HP 15c Scientific manual Uncertainty and the Display Format

Page 245

Appendix E: A Detailed Look at f 245

This approximation took about twice as long as the approximation in i 3 or i 2. In this case, the algorithm had to evaluate the function at about twice as many sample points as before in order to achieve an approximation of acceptable accuracy. Note, however, that you received a reward for your patience: the accuracy of this approximation is better, by almost two digits, than the accuracy of the approximation calculated using half the number of sample points.

The preceding examples show that repeating the approximation of an integral in a different display format sometimes will give you a more accurate answer, but sometimes it will not. Whether or not the accuracy is changed depends on the particular function, and generally can be determined only by trying it.

Furthermore, if you do get a more accurate answer, it will come at the cost of about double the calculation time. This unavoidable trade-off between accuracy and calculation time is important to keep in mind if you are considering decreasing the uncertainty in hopes of obtaining a more accurate answer.

The time required to calculate the integral of a given function depends not only on the number of digits specified in the display format, but also, to a certain extent on the limits of integration. When the calculation of an integral requires an excessive amount of time, the width of the interval of integration (that is, the difference of the limits) may be too large compared with certain features of the function being integrated. For most problems, however, you need not be concerned about the effects of the limits of integration on the calculation time. These conditions, as well as techniques for dealing with such situations, will be discussed later in this appendix.

Uncertainty and the Display Format

Because of round-off error, the subroutine you write for evaluating f(x) cannot calculate f(x) exactly, but rather calculates

ˆ = ± δ

f (x) f (x) 1(x),

where δ1 (x) is the uncertainty of f(x) caused by round-off error. If f(x) relates to a physical situation, then the function you would like to integrate is not f(x) but rather

Image 245

Contents HP Part Number 00015-90001 Edition 2.4, Sep HP-15C Owner’s HandbookLegal Notice Introduction Contents Display and Continuous Memory ContentsProgram Branching and Controls Program EditingSubroutines Calculating With Matrices Indirect Display ControlCalculating With Complex Numbers Numerical Integration Appendix C Memory Allocation Contents Appendix a Error ConditionsAppendix D a Detailed Look at Appendix E a Detailed Look at fFunction Summary and Index Contents Appendix F BatteriesProgramming Summary and Index Subject IndexQuick Look at HP-15C Problem SolverTo Compute Keystrokes Display Manual SolutionsKeystrokes Display Programmed Solutions001-42,21,11 KeystrokesDisplay002 003 004 005 006 007 008 009 8313 300.51HP-15C a Problem Solver Part l HP-15C Fundamentals Getting Started Power On and OffKeyboard Operation SectionChanging Signs Prefix KeysKeying in Exponents I O m ´ P I l F T s ? t H bClear Keys Clears only the last digit Display Clearing ` and −Digit entry not terminated One-Number Functions CalculationsTwo-Number Functions 653226.0000 22.0000 5000 17 +13.0000 78.0000Number Alteration Functions Numeric FunctionsGeneral Functions One-Number FunctionsPressing Calculates Trigonometric OperationsTime and Angle Conversions 7069 Degrees/Radians ConversionsRadians 40.5000Hyperbolic Functions Logarithmic FunctionsTwo-Number Functions Power FunctionPercentages To Calculate Keystrokes DisplayEnters the base number the price Polar and Rectangular Coordinate ConversionsCalculates 3% of $15.76 the tax Polar Conversion. PressingKeystrokes Display Automatic Memory Stack Stack Manipulation Automatic Memory Stack Last X, and Data StorageAutomatic Memory Stack Registers Always displayedLost Stack Manipulation FunctionsMemory Stack, Last X, and Data Storage Lost 287.0000 Last X Register and K22.2481 12.900020.6475 Calculator Functions and the Stack13.9 + +15 X15 Order of Entry and the v Key7 + Nested Calculations65.0000 69.0000Arithmetic Calculations With Constants 5 ‛15 Keys 000 Keystrokes Display Growth factor1000 Storing and Recalling Numbers Storage Register Operations322.5000 520.8750Storage and Recall Arithmetic Clearing Data Storage RegistersFor recall arithmetic For storage arithmeticOverflow and Underflow Problems24 l-0 15.0000Memory Stack, Last X, and Data Storage 60.0000 Statistics FunctionsProbability Calculations 270,725.0000 Random Number Generator5764 3422Registers Accumulating StatisticsRegister Contents Metric tons per Hectare, y 20.00 40.00 60.00 80.00 Kg per hectare20 z 61v 40 z 7.21 60 z 7.78 80 z l Σy220 w 20 z Correcting Accumulated Statistics40.00 MeanStandard Deviation 31.62 Linear RegressionStandard deviation about the mean nitrogen ApplicationLinear Estimation and Correlation Coefficient Statistics Functions 70 ´j Other ApplicationsFixed Decimal Display Display Continuous MemoryDisplay Control Engineering Notation Display Scientific Notation Display234568 234567Special Displays Round-Off ErrorMantissa Display AnnunciatorsDigit Separators Error Display12,345.67 12.345.6700Status Low-Power IndicationContinuous Memory Resetting Continuous Memory Page Part ll HP-15C Programming Mechanics Programming BasicsCreating a Program Loading a Program´b a Programming Basics002 003 004 005 006 007 008 Intermediate Program StopsRunning a Program 300.51 300.51 ´A How to Enter DataProgram Memory Totals Radius, r Height, h Base Area Volume Surface Area004 005 002007-44,40 010Or G a Instruction Coding Further InformationProgram Instructions Keycode 25 second row, fifth key Memory Configuration60 ´ m% Initial Memory Configuration´ m % Program Boundaries19 ´ m% 19.0000´bA ´b3 End of memory Unexpected Program StopsAbbreviated Key Sequences Polynomial Expressions and Horners Method User Mode¤ @ y ∕ LOG %001-42,21,12 Nonprogrammable Functions002 003 004 005 006 007 008 009 0000 12,691.0000Problems Moving to a Line in Program Memory Program EditingInserting Program Lines ExamplesDeleting Program Lines Or use Â Single-Step Operations Âhold Line PositionRelease ResultInitializing Calculator Status Insertions and DeletionsPV 1 + i n Interest+ i n ´bA D ´4 O0 2* O1 2÷ * ´ ´ l0 l1 ´r * n 100 270Branching Program Branching ControlsTest Conditional Tests n will clear flag number n FlagsExample Branching and Looping 013-43,30 010-45,20014 016-44,40Formula is Example Flags004-42,21,15 002-43005-43, 4 006-42,21250.0000 Go to48.0000 10,698.3049Conditional Branching LoopingSystem Flags Flags 8 Program Branching and Controls Go To Subroutine and Return SubroutinesSubroutine Execution ´b.1Subroutine Limits 002- R 000 001- ´b9003- O0 004´b.5 ´ b.4Nested Subroutines Subroutine Return106 Index Register Loop ControlV and % Keys Program Loop Control Indirect Program Control With the Index RegisterIndex Register Storage and Recall Index Register and Loop ControlIndirect Branching With Index Register ArithmeticExchanging the X-Register Loop Control With Counters I and e Indirect Flag Control WithIndirect Display Format Control With Start count at zero Count by twos Count up to Nnnnn x x x y y 5 0 0Iterations Examples Register OperationsStoring and Recalling Keystrokes Display 12.3456Storage Register Arithmetic Example Loop Control with eExchanging the X-Register −− 011- 42 Loop control number in R2012-42, 5 013- 2264.8420 0000 50.0000 Example Display Format Control15 O Index Register Contents Indirect Display Control Index Register and Loop Control 118 Part lll HP-15C Advanced Functions Calculating With Complex Numbers Complex Stack and Complex ModeCreating the Complex Stack 120Entering Complex Numbers Deactivating Complex ModeComplex Numbers and the Stack ´ % hold 8.0000 release Z 8 Y 7 X Keys Manipulating the Real and Imaginary Stacks Stack Lift in Complex ModeOr other operation Clearing a Complex Number− 4 v Continue with any operation Continue with any operation0000 17.0000 144.0000 Entering Complex Numbers with −. The clearing functions −´ %hold release Followed by another number Entering a Real Number´ Continue with any operation Entering a Pure Imaginary NumberStoring and Recalling Complex Numbers Operations With Complex Numbers´ O L 2 ´+ * ÷ y ¤x N o ∕ @ a7000 20000428 0491Complex Results from Real Numbers Polar and Rectangular Coordinate Conversions5708 ´ % hold Release1.5708Cos θ + i sin θ = re iθ Polar + ib = ∠ θ + 3.1434 84522981 872.0000 352.00002361 4721For Further Information 138 Calculating With Matrices= A-1B Keystrokes Display Deactivates Complex ModeRunning Matrix Dimensions11.2887 2496Number Rows Columns Dimensioning a MatrixChanging Matrix Dimensions Displaying Matrix Dimensions´mA Keystrokes l B DisplayStoring and Recalling All Elements in Order Storing and Recalling Matrix Elements⎡ a Checking and Changing Matrix Elements Individually Keystrokes Display Matrix Descriptors Matrix OperationsStoring a Number in All Elements of a Matrix Result Matrix One-Matrix Operations Copying a MatrixCalculating with Matrices LB b Scalar OperationsLA a Elements of Result MatrixLB b 2 LA a 2 Arithmetic OperationsKeystrokes Display Subtracts 1 from the elements Matrix Multiplication Keystrokes Display l a a = AT BSolving the Equation AX = B 2400 24 OA86 OA 8600274 OB 233 OB 331 OB 120.32 OB 112.96 OB 151.36 OB ´Á Week Cabbage kg 186 141 215 Broccoli kg 116 Calculating the ResidualCalculations With Complex Matrices Using Matrices in LU FormThen Z can be represented in the calculator by Storing the Elements of a Complex MatrixPressing Transforms Into LA a = ⎢Complex Transformations Between ZP and Z Inverting a Complex Matrix ´ a Multiplying Complex Matrices´U lC LC lC lC lC lC lC lC ´U Keystrokes lA lB Display Displays descriptor of matrix aZZ −1 Solving the Complex Equation AX = BAX = B 170.0000 200.00001311 03720437 1543Calculating with Matrices Miscellaneous Operations Involving Matrices Using a Matrix Element With Register OperationsUsing Matrix Descriptors in the Index Register Conditional Tests on Matrix Descriptors Stack Operation for Matrix CalculationsCalculating with Matrices Using Matrix Operations in a Program ´m a Summary of Matrix FunctionsKeystrokes Results Calculates residual in result matrix For Further Information 180 UsingFinding the Roots An Equation Clear program memory Finding the Roots of an Equation001-42,21 ´b0002 003 005 006 007Desired root Finding the Roots of an Equation´ bA Keystrokes ¥000 001-42,21,11 003 004Brings another t-value 5000 1 e tInto X-register 200 t000 001-42,21 002 003 004 005 When No Root Is FoundError Choosing Initial Estimates Label X + 8 003 004 005 007008 009 6 x + 8Finding the Roots of an Equation Using in a Program Memory Requirements Restriction on the Use194 Using fNumerical Integration 002 003 004 1416 7652 4040Begin subroutine with a label 4401 3825$ ÷ 6054 Accuracy of f ´ i ´ f 7091 8826382 Using f in a ProgramMemory Requirements Error 0 Improper Mathematics Operation Error ConditionsAppendix a 205Error 2 Improper Statistics Operation Error 1 Improper Matrix OperationError 4 Improper Line Number or Label Call Error 3 Improper Register Number or Matrix ElementError 5 Subroutine Level Too Deep Error 6 Improper Flag NumberPr Error Power Error Digit Entry Termination Stack Lift Last X RegisterStack Lift Appendix BEnabling Operations Disabling OperationsNeutral Operations Stack Stack Enabled. disabled 53.1301 No stack LiftAppendix B Stack Lift and the Last X Register Keys Nnn Clear u ¥\ k + H ∆ \ h ÷ À P* q r c ‘ / N z ∕ P\ o j Last X RegisterMemory Space Memory AllocationAppendix C RegistersAppendix C Memory Allocation M % Function Memory ReallocationMemory Status W 19 ´ m Restrictions on Reallocation´m% 1.0000 Whold 1 64 Automatic Program Memory Reallocation Program MemoryTwo-Byte Program Instructions Memory Requirements for the Advanced FunctionsIf executed TogetherAppendix C Memory Allocation How Works Detailed Look atAppendix D 220Appendix D a Detailed Look at Accuracy of the Root X4 = 000 006 007 008 009 010-43,30 011 012-43,30 013 17180681 Interpreting Results´ v B − 45 For 0 x Branch for x ≥ Test for x range3x 45x 2 + End subroutineInitial estimates 000.00001358 Possible rootAppendix D a Detailed Look at Bring x-value into X-register ´ b.0 001-42,21,.0 002 003 004 005007 008 009 010 013 014 015 01610 v ´ ‛ 20 017 018Finding Several Roots Error 0000 1250 5626002 003 004 005 006 007 Fx = xx a3 =6667 Second root Same initial estimatesStores root for deflation Deflated function valueDeflation for third root Limiting the Estimation Time Specifying a Tolerance For Advanced InformationCounting Iterations How f Works Detailed Look at fAppendix E 240X = π1 0π cos4θ − x sinθ dθ Accuracy, Uncertainty, and Calculation Time´ i ´ f 0000 1416´ Clear u Hold Keystrokes Display Return approximation toKeystrokes ´ i Display ´ f ´ Clear u hold7807 7858Uncertainty and the Display Format Functions values for example Δx = 0.5×10−n ×10m = aδx dxb = ab 0.5×10−n + m x dx Conditions That Could Cause Incorrect Results ∞ xe− xdx 001-42,21 002- 1 003 004 005 Appendix E a Detailed Look at f Appendix E a Detailed Look at f Conditions That Prolong Calculation Time Keys upper limit into Keys lower limit intoApproximation to integral UncertaintyAppendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Installing New Batteries Low-Power IndicationBatteries BatteriesAppendix F Batteries 2.C 3.H Verifying Proper Operation Self-TestsComplex Functions Function Summary and IndexConversions Digit EntryIndex Register Control Display ControlLogarithmic Exponential Functions Mantissa. Pressing146 MathematicsMatrix Functions To XT Number AlterationTo ZP page164 Probability PercentageStack Manipulation Clear uStorage StatisticsTrigonometry 269 Programming Summary and IndexProgramming Summary and Index 271 Subject IndexSubject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Modifications Product Regulatory Environment InformationFederal Communications Commission Notice Avis Canadien Canadian NoticeBody number is inserted between CE European Union Regulatory Notice

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