute the aggregate probability if BVN is the choice */
if (menuchoice=='F') {
Pnew=Pold+Aold*RefineMent /* compute the next point of the cumulative distro */;
Pold=Pnew;
fprintf(Destination2,"%0.6f %0.6f %0.6fn",xold,Aold,Pold);}
} /* next row */
8. The following algorithm, which generates sets of probability distributions along with their respective
expected values is defined in “CALC2011.cpp”…
/* Update the independent variable. */
xnew=xold+RefineMent;
Pnew1=Pold1+0.5*RefineMent*(pnew1+pold1);
Pnew2=Pold2+0.5*RefineMent*(pnew2+pold2);
Pnew3=Pold3+0.5*RefineMent*(pnew3+pold3);
Pnew4=Pold4+0.5*RefineMent*(pnew4+pold4);
Pnew5=Pold5+0.5*RefineMent*(pnew5+pold5);
EX1new1=EX1old1+0.5*RefineMent*(xold*pold1+xnew*pnew1);
EX1new2=EX1old2+0.5*RefineMent*(xold*pold2+xnew*pnew2);
EX1new3=EX1old3+0.5*RefineMent*(xold*pold3+xnew*pnew3);
EX1new4=EX1old4+0.5*RefineMent*(xold*pold4+xnew*pnew4);
EX1new5=EX1old5+0.5*RefineMent*(xold*pold5+xnew*pnew5);
EX2new1=EX2old1+0.5*RefineMent*(xold*xold*pold1+xnew*xnew*pnew1);
EX2new2=EX2old2+0.5*RefineMent*(xold*xold*pold2+xnew*xnew*pnew2);
EX2new3=EX2old3+0.5*RefineMent*(xold*xold*pold3+xnew*xnew*pnew3);
EX2new4=EX2old4+0.5*RefineMent*(xold*xold*pold4+xnew*xnew*pnew4);
EX2new5=EX2old5+0.5*RefineMent*(xold*xold*pold5+xnew*xnew*pnew5);
/* complete the feedback loop */
xold=xnew;
Pold1=Pnew1;
Pold2=Pnew2;
Pold3=Pnew3;
Pold4=Pnew4;
Pold5=Pnew5;
EX1old1=EX1new1;
EX1old2=EX1new2;
EX1old3=EX1new3;
EX1old4=EX1new4;
EX1old5=EX1new5;
EX2old1=EX2new1;
EX2old2=EX2new2;
EX2old3=EX2new3;
EX2old4=EX2new4;
EX2old5=EX2new5;
11. ...buggy algorithms produce buggy
results...
...finding an elusive bug in a stack of
code is like finding
a jewel in a dumpster...
The density functions reported by the
algorithm were in error by
a factor of two... The antiderivatives
reported however, were accurate.
Therefore, I shall prepare a report
containing the flawed algorithm,
as well as the erroneous data generated by
the algorithm in question,
with an analysis of the results, as well
as a solution. The discovery of
this flaw shall lead to the preparation of
a more robust and reliable
solution.
12. The following algorithm was defined in “CALC2001.cpp” and tabulated the density functions erroneously by a
factor of two…
xold=xmin; gdpold=BoundaryValue; ndpold=BoundaryValue;
while(xold<xmax) {
pointcount++;
/* UPDATE THE IDNEPENDENT VARIABLE. */
xnew=xold+refinement;
/* DEFINE THE PROBABILITY DENSITY FUNCTIONS. */
gdf=exp(-log(362880)-10*log(beta)+9*log(xold)-xold*exp(-log(beta)))+
exp(-log(362880)-10*log(beta)+9*log(xnew)-xnew*exp(-log(beta)));
ndf=exp(-0.5*(log(2*3.14159265*sigma*sigma)+exp(-log(sigma*sigma))*(xold-mu)*(xold-mu)))+
exp(-0.5*(log(2*3.14159265*sigma*sigma)+exp(-log(sigma*sigma))*(xnew-mu)*(xnew-
mu)));
/* PERFORM QUADRATURE USING THE TRAPEZOIDAL RULE. */
gdpnew=gdpold+0.5*refinement*(exp(-log(362880)-10*log(beta)+9*log(xold)-xold*exp(-log(beta)))+
exp(-log(362880)-
10*log(beta)+9*log(xnew)-xnew*exp(-log(beta))));
ndpnew=ndpold+0.5*refinement*(exp(-0.5*(log(2*3.14159265*sigma*sigma)+exp(-log(sigma*sigma))*(xold-
mu)*(xold-mu)))+
exp(-
0.5*(log(2*3.14159265*sigma*sigma)+exp(-log(sigma*sigma))*(xnew-mu)*(xnew-mu))));
/* UPDATE THE VARIABLES. */
ndpold=ndpnew;
gdpold=gdpnew;
xold=xnew;
The following erroneous results were produced by the previous algorithm… and
can be verified…
18. Let us show that the expression
fold = exp(-log(362880)-10*log(beta)+9*log(xold)-xold*exp(-log(beta)))
is indeed equal to
( )αβα
βα
Γ
−
−
x
ex 1
where α = 10.0 , and xold = x.
solution:
From the original expression,
20. That is;
( )βα ,0.10; ==Γ= xoldXfold .
Similarly, we can show
( )βα ,0.10; ==Γ= xnewXfnew .
Hence,
foldfnewfold 2≈+
since xnew=xold+refinement. Therefore, the error is established.
21. The buggy algorithm
was corrected by
redefining the
variables using the
updated library file
“density.kws”…
22. /* WRITE THE OUTPUT TABLE BASED ON THE MENU CHOICES. */
xold=xmin; gdpold=BoundaryValue; ndpold=BoundaryValue;
edpold=BoundaryValue; tdpold=BoundaryValue;fdpold=BoundaryValue;
while(xold<xmax) {
pointcount++;
/* UPDATE THE IDNEPENDENT VARIABLE. */
xnew=xold+refinement;
/* DEFINE THE PROBABILITY DENSITY FUNCTIONS. */
gdist=gdf(xold,alpha,beta);
ndist=ndf(xold,mu,sigma);
edf=theta*exp(-(xold*theta));
tdf=tdist(xold,v);
fdf=fdist(xold,v1,v2);
/* PERFORM QUADRATURE USING THE TRAPEZOIDAL RULE. */
gdpnew=gdpold+0.5*refinement*(gdf(xold,alpha,beta)+gdf(xnew,alpha,beta));
ndpnew=ndpold+0.5*refinement*(ndf(xold,mu,sigma)+ndf(xnew,mu,sigma));
edpnew=edpold+0.5*refinement*(theta*exp(-(theta*xold))+theta*exp(-(theta*xnew)));
tdpnew=tdpold+0.5*refinement*(tdist(xold,v)+tdist(xnew,v));
fdpnew=fdpold+0.5*refinement*(fdist(xold,v1,v2)+fdist(xnew,v1,v2));
/* UPDATE THE VARIABLES. */
ndpold=ndpnew;
gdpold=gdpnew;
edpold=edpnew;
tdpold=tdpnew;
fdpold=fdpnew;
xold=xnew;
/* ENTER THE DATA INTO THE REPORT TABLE. */
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Sky Atlas 2000.0 Companion, 2nd
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for all 2,700 Star Clusters, Nebulae, and Galaxies Shown in Sky
Atlas 2000.0, 2nd
Edition
Sky Publishing 2009 9780933346956
0-93-334695-6
Summers, Wilford I.
Croft, Terrell
Hartwell, Frederic
American Electricians’ Handbook (15th
edition) McGraw-Hill Companies, The 2008 9780071494625
0-07-149462-6
Swenson Getting Started w/Ti-89 Graphing Calculator (2nd
edition) John Wiley & Sons, Inc. 2005 9780471742067
0-47-174206-6
Swenson, Carl
Hughes-Hallett, Deborah
Gleason, Andrew M.
McCallum, William G.
Calculus Single and Multivariable: Graphing Calculator Guide for
the TI-89 (2nd
edition)
John Wiley & Sons, Inc. 2005 9780471732501
0-47-173250-8
Takeyh, Ray Guardians of the Revolution: Iran and the World in the Age of the
Ayatollahs
Oxford University Press, USA 2009 9780195327847
0-19-532784-5
Taylor, Edwin F. Exploring Black Holes: Introduction to General Relativity Benjamin Cummings 2000 9780201384239
0-20-138423-X
Thompson, A.J. Logarithmetica Britannica 2 volume set: Logarithms to 20 Decimal
Places 10,000-100,000
Cambridge University Press 2008 9780521740340
0-52-174034-7
Thompson, William J. Atlas for Computing Mathematical Functions: An Illustrated Guide
for Practitioners with Programs in Fortran 90 and Mathematica
John Wiley & Sons Inc. 1997 9780471181712
0-47-118171-4
Tirion, Wil Cambridge Star Atlas Cambridge University Press 2001 9780521800846
0-52-180084-6
Tirion, Wil
Sinnott, Roger W.
Sky Atlas 2000.0: Twenty-Six Star Charts, Covering Both
Hemispheres, and Seven Detailed Charts of Selected Regions (2nd
edition)
Sky Publishing Corporation 1998 9780933346871
0-93-334687-5
45. Triola, Mario F. Elementary Statistics / Edition 11 Addison Wesley 2008 9780321500243
0-32-150024-5
Tyn Myint-U
Debnath, Lokenath
Linear Partial Differential Equations for Scientists and Engineers Springer-Verlag, New York 2008 9780817643935
0-81-764393-1
Van Soelen, Wayne The Guidebook for Linemen and Cablemen Cengage Learning 2005 9781401899189
1-40-189918-8
Verzani, John Using R for Introductory Statistics Taylor & Francis, Inc. 2004 9781584884507
1-58-488450-9
Walpole, Ronald E. Probability and Statistics for Engineers and Scientists Prentice Hall 2006 9780131877115
0-13-187711-9
Warner, Seth Modern Algebra Dover Publications 1990 9780486663418
0-48-666341-8
Wheeler, Donald J. Understanding Variation, Second Edition: The Key to Managing
Chaos
SPC Press, Inc. 2000 9780945320531
0-94-532053-1
Widder, David V. Advanced Calculus Dover Publications 1989 9780486661032
0-48-666103-2
Wilson, Mark J.
Ford, Steven R.
The ARRL Handbook for Radio Communications [With CDROM]
(86th
edition)
American Radio Relay League, Incorporated 2008 9780872591394
0-87-259139-5
Woods, Randall B. LBJ: Architect of American Ambition Simon & Schuster Adult Publishing Group 2006 9780674026995
0-67-402699-3
Wrede, Robert C.
Spiegel, Murray R.
Schaum’s Outline of Advanced Calculus, Second Edition McGraw-Hill Companies 2002 9780071375672
0-07-137567-8
Zeidler, Eberhard Oxford Users’ Guide to Mathematics Oxford University Press, USA 2004 9780198507635
0-19-850763-1
Zill, Dennis Differential Equations with Boundary Value Problems Cengage Learning 2008 9780495108368
0-49-510836-7
Zill, Dennis G. A First Course in Differential Equations Cengage Learning 2008 9780495108245
0-49-510824-3
Zuur, Alain F.
Ieno, Elena N.
Meesters, Erik
A Beginner’s Guide To R Springer Verlag GmbH 2009 9780387938363
0-38-793836-2
48. APPLICATION INPUT
SPECIFICATIONS
DESCRIPTION WORKING DIRECTORY
BSSLCALC BSSLMENU GENERATES SOLUTIONS TO BESSEL EQUATIONS C:SMITTYWORKBOOK
CALC2000 MENU2000 GENERATES STATISTICAL TABLES AND DIFFERENCE TABLES C:SMITTYWORKBOOK
CALC2010 MENU2010 GENERATES SOLTUIONS TO ORDINARY DIFFERENTIAL
EQUATIONS
C:SMITTYWORKBOOK
CALC2011 MENU2011 GENERATES STATISTICAL TABLES C:SMITTYWORKBOOK
CALC2012 MENU2012 TABULATES LOGARITHMIC FUNCTIONS C:SMITTYWORKBOOK
*The statistical formulae used in these algorithms comes from “Schaum’s Outline of Probability and Statistics”.
49. NOTES FROM DAVID JEREMIAH, AND OTHERS FROM AFA/AFN
(SPRING 2010)
Rev. Ch. 3-4
Rapture occurs between Ch. 3 & Ch. 4 (Lycena)(Not a place, but an age…)
-Spirit of GOD has been transferred…
Ezekiel 10(14-17)…….[ Four Living Creatures]
Ezekiel 1()…………….””
(Angelic Protection)
(Execute Judgment)
(Lead Worship)
1 COR 15:
1 THESS 4: RAPTURE??
REV 4:
REV 21:21
1 COR 15:3-8 “THE CREED”
REV 4:6-8 (FOUR LIVING CREATURES)
EX 24:10 (SEA OF GLASS)
REV 21:21 (TRANSPARENT GLASS)
REV 4:5 (SEA OF GLASS)
ISRAEL = GOD PREVAILS/RULES
IS 11:…
“WHERE THE SPIRIT OFTHE LORD IS, THERE IS OUR LIBERTY…”
(THUNDERING, & LIGHTNING, & VOICES) (16:18)
50. Mom, thanks for pressing me to
settle for nothing less than the
best…
Dad, let’s go have some coffee…
53. The Lord’s Prayer:
Matthew 6:9-14
Luke 11:2-4
Our Father, which art in heaven,
Hallowed be thy name. Thy kingdom
come. Thy will be done in earth as it is
in heaven. Give us this day our daily
bread. And forgive us our debts, as we
forgive our debtors. And lead us not
into temptation, but deliver us from
evil: For thine is the kingdom, and the
power, and the glory, for ever.
Amen.
54. The Priestly Blessing:
Numbers 6:24-26
The LORD bless thee, and
keep thee:
The LORD make his face
shine upon thee, and be gracious
unto thee:
The LORD lift up his
countenance upon thee, and
give thee peace.
55. PSALMS 118:24
This is the day which the
LORD hath made; we will rejoice
and be glad in it.
56. I Am
Revelation 1:8
“I am the Alpha and Omega,
the beginning and the ending,
saith the Lord, which is, and
which was, and which is to
come, the Almighty.”
57. I Am
Revelation 1:8
“I am the Alpha and Omega,
the beginning and the ending,
saith the Lord, which is, and
which was, and which is to
come, the Almighty.”
