HP 15c Scientific manual Appendix D a Detailed Look at

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230 Appendix D: A Detailed Look at _

If the algorithm terminates its search near a local minimum of the function's magnitude, clear the Error 8 display and observe the numbers in the X-, Y-, and Z-registers by rolling down the stack. If the value of the function saved in the Z-register is relatively close to zero, it is possible that a root of your equation has been found – the number returned in the X-register may be a 10-digit number very close to a theoretical root. You

can explore this potential minimum further by rolling the stack until the returned estimates are back in the X- and Y-registers and then executing _ again using these numbers as initial estimates. If an actual minimum has been found, Error 8 will again be displayed and the number in the X-register will be approximately the same as before, but possibly closer to the actual location of the minimum.

Of course, you may deliberately use _ to find the location of a local minimum of the function's magnitude. However, in this case you must be careful to confine the search in the region of the minimum. Remember, _tries hard to find a zero of the function.

If the algorithm stops searching and displays Error 8 because it is working on a horizontal asymptote (when the value of the function is essentially constant for a large range of x), the estimates in X- and Y-registers usually are significantly different from each other. The number in the Z-register is the value of the potential asymptote. If you execute _ again using as initial estimates the numbers that were returned in the X- and Y-registers, a

horizontal asymptote may again cause Error 8, but with numbers in the X- and Y-registers that will differ from the previous numbers. The value of the function in the Z-register would then be about the same as that obtained previously.

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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 Matrices Indirect Display ControlCalculating With Complex Numbers 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 Clears only the last digit Display Clearing ` and −Digit entry not terminated Two-Number Functions CalculationsOne-Number Functions 653213.0000 17 +26.0000 22.0000 5000 78.0000Numeric Functions Number Alteration FunctionsOne-Number Functions General FunctionsPressing Calculates Trigonometric OperationsTime and Angle Conversions 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 displayedLost Stack Manipulation FunctionsMemory Stack, Last X, and Data Storage Lost 22.2481 Last X Register and K287.0000 12.900020.6475 Calculator Functions and the Stack13.9 + Order of Entry and the v Key +15 X1565.0000 Nested Calculations7 + 69.0000Arithmetic Calculations With Constants 5 ‛15 Keys 000 Keystrokes Display Growth factor1000 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 60.0000 Statistics FunctionsProbability Calculations 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 z40.00 MeanStandard Deviation Standard deviation about the mean nitrogen Linear Regression31.62 ApplicationLinear Estimation and Correlation Coefficient Statistics Functions Other Applications 70 ´jFixed Decimal Display Display Continuous MemoryDisplay Control 234568 Scientific Notation DisplayEngineering Notation Display 234567Mantissa Display Round-Off ErrorSpecial Displays Annunciators12,345.67 Error DisplayDigit Separators 12.345.6700Status Low-Power IndicationContinuous Memory Resetting Continuous Memory Page Part ll HP-15C Programming Creating a Program Programming BasicsMechanics Loading a ProgramProgramming Basics ´b a002 003 004 005 006 007 008 Intermediate Program StopsRunning a Program 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 Instruction Coding Further InformationProgram Instructions Memory Configuration Keycode 25 second row, fifth keyInitial Memory Configuration 60 ´ m%19 ´ m% Program Boundaries´ m % 19.0000´bA ´b3 End of memory Unexpected Program StopsAbbreviated Key Sequences ¤ @ 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 MemoryInserting Program Lines ExamplesDeleting Program Lines Or use Â Single-Step Operations Release Line PositionÂhold ResultInsertions and Deletions Initializing Calculator StatusPV 1 + i n Interest+ 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 Subroutines106 Index Register Loop ControlV and % Keys Index Register Storage and Recall Indirect Program Control With the Index RegisterProgram Loop Control 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 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 12.3456Storage Register Arithmetic Example Loop Control with eExchanging the X-Register 012-42, 5 Loop control number in R2−− 011- 42 013- 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 Creating the Complex Stack Complex Stack and Complex ModeCalculating With Complex Numbers 120Entering Complex Numbers Deactivating Complex ModeComplex Numbers and the Stack ´ % 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 operation0000 17.0000 144.0000 Entering Complex Numbers with −. The clearing functions −´ %hold release 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 + 3.1434 84522981 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 Matrix Descriptors Matrix OperationsStoring a Number in All Elements of a Matrix Result Matrix Copying a Matrix One-Matrix OperationsCalculating with Matrices Scalar Operations LB bElements of Result Matrix LA aLB b 2 LA a 2 Arithmetic OperationsKeystrokes Display Subtracts 1 from the elements 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 Miscellaneous Operations Involving Matrices Using a Matrix Element With Register OperationsUsing Matrix Descriptors in the Index Register Stack Operation for Matrix Calculations Conditional Tests on Matrix DescriptorsCalculating 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 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 Requirements194 Using fNumerical Integration 002 003 004 4040 1416 7652Begin subroutine with a label 4401 3825$ ÷ 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 M % Function Memory ReallocationMemory Status W 19 ´ m Restrictions on Reallocation´m% 1.0000 Whold 1 64 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 0130681 Interpreting Results´ v B − 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 Specifying a Tolerance For Advanced InformationCounting Iterations 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. Pressing146 MathematicsMatrix Functions To XT Number AlterationTo ZP page164 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 Modifications Product Regulatory Environment InformationFederal Communications Commission Notice Canadian Notice Avis CanadienEuropean Union Regulatory Notice Body number is inserted between CE

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