Keystrokes, ´ i ´ f | HP 15c Scientific instruction

Section 14: Numerical Integration 201

Because the accuracy of any integral is limited by the accuracy of the function (as indicated in the display format), the calculator cannot compute the value of an integral exactly, but rather only approximates it. The HP-15C places the uncertainty* of an integral's approximation in the Y- register at the same time it places the approximation in the X-register. To determine the accuracy of an approximation, check its uncertainty by pressing ®.

Example: With the display format set to i 2, calculate the integral in the expression for J1(1) (from the example on page 197).

Keystrokes

0 v

$

R

´i 2

´f 1

Display
0.0000	Key lower limit into
	Y-register.
3.1416	Key upper limit into
	X-register.
3.1416	(If not already in Radians mode.)

3.1400 Set display format to i2.

1.300 Integral approximated in i2.

8

® 1.8 - Uncertainty of i2

803 approximation.

The integral is 1.38 ± 0.00188. Since the uncertainty would not affect the approximation until its third decimal place, you can consider all the displayed digits in this approximation to be accurate. In general, though, it is difficult to anticipate how many digits in an approximation will be unaffected by its uncertainty. This depends on the particular function being integrated, the limits of integration, and the display format.

*No algorithm for numerical integration can compute the exact difference between its approximation and the actual integral. But the algorithm in the HP-15C estimates an ―upper bound‖ on this difference, which is the uncertainty of the approximation. For example, if the integral Si (2) is 1.6054 ± 0.0001, the approximation to the integral is 1.6054 and its uncertainty is 0.0001. This means that while we don't know the exact difference between the actual integral and its approximation, we do know that it is highly unlikely that the difference is bigger than 0.0001. (Note the first footnote on page 200.)

Image 201

HP 15c Scientific manual Keystrokes, ´ i ´ f

Contents

HP Part Number 00015-90001 Edition 2.4, Sep HP-15C Owner’s Handbook Legal Notice Introduction Contents Display and Continuous Memory Contents Program Branching and Controls Program Editing Subroutines Indirect Display Control Calculating With Complex NumbersCalculating With Matrices Numerical Integration Appendix C Memory Allocation Contents Appendix a Error ConditionsAppendix D a Detailed Look at Appendix E a Detailed Look at f Function Summary and Index Contents Appendix F BatteriesProgramming Summary and Index Subject Index Quick Look at HP-15C Problem Solver To Compute Keystrokes Display Manual Solutions Keystrokes Display Programmed Solutions 001-42,21,11 KeystrokesDisplay002 003 004 005 006 007 008 009 8313 300.51 HP-15C a Problem Solver Part l HP-15C Fundamentals Getting Started Power On and OffKeyboard Operation Section Changing Signs Prefix KeysKeying in Exponents I O m ´ P I l F T s ? t H b Clear Keys Display Clearing ` and − Digit entry not terminatedClears only the last digit One-Number Functions CalculationsTwo-Number Functions 6532 26.0000 22.0000 5000 17 +13.0000 78.0000 Number Alteration Functions Numeric Functions General Functions One-Number Functions Trigonometric Operations Time and Angle ConversionsPressing Calculates 7069 Degrees/Radians ConversionsRadians 40.5000 Hyperbolic Functions Logarithmic Functions Two-Number Functions Power FunctionPercentages To Calculate Keystrokes Display Enters the base number the price Polar and Rectangular Coordinate ConversionsCalculates 3% of $15.76 the tax Polar Conversion. Pressing Keystrokes Display Automatic Memory Stack Stack Manipulation Automatic Memory Stack Last X, and Data StorageAutomatic Memory Stack Registers Always displayed Stack Manipulation Functions Memory Stack, Last X, and Data StorageLost Lost 287.0000 Last X Register and K22.2481 12.9000 Calculator Functions and the Stack 13.9 +20.6475 +15 X15 Order of Entry and the v Key 7 + Nested Calculations65.0000 69.0000 Arithmetic Calculations With Constants 5 ‛15 Keys Keystrokes Display Growth factor 1000000 Storing and Recalling Numbers Storage Register Operations322.5000 520.8750 Storage and Recall Arithmetic Clearing Data Storage Registers For recall arithmetic For storage arithmetic Overflow and Underflow Problems24 l-0 15.0000 Memory Stack, Last X, and Data Storage Statistics Functions Probability Calculations60.0000 270,725.0000 Random Number Generator5764 3422 Registers Accumulating Statistics Register 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 Σy2 20 w 20 z Correcting Accumulated Statistics Mean Standard Deviation40.00 31.62 Linear RegressionStandard deviation about the mean nitrogen Application Linear Estimation and Correlation Coefficient Statistics Functions 70 ´j Other Applications Display Continuous Memory Display ControlFixed Decimal Display Engineering Notation Display Scientific Notation Display234568 234567 Special Displays Round-Off ErrorMantissa Display Annunciators Digit Separators Error Display12,345.67 12.345.6700 Low-Power Indication Continuous MemoryStatus Resetting Continuous Memory Page Part ll HP-15C Programming Mechanics Programming BasicsCreating a Program Loading a Program ´b a Programming Basics Intermediate Program Stops Running a Program002 003 004 005 006 007 008 300.51 300.51 ´A How to Enter Data Program Memory Totals Radius, r Height, h Base Area Volume Surface Area 004 005 002007-44,40 010 Or G a Further Information Program InstructionsInstruction Coding Keycode 25 second row, fifth key Memory Configuration 60 ´ m% Initial Memory Configuration ´ m % Program Boundaries19 ´ m% 19.0000 Unexpected Program Stops Abbreviated Key Sequences´bA ´b3 End of memory Polynomial Expressions and Horners Method User Mode¤ @ y ∕ LOG % 001-42,21,12 Nonprogrammable Functions002 003 004 005 006 007 008 009 0000 12,691.0000 Problems Moving to a Line in Program Memory Program Editing Examples Deleting Program LinesInserting Program Lines Or use Â Single-Step Operations Âhold Line PositionRelease Result Initializing Calculator Status Insertions and Deletions Interest + i nPV 1 + i n ´bA D ´4 O0 2* O1 2÷ * ´ ´ l0 l1 ´r * n 100 270 Branching Program Branching Controls Test Conditional Tests  n will clear flag number n Flags Example Branching and Looping 013-43,30 010-45,20014 016-44,40 Formula is Example Flags 004-42,21,15 002-43005-43, 4 006-42,21 250.0000 Go to48.0000 10,698.3049 Conditional Branching Looping System Flags Flags 8 Program Branching and Controls Go To Subroutine and Return SubroutinesSubroutine Execution ´b.1 Subroutine Limits 002- R 000 001- ´b9003- O0 004 ´b.5 ´ b.4 Nested Subroutines Subroutine Return Index Register Loop Control V and % Keys106 Program Loop Control Indirect Program Control With the Index RegisterIndex Register Storage and Recall Index Register and Loop Control Index Register Arithmetic Exchanging the X-RegisterIndirect Branching With Indirect Flag Control With Indirect Display Format Control WithLoop Control With Counters I and e Start count at zero Count by twos Count up to Nnnnn x x x y y 5 0 0 Iterations Examples Register OperationsStoring and Recalling Keystrokes Display 12.3456 Example Loop Control with e Exchanging the X-RegisterStorage Register Arithmetic −− 011- 42 Loop control number in R2012-42, 5 013- 22 Example Display Format Control 15 O64.8420 0000 50.0000 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 120 Deactivating Complex Mode Complex Numbers and the StackEntering Complex Numbers ´ % hold 8.0000 release Z 8 Y 7 X Keys Manipulating the Real and Imaginary Stacks Stack Lift in Complex Mode Or other operation Clearing a Complex Number − 4 v Continue with any operation Continue with any operation Entering Complex Numbers with −. The clearing functions − ´ %hold release0000 17.0000 144.0000 Followed by another number Entering a Real Number ´ Continue with any operation Entering a Pure Imaginary Number Storing and Recalling Complex Numbers Operations With Complex Numbers´ O L 2 ´ + * ÷ y ¤x N o ∕ @ a 7000 20000428 0491 Complex Results from Real Numbers Polar and Rectangular Coordinate Conversions5708 ´ % hold Release1.5708 Cos θ + i sin θ = re iθ Polar + ib = ∠ θ 8452 2981+ 3.1434 872.0000 352.00002361 4721 For Further Information 138 Calculating With Matrices = A-1B Keystrokes Display Deactivates Complex Mode Running Matrix Dimensions11.2887 2496 Number Rows Columns Dimensioning a Matrix Changing Matrix Dimensions Displaying Matrix Dimensions´mA Keystrokes l B Display Storing and Recalling All Elements in Order Storing and Recalling Matrix Elements ⎡ a Checking and Changing Matrix Elements Individually Keystrokes Display Matrix Operations Storing a Number in All Elements of a MatrixMatrix Descriptors Result Matrix One-Matrix Operations Copying a Matrix Calculating with Matrices LB b Scalar Operations LA a Elements of Result Matrix Arithmetic Operations Keystrokes Display Subtracts 1 from the elementsLB b 2 LA a 2 Matrix Multiplication Keystrokes Display l a a = AT B Solving the Equation AX = B 2400 24 OA86 OA 8600 274 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 Residual Calculations With Complex Matrices Using Matrices in LU Form Then Z can be represented in the calculator by Storing the Elements of a Complex Matrix Pressing 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 a ZZ −1 Solving the Complex Equation AX = B AX = B 170.0000 200.0000 1311 03720437 1543 Calculating with Matrices Using a Matrix Element With Register Operations Using Matrix Descriptors in the Index RegisterMiscellaneous Operations Involving Matrices Conditional Tests on Matrix Descriptors Stack Operation for Matrix Calculations Calculating with Matrices Using Matrix Operations in a Program Summary of Matrix Functions Keystrokes Results´m a Calculates residual in result matrix For Further Information Using Finding the Roots An Equation180 Clear program memory Finding the Roots of an Equation 001-42,21 ´b0002 003 005 006 007 Desired root Finding the Roots of an Equation ´ bA Keystrokes ¥000 001-42,21,11 003 004 Brings another t-value 5000 1 e tInto X-register 200 t 000 001-42,21 002 003 004 005 When No Root Is Found Error Choosing Initial Estimates Label X + 8 003 004 005 007008 009 6 x + 8 Finding the Roots of an Equation Using in a Program Memory Requirements Restriction on the Use Using f Numerical Integration194 002 003 004 1416 7652 4040 Begin subroutine with a label 3825 $ ÷4401 6054 Accuracy of f ´ i ´ f 7091 8826 382 Using f in a Program Memory Requirements Error 0 Improper Mathematics Operation Error ConditionsAppendix a 205 Error 2 Improper Statistics Operation Error 1 Improper Matrix Operation Error 4 Improper Line Number or Label Call Error 3 Improper Register Number or Matrix ElementError 5 Subroutine Level Too Deep Error 6 Improper Flag Number Pr Error Power Error Digit Entry Termination Stack Lift Last X RegisterStack Lift Appendix B Enabling Operations Disabling Operations Neutral 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 Register Memory Space Memory AllocationAppendix C Registers Appendix C Memory Allocation Memory Reallocation Memory Status WM % Function Restrictions on Reallocation ´m% 1.0000 Whold 1 6419 ´ m Automatic Program Memory Reallocation Program Memory Two-Byte Program Instructions Memory Requirements for the Advanced FunctionsIf executed Together Appendix C Memory Allocation How Works Detailed Look atAppendix D 220 Appendix D a Detailed Look at Accuracy of the Root X4 = 000 006 007 008 009 010-43,30 011 012-43,30 013 1718 Interpreting Results ´ v B0681 − 45 For 0 x Branch for x ≥ Test for x range3x 45x 2 + End subroutine Initial estimates 000.00001358 Possible root Appendix D a Detailed Look at Bring x-value into X-register ´ b.0 001-42,21,.0 002 003 004 005007 008 009 010 013 014 015 016 10 v ´ ‛ 20 017 018 Finding Several Roots Error 0000 1250 5626 002 003 004 005 006 007 Fx = xx a3 = 6667 Second root Same initial estimatesStores root for deflation Deflated function value Deflation for third root Limiting the Estimation Time For Advanced Information Counting IterationsSpecifying a Tolerance How f Works Detailed Look at fAppendix E 240 X = 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 hold 7807 7858 Uncertainty 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 Uncertainty Appendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Installing New Batteries Low-Power IndicationBatteries Batteries Appendix F Batteries 2.C 3.H Verifying Proper Operation Self-Tests Complex Functions Function Summary and IndexConversions Digit Entry Index Register Control Display ControlLogarithmic Exponential Functions Mantissa. Pressing Mathematics Matrix Functions146 Number Alteration To ZP page164To XT Probability PercentageStack Manipulation Clear u Storage Statistics Trigonometry 269 Programming Summary and Index Programming Summary and Index 271 Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Product Regulatory Environment Information Federal Communications Commission NoticeModifications Avis Canadien Canadian Notice Body number is inserted between CE European Union Regulatory Notice