HP 15c Scientific manual Limiting the Estimation Time

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238 Appendix D: A Detailed Look at _

Using the same initial estimates each time, you have found four roots for this equation involving a fourth-degree polynomial. However, the last two roots are quite close to each other and are actually one root (with a multiplicity of 2). That is why the root was not eliminated when you tried deflation once at this root. (Round-off error causes the original function to have small positive and negative values for values of x between 8.4999 and 8.5001; for x = 8.5 the function is exactly zero.)

In general, you will not know in advance the multiplicity of the root you are trying to eliminate. If, after you have attempted to eliminate a root, _ finds that same root again, you can proceed in a number of ways:

∙Use different initial estimates with the deflated function in an attempt to search for a different root.

∙Use deflation again in an attempt to eliminate a multiple root. If you do not know the multiplicity of the root, you may need to repeat this a number of times.

∙Examine the behavior of the deflated function at x-values near the known root. If the function's calculated values cross the x-axis smoothly, either another root or a greater multiplicity is indicated.

∙Analyze the original function and its derivatives algebraically. It may be possible to determine its behavior for x-values near the known root. (A Taylor series representation, for example, may indicate the multiplicity of a root.)

Limiting the Estimation Time

Occasionally, you may desire to limit the time used by _ to find a root. You can use two possible techniques to do this – counting iterations and specifying a tolerance.

Image 238

Contents HP-15C Owner’s Handbook HP Part Number 00015-90001 Edition 2.4, SepLegal Notice Introduction Contents Contents Display and Continuous MemoryProgram Editing Program Branching and ControlsSubroutines Calculating With Complex Numbers Indirect Display ControlCalculating With Matrices Numerical Integration Appendix D a Detailed Look at Contents Appendix a Error ConditionsAppendix C Memory Allocation Appendix E a Detailed Look at fProgramming Summary and Index Contents Appendix F BatteriesFunction Summary and Index Subject IndexHP-15C Problem Solver Quick Look atManual Solutions To Compute Keystrokes DisplayProgrammed Solutions Keystrokes Display002 003 004 005 006 007 008 009 8313 KeystrokesDisplay001-42,21,11 300.51HP-15C a Problem Solver Part l HP-15C Fundamentals Keyboard Operation Power On and OffGetting Started SectionKeying in Exponents Prefix KeysChanging Signs I O m ´ P I l F T s ? t H bClear Keys Digit entry not terminated Display Clearing ` and −Clears only the last digit Two-Number Functions CalculationsOne-Number Functions 653213.0000 17 +26.0000 22.0000 5000 78.0000Numeric Functions Number Alteration FunctionsOne-Number Functions General FunctionsTime and Angle Conversions Trigonometric OperationsPressing Calculates Radians Degrees/Radians Conversions7069 40.5000Logarithmic Functions Hyperbolic FunctionsPercentages Power FunctionTwo-Number Functions To Calculate Keystrokes DisplayCalculates 3% of $15.76 the tax Polar and Rectangular Coordinate ConversionsEnters the base number the price Polar Conversion. PressingKeystrokes Display Automatic Memory Stack Registers Automatic Memory Stack Last X, and Data StorageAutomatic Memory Stack Stack Manipulation Always displayedMemory Stack, Last X, and Data Storage Stack Manipulation FunctionsLost Lost 22.2481 Last X Register and K287.0000 12.900013.9 + Calculator Functions and the Stack20.6475 Order of Entry and the v Key +15 X1565.0000 Nested Calculations7 + 69.0000Arithmetic Calculations With Constants 5 ‛15 Keys 1000 Keystrokes Display Growth factor000 322.5000 Storage Register OperationsStoring and Recalling Numbers 520.8750Clearing Data Storage Registers Storage and Recall ArithmeticFor storage arithmetic For recall arithmetic24 l-0 ProblemsOverflow and Underflow 15.0000Memory Stack, Last X, and Data Storage Probability Calculations Statistics Functions60.0000 5764 Random Number Generator270,725.0000 3422Accumulating Statistics RegistersRegister Contents 20 z 61v 40 z 7.21 60 z 7.78 80 z l 20.00 40.00 60.00 80.00 Kg per hectareMetric tons per Hectare, y Σy2Correcting Accumulated Statistics 20 w 20 zStandard Deviation Mean40.00 Standard deviation about the mean nitrogen Linear Regression31.62 ApplicationLinear Estimation and Correlation Coefficient Statistics Functions Other Applications 70 ´jDisplay Control Display Continuous MemoryFixed Decimal Display 234568 Scientific Notation DisplayEngineering Notation Display 234567Mantissa Display Round-Off ErrorSpecial Displays Annunciators12,345.67 Error DisplayDigit Separators 12.345.6700Continuous Memory Low-Power IndicationStatus Resetting Continuous Memory Page Part ll HP-15C Programming Creating a Program Programming BasicsMechanics Loading a ProgramProgramming Basics ´b aRunning a Program Intermediate Program Stops002 003 004 005 006 007 008 How to Enter Data 300.51 300.51 ´AProgram Memory Radius, r Height, h Base Area Volume Surface Area Totals007-44,40 002004 005 010Or G a Program Instructions Further InformationInstruction Coding Memory Configuration Keycode 25 second row, fifth keyInitial Memory Configuration 60 ´ m%19 ´ m% Program Boundaries´ m % 19.0000Abbreviated Key Sequences Unexpected Program Stops´bA ´b3 End of memory ¤ @ y ∕ User ModePolynomial Expressions and Horners Method LOG %002 003 004 005 006 007 008 009 0000 Nonprogrammable Functions001-42,21,12 12,691.0000Problems Program Editing Moving to a Line in Program MemoryDeleting Program Lines ExamplesInserting Program Lines Or use Â Single-Step Operations Release Line PositionÂhold ResultInsertions and Deletions Initializing Calculator Status+ i n InterestPV 1 + i n 100 270 ´bA D ´4 O0 2* O1 2÷ * ´ ´ l0 l1 ´r * nProgram Branching Controls BranchingConditional Tests TestFlags  n will clear flag number nExample Branching and Looping 014 010-45,20013-43,30 016-44,40Example Flags Formula is005-43, 4 002-43004-42,21,15 006-42,2148.0000 Go to250.0000 10,698.3049Looping Conditional BranchingSystem Flags Flags 8 Program Branching and Controls Subroutine Execution SubroutinesGo To Subroutine and Return ´b.1Subroutine Limits 003- O0 000 001- ´b9002- R 004´ b.4 ´b.5Subroutine Return Nested SubroutinesV and % Keys Index Register Loop Control106 Index Register Storage and Recall Indirect Program Control With the Index RegisterProgram Loop Control Index Register and Loop ControlExchanging the X-Register Index Register ArithmeticIndirect Branching With Indirect Display Format Control With Indirect Flag Control WithLoop Control With Counters I and e Nnnnn x x x y y 5 0 0 Start count at zero Count by twos Count up toStoring and Recalling Keystrokes Display Examples Register OperationsIterations 12.3456Exchanging the X-Register Example Loop Control with eStorage Register Arithmetic 012-42, 5 Loop control number in R2−− 011- 42 013- 2215 O Example Display Format Control64.8420 0000 50.0000 Index Register Contents Indirect Display Control Index Register and Loop Control 118 Part lll HP-15C Advanced Functions Creating the Complex Stack Complex Stack and Complex ModeCalculating With Complex Numbers 120Complex Numbers and the Stack Deactivating Complex ModeEntering Complex Numbers ´ % hold 8.0000 release Z 8 Y 7 X Keys Stack Lift in Complex Mode Manipulating the Real and Imaginary StacksClearing a Complex Number Or other operationContinue with any operation − 4 v Continue with any operation´ %hold release Entering Complex Numbers with −. The clearing functions −0000 17.0000 144.0000 Entering a Real Number Followed by another numberEntering a Pure Imaginary Number ´ Continue with any operation´ O Operations With Complex NumbersStoring and Recalling Complex Numbers L 2 ´¤x N o ∕ @ a + * ÷ y0428 20007000 04915708 Polar and Rectangular Coordinate ConversionsComplex Results from Real Numbers ´ % hold Release1.5708Cos θ + i sin θ = re iθ Polar + ib = ∠ θ 2981 8452+ 3.1434 2361 352.0000872.0000 4721For Further Information Calculating With Matrices 138Keystrokes Display Deactivates Complex Mode = A-1B11.2887 Matrix DimensionsRunning 2496Dimensioning a Matrix Number Rows Columns´mA Displaying Matrix DimensionsChanging Matrix Dimensions Keystrokes l B DisplayStoring and Recalling Matrix Elements Storing and Recalling All Elements in Order⎡ a Checking and Changing Matrix Elements Individually Keystrokes Display Storing a Number in All Elements of a Matrix Matrix OperationsMatrix Descriptors Result Matrix Copying a Matrix One-Matrix OperationsCalculating with Matrices Scalar Operations LB bElements of Result Matrix LA aKeystrokes Display Subtracts 1 from the elements Arithmetic OperationsLB b 2 LA a 2 Matrix Multiplication = AT B Keystrokes Display l a aSolving the Equation AX = B 86 OA 24 OA2400 8600274 OB 233 OB 331 OB 120.32 OB 112.96 OB 151.36 OB ´Á Calculating the Residual Week Cabbage kg 186 141 215 Broccoli kg 116Using Matrices in LU Form Calculations With Complex MatricesStoring the Elements of a Complex Matrix Then Z can be represented in the calculator byPressing Transforms Into = ⎢ LA aComplex Transformations Between ZP and Z Inverting a Complex Matrix Multiplying Complex Matrices ´ aKeystrokes lA lB Display Displays descriptor of matrix a ´U lC LC lC lC lC lC lC lC ´USolving the Complex Equation AX = B ZZ −1AX = B 200.0000 170.00000437 03721311 1543Calculating with Matrices Using Matrix Descriptors in the Index Register Using a Matrix Element With Register OperationsMiscellaneous Operations Involving Matrices Stack Operation for Matrix Calculations Conditional Tests on Matrix DescriptorsCalculating with Matrices Using Matrix Operations in a Program Keystrokes Results Summary of Matrix Functions´m a Calculates residual in result matrix For Further Information Finding the Roots An Equation Using180 Finding the Roots of an Equation Clear program memory002 003 ´b0001-42,21 005 006 007Finding the Roots of an Equation Desired root000 001-42,21,11 Keystrokes ¥´ bA 003 004Into X-register 5000 1 e tBrings another t-value 200 tWhen No Root Is Found 000 001-42,21 002 003 004 005Error Choosing Initial Estimates Label 008 009 003 004 005 007X + 8 6 x + 8Finding the Roots of an Equation Using in a Program Restriction on the Use Memory RequirementsNumerical Integration Using f194 002 003 004 4040 1416 7652Begin subroutine with a label $ ÷ 38254401 6054 Accuracy of f ´ i ´ f 8826 7091Using f in a Program 382Memory Requirements Appendix a Error ConditionsError 0 Improper Mathematics Operation 205Error 1 Improper Matrix Operation Error 2 Improper Statistics OperationError 5 Subroutine Level Too Deep Error 3 Improper Register Number or Matrix ElementError 4 Improper Line Number or Label Call Error 6 Improper Flag NumberPr Error Power Error Stack Lift Stack Lift Last X RegisterDigit Entry Termination Appendix BDisabling Operations Enabling OperationsAppendix B Stack Lift and the Last X Register Keys Stack Stack Enabled. disabled 53.1301 No stack LiftNeutral Operations Nnn Clear u ¥Last X Register \ k + H ∆ \ h ÷ À P* q r c ‘ / N z ∕ P\ o jAppendix C Memory AllocationMemory Space RegistersAppendix C Memory Allocation Memory Status W Memory ReallocationM % Function ´m% 1.0000 Whold 1 64 Restrictions on Reallocation19 ´ m Program Memory Automatic Program Memory ReallocationIf executed Memory Requirements for the Advanced FunctionsTwo-Byte Program Instructions TogetherAppendix C Memory Allocation Appendix D Detailed Look atHow Works 220Appendix D a Detailed Look at Accuracy of the Root X4 = 000 1718 006 007 008 009 010-43,30 011 012-43,30 013´ v B Interpreting Results0681 − 45 For 0 x 3x 45x 2 + Test for x rangeBranch for x ≥ End subroutine1358 000.0000Initial estimates Possible rootAppendix D a Detailed Look at 007 008 009 010 ´ b.0 001-42,21,.0 002 003 004 005Bring x-value into X-register 013 014 015 016017 018 10 v ´ ‛ 20Error 0000 1250 5626 Finding Several RootsFx = xx a3 = 002 003 004 005 006 0076667 Stores root for deflation Same initial estimatesSecond root Deflated function valueDeflation for third root Limiting the Estimation Time Counting Iterations For Advanced InformationSpecifying a Tolerance Appendix E Detailed Look at fHow f Works 240Accuracy, Uncertainty, and Calculation Time X = π1 0π cos4θ − x sinθ dθ0000 1416 ´ i ´ fKeystrokes ´ i Display Keystrokes Display Return approximation to´ Clear u Hold ´ f ´ Clear u hold7858 7807Uncertainty 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 Approximation to integral Keys lower limit intoKeys upper limit into UncertaintyAppendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Batteries Low-Power IndicationInstalling New Batteries BatteriesAppendix F Batteries Verifying Proper Operation Self-Tests 2.C 3.HConversions Function Summary and IndexComplex Functions Digit EntryLogarithmic Exponential Functions Display ControlIndex Register Control Mantissa. PressingMatrix Functions Mathematics146 To ZP page164 Number AlterationTo XT Stack Manipulation PercentageProbability Clear uStatistics StorageTrigonometry Programming Summary and Index 269Programming Summary and Index Subject Index 271Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Federal Communications Commission Notice Product Regulatory Environment InformationModifications Canadian Notice Avis CanadienEuropean Union Regulatory Notice Body number is inserted between CE

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