Calculus-level Coding Overview

Improve Math Models &Improve Math Models &
Increase ProductivityIncrease Productivity
By Phil BrubakerBy Phil Brubaker
forfor
Scientists,Scientists,
Engineers & ManagementEngineers & Management
Optimal Designs EnterpriseOptimal Designs Enterprise
goal-driven.netgoal-driven.net

Improve Models &Improve Models &
Increase ProductivityIncrease Productivity
AgendaAgenda
Design ObjectivesDesign Objectives
Language BackgroundLanguage Background
Example CodeExample Code
Optimization of OptimizationsOptimization of Optimizations

Code DefinitionCode Definition
OptimizationOptimization
==
SimulationSimulation
++
Objective (Function)Objective (Function)

Why OptimizationWhy Optimization
over Simulation?over Simulation?
Maximize Parameter Tolerance for Mfg.Maximize Parameter Tolerance for Mfg.
Solutions are at min./max. locations whereSolutions are at min./max. locations where
their partial derivatives = 0 & thus allow largesttheir partial derivatives = 0 & thus allow largest
delta errors.delta errors.
Minimize # of Executions for a solutionMinimize # of Executions for a solution

IntroductionIntroduction
toto
FortranCalculusFortranCalculus
a Calculus-level Computer Languagea Calculus-level Computer Language
1974 First Commercial Release (PROSE)1974 First Commercial Release (PROSE)
I taught PROSE in 1975-79.I taught PROSE in 1975-79.
Solved problems within 4 hours (each)Solved problems within 4 hours (each)
Present version is FortranCalculusPresent version is FortranCalculus
Main obstacle: no design objectiveMain obstacle: no design objective

Objective-Driven EngineeringObjective-Driven Engineering
What are yourWhat are your
Goals/ObjectivesGoals/Objectives
forfor
a given Project?a given Project?

What are you building?What are you building?
Mr. Arithmetic Mr. Algebra Mr. CalculusMr. Arithmetic Mr. Algebra Mr. Calculus
------------------- --------------- ----------------------------------- --------------- ----------------
Slide Rule Simulate OptimizeSlide Rule Simulate Optimize

Objective-Driven CutsObjective-Driven Cuts
Sawmill's operation
Objective for cutting log?Objective for cutting log?

Sawmill's operation
Objective for cutting log?Objective for cutting log?
Maximize company's profitsMaximize company's profits
Minimize pollutionMinimize pollution
Minimize 'waste'Minimize 'waste'
Maximize 2" x 4"sMaximize 2" x 4"s

Sawmill's operationSawmill's operation
Parameters to consider:Parameters to consider:
Log-to-lumber Processing:Log-to-lumber Processing:
Size of logSize of log: diameter, length, taper, knots, etc.: diameter, length, taper, knots, etc.
Time requiredTime required to: cut log, sharpen blades, lubricateto: cut log, sharpen blades, lubricate
machinery, etc.machinery, etc.
Strength or flexibilityStrength or flexibility desired of various sizes.desired of various sizes.
InventoryInventory
Market trendsMarket trends

Chaotic Design ProcessChaotic Design Process
Control BoxControl Box
Bingo, a design!Bingo, a design!

Objective-Driven DesignObjective-Driven Design
Surveillance Design Objective:Surveillance Design Objective:
Maximize surveillance coverageMaximize surveillance coverage
Minimize number of satellitesMinimize number of satellites
Solution? (Not realistic; Objective needs work.)Solution? (Not realistic; Objective needs work.)

Thin-Film-Head (TFH) for Magnetic RecordingThin-Film-Head (TFH) for Magnetic Recording
TFH with coilTFH with coil
Disc platter cross-sectionDisc platter cross-section
Typical Readback PulseTypical Readback Pulse Optimum pulse shapeOptimum pulse shape
versusversus

Thin-Film-Head (TFH) for Magnetic Recording
Design Objective?Design Objective?
Maximize ProfitMaximize Profit
Maximize Pulse SymmetryMaximize Pulse Symmetry
Minimize Pulse WidthMinimize Pulse Width
Minimize PollutionMinimize Pollution
Maximize User SatisfactionMaximize User Satisfaction
??????
Solution: Determine TFH geometrySolution: Determine TFH geometry
parametersparameters A, B, CA, B, C, etc. to achieve objective., etc. to achieve objective.

Matched Filter for Magnetic RecordingMatched Filter for Magnetic Recording
Electrical FilterElectrical Filter
Yin(t) ==>Yin(t) ==>
TransferTransfer
FunctionFunction
--------------------------
H(s)H(s)
==> Yout(t)==> Yout(t)
versusversus
Typical Input Pulse, Yin(t)Typical Input Pulse, Yin(t) Desired Output Pulse, Yout(t)Desired Output Pulse, Yout(t)

Matched Filter for Magnetic RecordingMatched Filter for Magnetic Recording
Results:Results:
Textbook problem solved in 4 hoursTextbook problem solved in 4 hours
Design required 2 years to acquire a true practicalDesign required 2 years to acquire a true practical
objective functionobjective function
Development time droppedDevelopment time dropped from 12 to 1 weekfrom 12 to 1 week
Design was mathematically optimalDesign was mathematically optimal

Explicit & Implicit EquationsExplicit & Implicit Equations
Inverse & Optimization ProblemsInverse & Optimization Problems
Differential EquationsDifferential Equations
IVP & BVP ProblemsIVP & BVP Problems
Limits & ConstraintsLimits & Constraints
FortranCalculusFortranCalculus LanguageLanguage
Example CodeExample Code
AgendaAgenda

Complete Example CodeComplete Example Code
global allglobal all
problem rocket ! three stage rocket design optimizationproblem rocket ! three stage rocket design optimization
dimension spi(3),spivac(3),tburn(3),thrust(3),xip(3),wprop(3),dimension spi(3),spivac(3),tburn(3),thrust(3),xip(3),wprop(3),
& ratio(3),wstage(3),strfac(3),delv(3),g(2)& ratio(3),wstage(3),strfac(3),delv(3),g(2)
thrust(1)=350 : thrust(2)=1500 : thrust(3)=4100thrust(1)=350 : thrust(2)=1500 : thrust(3)=4100
tburn(1)=110 : tburn(2)=100 : tburn(3)=180tburn(1)=110 : tburn(2)=100 : tburn(3)=180
xip(1)=5d-3 : xip(2)=0 : xip(3)=0xip(1)=5d-3 : xip(2)=0 : xip(3)=0
spivac(1)=315 : spivac(2)=315 : spivac(3)=315spivac(1)=315 : spivac(2)=315 : spivac(3)=315
FINDFIND thrust(1),thrust(2),tburn(2),tburn(3); in stages;thrust(1),thrust(2),tburn(2),tburn(3); in stages;
* by Hera; reporting dlvtot,tbtot; to minimize weight* by Hera; reporting dlvtot,tbtot; to minimize weight
endend
model stagesmodel stages
FindFind thrust(3),tburn(1); in eqns; by Ajax; to match gthrust(3),tburn(1); in eqns; by Ajax; to match g
endend
model eqnsmodel eqns
data gc,wpayld,delvip,tbip/32.174,50,2.8e4,400/data gc,wpayld,delvip,tbip/32.174,50,2.8e4,400/
dlvtot=0 : tbtot=0dlvtot=0 : tbtot=0
weight=wpayldweight=wpayld
do 10 i=1,3do 10 i=1,3
spi(i)=spivac(i)*(1-xip(i))spi(i)=spivac(i)*(1-xip(i))
wprop(i)=thrust(i)*tburn(i)/spi(i)wprop(i)=thrust(i)*tburn(i)/spi(i)
wstage(i)=0.0234*thrust(i)+wprop(i)+1.255*wprop(i)**0.704+4wstage(i)=0.0234*thrust(i)+wprop(i)+1.255*wprop(i)**0.704+4
strfac(i)=wprop(i)/wstage(i)strfac(i)=wprop(i)/wstage(i)
weight=weight+wstage(i)weight=weight+wstage(i)
ratio(i)=weight/(weight-wprop(i))ratio(i)=weight/(weight-wprop(i))
delv(i)=gc*spi(i)*log(ratio(i))delv(i)=gc*spi(i)*log(ratio(i))
dlvtot=dlvtot+delv(i)dlvtot=dlvtot+delv(i)
tbtot=tbtot+tburn(i)tbtot=tbtot+tburn(i)
10 continue10 continue
g(1)=dlvtot-delvip ! total delta v constraintg(1)=dlvtot-delvip ! total delta v constraint
g(2)=tbtot-tbipg(2)=tbtot-tbip ! total burn time constraint! total burn time constraint
endend
MathMath
ModelModel

Example Convergence ReportExample Convergence Report
o o oo o o
LOOP NUMBER .... [INITIAL] 5 6LOOP NUMBER .... [INITIAL] 5 6
UNKNOWNSUNKNOWNS
A ( 1) 1.000000E+00 4.432149E-01 3.737358E-01A ( 1) 1.000000E+00 4.432149E-01 3.737358E-01
B ( 1) 1.000000E+00 4.040783E+00 4.284183E+00B ( 1) 1.000000E+00 4.040783E+00 4.284183E+00
C ( 1) 1.230000E+02 4.305000E+02 4.920000E+02C ( 1) 1.230000E+02 4.305000E+02 4.920000E+02
OBJECTIVEOBJECTIVE
ERRSUM 8.189812E+00ERRSUM 8.189812E+00 4.870502E-02 3.879211E-024.870502E-02 3.879211E-02
o o oo o o

Explicit EquationExplicit Equation
o o oo o o
FindFind A,B,CA,B,C; In; In EngineEngine; to ...; to ...
o o oo o o
ModelModel EngineEngine
Y = Function( X;Y = Function( X; A,B,CA,B,C))
End ModelEnd Model
MathMath
ModelModel

Inverse ProblemInverse Problem
o o oo o o
Ydesired = 123.456 ! target valueYdesired = 123.456 ! target value
FindFind A,B,CA,B,C; In; In EngineEngine; to; to MatchMatch GG
o o oo o o
Y = Function( X;Y = Function( X; A,B,CA,B,C))
GG = Ydesired - Y= Ydesired - Y
End ModelEnd Model

Optimization ProblemOptimization Problem
o o oo o o
FindFind A,B,CA,B,C; In; In EngineEngine; to; to MinimizeMinimize GG
o o oo o o
GG = Y - Function( X;= Y - Function( X; A,B,CA,B,C))
End ModelEnd Model

Explicit Differential EquationsExplicit Differential Equations
o o oo o o
InitiateInitiate ISISISIS; For; For EngineEngine;;
EquationsEquations Y2Dot/YDot, YDot/Y; ...Y2Dot/YDot, YDot/Y; ...
o o oo o o
IntegrateIntegrate EngineEngine; By; By ISISISIS
o o oo o o
Y2DotY2Dot = Function( YDot, Y= Function( YDot, Y))
End ModelEnd Model

Initial Value ProblemsInitial Value Problems
Explicit EquationsExplicit Equations
InitiateInitiate
YDot = 123.456 : Y2Dot = 234.567 : Y3Dot = ...YDot = 123.456 : Y2Dot = 234.567 : Y3Dot = ... ! Initial Values! Initial Values
o o oo o o
IntegrateIntegrate ......
o o oo o o
Model ...Model ...
YnDotYnDot = Function( Yn= Function( Yn11Dot, ... , YDot, YDot, ... , YDot, Y))
End ModelEnd Model

Boundary Value ProblemsBoundary Value Problems
Explicit EquationsExplicit Equations
o o oo o o
FindFind YDot0, ... To MatchYDot0, ... To Match HH
o o oo o o
o o oo o o
HH = (Y0 - Y0_desired)**2 += (Y0 - Y0_desired)**2 +
(Ylast - Ylast_desired)**2 ! boundary values(Ylast - Ylast_desired)**2 ! boundary values

Limits & Inequality ConstraintsLimits & Inequality Constraints
o o oo o o
Find ...Find ... WithWith LowersLowers ... And... And UppersUppers ... Holding ...... Holding ...
o o oo o o

TweakTweak TweakTweak TweakTweak
o o oo o o
FindFind E,F,GE,F,G ...... ! Add to any problem! Add to any problem
o o oo o o ! in order to tweak! in order to tweak E,F,GE,F,G ......

Nested Calculus ProcessesNested Calculus Processes
o o oo o o
FindFind ......
o o oo o o
o o oo o o
o o oo o o
FindFind ......

FindFind Statement:Statement:
Key for Calculus-level language Simplicity!Key for Calculus-level language Simplicity!
..

FindFind independentsindependents; In; In modelNamemodelName; By; By solversolver;;
Find StatementFind Statement
The Workhorse of languageThe Workhorse of language
{{With|And} Lower{s}{{With|And} Lower{s} bottomsbottoms;};}
{{With|And} Upper{s}{{With|And} Upper{s} topstops;};}
{{With|And} Holding{{With|And} Holding inequalitiesinequalities;};}
{{With|And} Matching{{With|And} Matching equalitiesequalities;};}
ToTo criterioncriterion

Handling ConstraintsHandling Constraints
o o oo o o
Find a, b, c, etc.; In MyXYZmodel;Find a, b, c, etc.; In MyXYZmodel;
with Holdingwith Holding h1, h2, h3h1, h2, h3,, o o oo o o
EndEnd
Model MyXYZmodelModel MyXYZmodel
h1h1 = a - 4 * w + 100 != a - 4 * w + 100 ! for a > 4w - 100for a > 4w - 100
EndEnd

User Control of SolverUser Control of Solver
o o oo o o
Find a, b, c, etc.; In ABCmodel;Find a, b, c, etc.; In ABCmodel;
By Solver(By Solver( MyCtrlMyCtrl); To); To o o oo o o
EndEnd
ControllerController MyCtrlMyCtrl( Solver)( Solver)
Summary=1 ! produces a summary reportSummary=1 ! produces a summary report
Detail=n ! produce a detailed report everyDetail=n ! produce a detailed report every
! 'n' iterations plus 1st & last.! 'n' iterations plus 1st & last.
o o oo o o
EndEnd

Integrating Differential EquationsIntegrating Differential Equations
o o oo o o
InitiateInitiate solversolver;; {{With} Flag{{With} Flag flagflag;};}
{{With|And} Lower{s}{{With|And} Lower{s} bottomsbottoms;};}
{{With|And} Upper{s}{{With|And} Upper{s} topstops;};}
ForFor modelmodel; Equations; Equations rates/statesrates/states;;
OfOf independentindependent; Step; Step incrementincrement; To; To limitlimit
o o oo o o
IntegrateIntegrate modelmodel

How are problems solved?How are problems solved?
o o oo o o
FindFind ...... byby 'Solver''Solver'
o o oo o o
where 'Solver' is a numerical methodwhere 'Solver' is a numerical method
using Automatic Differention tousing Automatic Differention to
calculate necessary derivatives. Thecalculate necessary derivatives. The
available solvers are in a FC library; e.g.available solvers are in a FC library; e.g.
Ajax, Mars, Neptune, etc.Ajax, Mars, Neptune, etc.

Objective-Driven EngineeringObjective-Driven Engineering
OptimizationsOptimizations
withinwithin
OptimizationOptimization
A nesting exampleA nesting example

Automotive Mfg. CompanyAutomotive Mfg. Company
Optimization LevelsOptimization Levels
Level 1
Company
Company
Design
Dept.
Mfg.
Dept.
Engine
Mfg.
Power
Train
Engine
Design
Power
Train
Level 2
Dept.s
Level 3
Groups

Design Department'sDesign Department's
Engine Design CodeEngine Design Code
(Get Iron, Rubber, etc values from Co. database)(Get Iron, Rubber, etc values from Co. database)
FindFind EngineSizeEngineSize, etc; In, etc; In EngineEngine; to Minimize; to Minimize
PollutionPollution; and Maximize; and Maximize GasEfficiencyGasEfficiency......
o o oo o o
HorsePower = ... Iron ...HorsePower = ... Iron ... EngineSizeEngineSize ......
GasEfficiencyGasEfficiency = ... TerrainType ... HorsePower= ... TerrainType ... HorsePower
PollutionPollution = ... CarWeight ... HorsePower ...= ... CarWeight ... HorsePower ...
GasInEfficiency ... RubberGasInEfficiency ... Rubber
End ModelEnd Model
Level 3

Design Department's CodeDesign Department's Code
(Get Iron, Rubber, etc values from Co. database)(Get Iron, Rubber, etc values from Co. database)
FindFind CarWeightCarWeight, etc; In, etc; In CarDesignCarDesign; to Minimize; to Minimize
CarPollutionCarPollution; and Maximize; and Maximize CarSafetyCarSafety ......
o o oo o o
ModelModel CarDesignCarDesign
Call EngineDesignCall EngineDesign !! Another OptimizationAnother Optimization
Call PowerTrainCall PowerTrain !! Another OptimizationAnother Optimization
CarPollutionCarPollution = ...= ... CarWeightCarWeight ... HorsePower ...... HorsePower ...
GasInEfficiency ... Rubber ... CoalGasInEfficiency ... Rubber ... Coal
CarSafetyCarSafety = ... Iron ... CarWeight ...= ... Iron ... CarWeight ...
End ModelEnd Model
Level 2

Company's CodeCompany's Code
(Get present Iron, Rubber, etc values from Co. database)(Get present Iron, Rubber, etc values from Co. database)
FindFind IronIron,, RubberRubber, etc; In, etc; In CompanyCompany; to Minimize; to Minimize
Time2MarketTime2Market; and Maximize; and Maximize ProfitProfit ......
o o oo o o
ModelModel CompanyCompany
Call DesignCall Design !! Another OptimizationAnother Optimization
Call ManufacturingCall Manufacturing !! Another OptimizationAnother Optimization
Call SalesCall Sales !! Another OptimizationAnother Optimization
Time2MarketTime2Market = ...= ... IronIron ......
ProfitProfit = ...= ... IronIron ...... RubberRubber ... Coal ...... Coal ...
End ModelEnd Model
Level 1

Code for OptimizationsCode for Optimizations
Level 3 Code ... GroupsLevel 3 Code ... Groups
Most Important ... fundamental equationsMost Important ... fundamental equations
Can Run IndependentlyCan Run Independently
Contains Math Models to Simulate Design/Mfg.Contains Math Models to Simulate Design/Mfg.
Level 2 Code ... Dept.sLevel 2 Code ... Dept.s
Runs Latest Level 3 code tooRuns Latest Level 3 code too
Level 1 Code ... CompanyLevel 1 Code ... Company
Runs Latest of All LevRuns Latest of All Levelsels
All L 3s
All L 2s
L 1
Some L 3s
l L 2
1 L 3
Easy to UpdateEasy to Update

Building ApplicationsBuilding Applications
o o oo o o
Find a, b, c, etc.;Find a, b, c, etc.; o o oo o o
call MyOutputcall MyOutput
EndEnd
ModelModel xxxxxx ! derivatives! derivatives ActiveActive
o o oo o o
EndEnd
SubroutineSubroutine MyOutputMyOutput ! derivatives! derivatives NOT ActiveNOT Active
o o oo o o
EndEnd

Variable DefinitionsVariable Definitions
64-bit words ... Local by default64-bit words ... Local by default
GlobalGlobal classesclasses {; In{; In proceduresprocedures }}
Global AllGlobal All
Global realGlobal real
Global real; in abc, def, xyzGlobal real; in abc, def, xyz

Input Variable AccuracyInput Variable Accuracy
for Optimization runsfor Optimization runs
pi = 3.14pi = 3.14 ! Slide rule accuracy, not good!! Slide rule accuracy, not good!
abc = 1.2345678abc = 1.2345678 ! 8+ digits, Good! 8+ digits, Good
def = x.xx...xdef = x.xx...x ! 15 digits, excellent!! 15 digits, excellent!

Code for XY_PlotsCode for XY_Plotsoooooo
C =============== Example code for writing ODE-XYplot files ===============C =============== Example code for writing ODE-XYplot files ===============
@aplot( 1)@aplot( 1) ! any 6 char.s or less for 'od-plt' ... temp filename! any 6 char.s or less for 'od-plt' ... temp filename
endend
procedure aplot( iplot)procedure aplot( iplot)
character*8 fn*12, string*66, date7, time7character*8 fn*12, string*66, date7, time7
C ---------------------------------------------------------------C ---------------------------------------------------------------
C ... Do NOT use stmt. labels numbered less than 11! A bug in FCC ... Do NOT use stmt. labels numbered less than 11! A bug in FC
C ---------------------------------------------------------------C ---------------------------------------------------------------
11 format( 3X, 1pe11.3)11 format( 3X, 1pe11.3)
date7 = "trick 2":date7 = "trick 2": time7 = "pass it"time7 = "pass it"
call getdate( date7, time7)call getdate( date7, time7)
fn = "~4plots.plt":fn = "~4plots.plt": ifile = 30+iplotifile = 30+iplot
C if more than one plot (iplot), change filetype as shown on next line.C if more than one plot (iplot), change filetype as shown on next line.
if( iplot .gt. 1) write(fn, '("~4plots.", i1, "lt")') iplotif( iplot .gt. 1) write(fn, '("~4plots.", i1, "lt")') iplot
!! print *, "ifile, fn = ", ifile, fnprint *, "ifile, fn = ", ifile, fn
!! print *, " "print *, " "
open( ifile, FILE=fn, Status='unknown')open( ifile, FILE=fn, Status='unknown')
C 1st line is file name for this problem.C 1st line is file name for this problem.
! file name of file that created this output plot.! file name of file that created this output plot.
write(ifile, *) "=== enter your file name that calculated these points ==="write(ifile, *) "=== enter your file name that calculated these points ==="
write(ifile,11) errPoswrite(ifile,11) errPos ! 'errsum' value ... gives user idea of convergence.! 'errsum' value ... gives user idea of convergence.
C change 'Bang Bang 2a, Fixed Time' in following 3 lines to your program name.C change 'Bang Bang 2a, Fixed Time' in following 3 lines to your program name.
string = "=== enter problem title here === ... Sol'n: " // date7 //string = "=== enter problem title here === ... Sol'n: " // date7 //
+ " @ " // time7+ " @ " // time7
write(ifile, *) stringwrite(ifile, *) string
C ModelX ... must be > 20 ... hmmm, not sure how to find this.C ModelX ... must be > 20 ... hmmm, not sure how to find this.
modelx = 21modelx = 21
C nTerms is number of terms in series.C nTerms is number of terms in series.
nterms = 0nterms = 0
C nPoints is number of points to be read and then plotted.C nPoints is number of points to be read and then plotted.
nPoints = nptsnPoints = npts ! you need to calculate or enter value for 'npts' here! you need to calculate or enter value for 'npts' here
C nCurves is number of curves in table ... 1 or 2?C nCurves is number of curves in table ... 1 or 2?
nCurves = 1nCurves = 1
C nArrays is number of Arrays in series.C nArrays is number of Arrays in series.
nArrays = 0nArrays = 0
write(ifile, *) modelx, nterms, nPoints, nCurves, nArrayswrite(ifile, *) modelx, nterms, nPoints, nCurves, nArrays
write(ifile, *) " "write(ifile, *) " " ! filename if you want it, blank if not.! filename if you want it, blank if not.
y0 = 0y0 = 0 ! x-offset value! x-offset value
write(ifile, *) y0write(ifile, *) y0
write(ifile,*) "=== your plot title goes here ===" ! plot titlewrite(ifile,*) "=== your plot title goes here ===" ! plot title
write(ifile,*) "Time (seconds)" ! x-axis titlewrite(ifile,*) "Time (seconds)" ! x-axis title
write(ifile,*) "Velocity (mph)" ! y-axis titlewrite(ifile,*) "Velocity (mph)" ! y-axis title
print *, ' 'print *, ' '
print *,' ================ End of Run ==========='print *,' ================ End of Run ==========='
do 90 i=1, nPointsdo 90 i=1, nPoints
x = time(i): y1= velocity(i):x = time(i): y1= velocity(i): y2 = 0y2 = 0
C set 'y2' to zero if not used as shown above OR remove y2 from next line.C set 'y2' to zero if not used as shown above OR remove y2 from next line.
write(31, 11) x, y1, y2write(31, 11) x, y1, y2
close( 31)close( 31)
endend
subroutine getdate( date7, time7)subroutine getdate( date7, time7)
character*(*) date7, time7character*(*) date7, time7
call date( date7)call date( date7)
call time( time7)call time( time7)
returnreturn
endend
See
fc-compilertemplateXYplot-code.fc0
file for this plot code

Helpful TipsHelpful Tips
Normalize your data ... between -1 & 1 or 0 & 1Normalize your data ... between -1 & 1 or 0 & 1
To start ... use '1' amplitude valuesTo start ... use '1' amplitude values
Once model is looking good, round last results toOnce model is looking good, round last results to
2 or 3 digits and try again. Think about your2 or 3 digits and try again. Think about your
solution terrain, any dead-ends?solution terrain, any dead-ends?
Any parametersAny parameters
dependent on others?dependent on others?
If so, tryIf so, try JupiterJupiter solver ...solver ...
usesuses HessianHessian matrix to converge on solution.matrix to converge on solution.

Sinusoidal Curve fit to dataSinusoidal Curve fit to data
'n' Sine functions each with 3 parameters'n' Sine functions each with 3 parameters
A side note:

Some ParametersSome Parameters
areare NOTNOT
Independent!Independent!
ffii andand thetathetaii are dependent parameters!are dependent parameters!
They depend uponThey depend upon aaii
So start search with 'large'So start search with 'large' aaii valuesvalues
to insure partial(to insure partial(ffii, t) & partial(, t) & partial(thetathetaii, t) will carry some weight., t) will carry some weight.
Try another Solver!Try another Solver! JUPITERJUPITER is a keyis a key
solver for tuff problems.solver for tuff problems.

Quiz TimeQuiz Time
A plane crashed!A plane crashed!
'n' sites have debris'n' sites have debris
Find time-line of debris trail & other sightingsFind time-line of debris trail & other sightings

Plane Crash Time-linePlane Crash Time-line
o o oo o o
locPlane=???: velPlane= Err=0: g= 32.??? ...locPlane=???: velPlane= Err=0: g= 32.??? ...
FindFind timetime, height, velocity, etc. ..., height, velocity, etc. ... To MinimizeTo Minimize ErrErr
o o oo o o
ErrH=0: dt= tf=ErrH=0: dt= tf=
InitiateInitiate ISIS;ISIS; forfor abcModel;abcModel; equationsequations dydt / y,dydt / y, ... ;... ; ofof tt;; stepstep dt;dt; toto tftf
do i = 1, nSitesdo i = 1, nSites
tt==time(i)time(i): h=height(i): v=velocity(i) ...: h=height(i): v=velocity(i) ...
do while (do while (hObjecthObject .ge. hDat(i)).ge. hDat(i))
IntegrateIntegrate abcModel;abcModel; byby ISISISIS
! print *,! print *, tt, dydt, y, h, dydt, y, h
end doend do
time(i)time(i) == tt : ErrH = ErrH + (locPlane + velPlane * t: - hDat(i))**2: ErrH = ErrH + (locPlane + velPlane * t: - hDat(i))**2
end doend do
ErrErr = ErrH + ErrVel + ErrLoc + ...= ErrH + ErrVel + ErrLoc + ... ! Recommend normalizing your ErrXXX variables; Direction vs. Velocity Error Importance?! Recommend normalizing your ErrXXX variables; Direction vs. Velocity Error Importance?
EndEnd
Model abcModelModel abcModel
hObjecthObject = h - g *= h - g * tt ** tt / 2/ 2 ! stop fall when h = height / elevation of debris! stop fall when h = height / elevation of debris
EndEnd ! save time 't' for fall.! save time 't' for fall.

Variable FlowVariable Flow
e.g. finding Planee.g. finding Plane DirectionDirection
o o oo o o
<dirPlane> = -999<dirPlane> = -999 ! '-999' flag for no data available! '-999' flag for no data available
FindFind timetime,, ... To Minimize Err... To Minimize Err
o o oo o o
ErrDirErrDir=0=0
InitiateInitiate ISIS;ISIS; forfor abcModel; ...abcModel; ...
o o oo o o
if( dirObj(i) .eq. -999.) go to 22 ! no data available!if( dirObj(i) .eq. -999.) go to 22 ! no data available!
dirPlane(i) = dirObj(i)dirPlane(i) = dirObj(i)
ErrDirErrDir == ErrDirErrDir + ( dirPlane(i) - dirObj(i))**2+ ( dirPlane(i) - dirObj(i))**2
o o oo o o
end doend do
Err = ErrH + ErrVel + ErrLoc +Err = ErrH + ErrVel + ErrLoc + ErrDirErrDir ...... ! Recommend normalize!! Recommend normalize!
EndEnd

Quiz: Variable FlowQuiz: Variable Flow
e.g. finding Planee.g. finding Plane VelocityVelocity
o o oo o o
<velPlane> = -999<velPlane> = -999 ! '-999' flag for no data available! '-999' flag for no data available
FindFind timetime,, ... To Minimize Err... To Minimize Err
o o oo o o
ErrVelErrVel=0=0
InitiateInitiate ISIS;ISIS; forfor abcModel; ...abcModel; ...
! your turn, what code goes here in order to find plane velocity?! your turn, what code goes here in order to find plane velocity?
! any additions in remainder of code for plane velocity?! any additions in remainder of code for plane velocity?

Crash Conclusion ProceduresCrash Conclusion Procedures
Now you should have a plane time-line of events. Plot planeNow you should have a plane time-line of events. Plot plane
height, velocity, direction, etc. over time-line in order to analysisheight, velocity, direction, etc. over time-line in order to analysis
flight.flight.
With the 2 last event points, calculate their slope. Is the planeWith the 2 last event points, calculate their slope. Is the plane
heading for a crash? If so, when ... how long (time & distance)heading for a crash? If so, when ... how long (time & distance)
before crash?before crash?
Try to extrapolate, from the last few data points, where theTry to extrapolate, from the last few data points, where the
plane may have crashed.plane may have crashed.

What are you building?What are you building?
Mr. Arithmetic Mr. Algebra Mr. CalculusMr. Arithmetic Mr. Algebra Mr. Calculus
------------------- --------------- ----------------------------------- --------------- ----------------
Slide Rule Simulation OptimizeSlide Rule Simulation Optimize

ConclusionConclusion
UseUse
FortranCalculusFortranCalculus
to improveto improve
Math ModelsMath Models
Design ProductivityDesign Productivity
Manufacturing TolerancesManufacturing Tolerances

Comments & FeedbackComments & Feedback
Have a new Curve Fit Model?Have a new Curve Fit Model?
Have data set for Human Heart Beat, 1-cycle?Have data set for Human Heart Beat, 1-cycle?
If so, please contact us atIf so, please contact us at optim.designs@gmail.comoptim.designs@gmail.com
Once model is looking good, round last results toOnce model is looking good, round last results to
2 or 3 digits and try again.2 or 3 digits and try again.

Calculus-level Coding Overview

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