More about Solving, How Solve Finds a Root | HP 35s Scientific

D

More about Solving

This appendix provides information about the SOLVE operation beyond that given in chapter 7.

How SOLVE Finds a Root

SOLVE first attempts to solve the equation directly for the unknown variable. If the attempt fails, SOLVE changes to an iterative(repetitive) procedure. The iterative operation is to execute repetitively the specified equation. The value returned by the equation is a function f(x) of the unknown variable x. (f(x) is mathematical shorthand for a function defined in terms of the unknown variable x.) SOLVE starts with an estimate for the unknown variable, x, and refines that estimate with each successive execution of the function, f(x).

If any two successive estimates of the function f(x) have opposite signs, then SOLVE presumes that the function f(x) crosses the x–axis in at least one place between the two estimates. This interval is systematically narrowed until a root is found.

For SOLVE to find a root, the root has to exist within the range of numbers of the calculator, and the function must be mathematically defined where the iterative search occurs. SOLVE always finds a root, provided one exists (within the overflow bounds), if one or more of these conditions are met:

Two estimates yield f(x) values with opposite signs, and the function's graph crosses the x–axis in at least one place between those estimates (figure a, below).

f(x) always increases or always decreases as x increases (figure b, below).

The graph of f(x) is either concave everywhere or convex everywhere (figure c, below).

More about Solving D-1

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HP 35s Scientific manual How Solve Finds a Root, More about Solving D-1

Contents

Users guide HP 35s scientific calculator Printing History Contents Storing Data into Variables Viewing a Variable ContentsRPN The Automatic Memory Stack Hyperbolic Functions Real-Number FunctionsQuotient and Remainder of Division Trigonometry Entering π Entering and Evaluating Equations Fractions Vector Arithmetic 10-1 Solving EquationsIntegrating Equations Operations with Complex Numbers Creating Vectors from Variables or Registers 10-8 Base Conversions and Arithmetic and Logic 11-1Statistical Operations 12-1 10-7 Simple Programming 13-1 Part 2. Programming Programming Techniques 14-1 Statistics Programs 16-1 Support, Batteries, and ServicePart 3. Appendixes and Reference Solving and Integrating Programs 15-1 More about Solving User Memory and the StackALG Summary How the Integral Is Evaluated Round-Off ErrorMore about Integration Messages Operation Index Contents Part Page Turning the Calculator On and Off Adjusting Display ContrastGetting Started Important Preliminaries 2Getting Started Highlights of the Keyboard and DisplayShifted Keys Left-shifted function Right-shifted Letter for alphabetic Alpha KeysCursor Keys Function Key 4Getting Started  Clear or CancelBackspacing and Clearing Keys for Clearing Clear menu       Key Description 6Getting Started Using Menusˆ ˆ    HP 35s Menus Chapter Name Other functions Programming Instructions 8 Exiting Menusor   KeysDisplay RPN and ALG Modes Keys Display 10Getting Started RPN mode ALG mode Undo key 12Getting Started Display and AnnunciatorsFirst Line Second Line Annunciators EQN HP 35s Annunciators Meaning Chapter 14Getting Started Exponents of Ten Keying in NumbersMaking Numbers Negative 16Getting Started z  Understanding Entry CursorRange of Numbers and Overflow  18Getting Started Performing Arithmetic CalculationsSingle Argument or Unary Operations Key RPN,RPN Program ALG, Equation, ALG Program Two Argument or Binary Operations Õ   Key RPN, RPN Program ALG, Equation, ALG Program Controlling the Display Format 22Getting Started Periods and Commas in Numbers   24Getting Started Complex number display format  ,  , 8 8  SHOWing Full 12-Digit Precision8   8× 26Getting Started FractionsEntering Fractions  Messages  É 28Getting Started Calculator MemoryChecking Available Memory Clearing All of Memory 30Getting Started RPN The Automatic Memory Stack RPN The Automatic Memory StackWhat the Stack Is 2RPN The Automatic Memory Stack Reviewing the Stack X and Y-Registers are in the DisplayClearing the X-Register 4RPN The Automatic Memory Stack Exchanging the X- and Y-Registers in the StackPress  Arithmetic How the Stack Does It 6RPN The Automatic Memory Stack How Enter Works Filling the stack with a constant How to Clear the Stack 8RPN The Automatic Memory Stack Last X Register Correcting Mistakes with Last  Reusing Numbers with Last Ù     Chain Calculations in RPN ModeWork from the Parentheses Out       ExercisesOrder of Calculation        More Exercises     18RPN The Automatic Memory Stack Storing Data into Variables AKeys DisplayDescription Press letter-key  Storing and Recalling Numbers2Storing Data into Variables Press letter-key    AG  4Storing Data into Variables Using the MEM CatalogViewing a Variable VAR catalog D E C Storage arithmetic uses , , , or  Arithmetic with Stored VariablesStorage Arithmetic 6Storing Data into Variables Recall Arithmetic E Exchanging x with Any Variable D D A Variables I and J 10Storing Data into Variables Real-Number Functions Exponential and Logarithmic Functions   Power FunctionsQuotient and Remainder of Division   Entering π Trigonometry 4Real-Number Functions Setting the Angular ModeTrigonometric Functions Option Description Annunciator   Percentage Functions Hyperbolic Functions     8Real-Number Functions Physics ConstantsConst Menu Items Description Value Items Description Value 10Real-Number Functions Conversion FunctionsRectangular/Polar Conversions 30o 12Real-Number Functions 86  Time ConversionsAngle Conversions Unit Conversions Probability Probability FunctionsFactorial Gamma 16Real-Number Functions  Absolute value Parts of NumbersInteger part Fractional part 18Real-Number Functions To calculatePressDisplay   FractionsEntering Fractions 2Fractions Fractions in the DisplayDisplay Rules Entered Value Internal Value Displayed Fraction Accuracy Indicators 4Fractions Setting the Maximum DenominatorChanging the Fraction Display  6Fractions Choosing a Fraction Format To Get This Fraction Format Change These Flags   j Examples of Fraction Displays Rounding Fractions Fractions in Equations 10Fractions Fractions in Programs = .25 π d2 l Entering and Evaluating EquationsHow You Can Use Equations Entering and Evaluating Equations Value      D 2Entering and Evaluating Equations Key Operation Summary of Equation Operations 4Entering and Evaluating Equations Entering Equations into the Equation ListVariables in Equations Functions in Equations Numbers in Equations  Displaying and Selecting EquationsParentheses in Equations   Ö  8Entering and Evaluating Equations Editing and Clearing Equations Types of Equations 10Entering and Evaluating Equations Evaluating Equations Type of Equation Result for  Result for  Using Enter for Evaluation 12Entering and Evaluating Equations Using XEQ for Evaluation       Responding to Equation Prompts 14Entering and Evaluating Equations Syntax of EquationsOperator Precedence Equations Meaning Order Operation Example Σxy Equation Functions16Entering and Evaluating Equations Perimeter = a + b + h sin1θ + sin1φ RPN Operation Equation function Area = 18Entering and Evaluating Equations  Õ Syntax ErrorsVerifying Equations 20Entering and Evaluating Equations Markup ⋅ Cost = Price Solving EquationsSolving an Equation Solving Equations = v0 t + 1/2 g t 2Solving Equations   G      P    P  6Solving Equations Solving built-in Equation   Understanding and Controlling SolveVerifying the Result 8Solving Equations Interrupting a Solve CalculationChoosing Initial Guesses for Solve Solving Equations Equation 10Solving Equations= 40 H ⋅ 20 H ⋅ 4 ⋅ H Type in the equation H  value 4H   12Solving Equations For More Information Integrating Equations Integrating Equations 2Integrating Equations Integrating Equations FN   J0 x = π 0 cosx sin t dt 4Integrating Equations = t X SinxX  6Integrating Equations Accuracy of IntegrationSpecifying Accuracy Interpreting Accuracy 8        8Integrating Equations 8 Form  Operations with Complex Numbers 2Operations with Complex Numbers Complex StackComplex Operations Functions for One Complex Number, z To Calculate Press 4Operations with Complex Numbers 8  68  6 6 6 Display Description Using Complex Numbers in Polar Notation 6Operations with Complex Numbers ?  ?  ?  6 ? ? Complex Numbers in Equations        Program lines ALG modeComplex Number in Program    Vector Arithmetic Vector ArithmeticVector operations     Õ 3  3  Absolute value of the vector 10-4Vector Arithmetic Dot product   3 3 3   Angle between vectors Õ   10-6Vector Arithmetic Vectors in Equations3   Program lines Description Vectors in Programs 10-8Vector Arithmetic Creating Vectors from Variables or RegistersThen hC Õ hA Base Conversions and Arithmetic and Logic Base Conversions and Arithmetic and LogicBase Menu Menu label Description 11-2Base Conversions and Arithmetic and Logic    Õ  11-4Base Conversions and Arithmetic and Logic Arithmetic in Bases 2, 8,Logic Menu           Representation of NumbersNegative Numbers   Range of Numbers    Using base in program and equationsWindows for Long Binary Numbers  Statistical Operations Statistical OperationsEntering Statistical Data 12-2Statistical Operations Correcting Errors in Data EntryEntering One-Variable Data Entering Two-Variable Data Initial x, y Corrected x, y   Menu Key Description Statistical CalculationsMean Statistics Menus         Sample Standard Deviation  Linear Regression Population Standard Deviation 12-8Statistical Operations  ˆ Õ ˆLinear Regression Menu        12-10Statistical Operations Limitations on Precision of Data Summation Statistics Summation Values and the Statistics Registers 12-12Statistical Operations Access to the Statistics Registers Statistical Operations 12-14Statistical Operations Programming Page Simple Programming Simple Programming Area = πx2 13-2Simple Programming  X  Selecting a ModeKeysDisplay In ALG mode Designing a Program  letter-key Using RPN, ALG and Equations in ProgramsProgram Boundaries LBL and RTN 13-4Simple Programming Strengths of RPN Operations Data Input and Output 13-6Simple Programming Entering a Program Clear functions and backspace key 13-8Simple Programming Function Names in ProgramsA  R  RPN modeE R 13-10Simple Programming Running a ProgramExecuting a Program XEQ   Testing a ProgramKeys Display Description RPN mode 13-12Simple Programming Entering and Displaying Data RPN mode ALG mode Using Input for Entering Data 13-14Simple Programming Using View for Displaying Data 13-16Simple Programming Using Equations to Display Messages  ÇR ÇH  V  S     Keys RPN mode R H V  R R H   V OL  A RE A   C  value Displaying Information without Stopping  13-18Simple Programming Interrupting a Running Program Error StopsStopping or Interrupting a Program Programming a Stop or Pause STOP, PSE 13-20Simple Programming Editing a Program Viewing Program Memory Program Memory 13-22Simple Programming Memory UsageCatalog of Programs MEM Checksum Clearing One or More Programs   Nonprogrammable FunctionsProgramming with Base  Numbers Entered in Program Lines Selecting a Base Mode in a ProgramDecimal mode set Ax4 + Bx3 + Cx2 + Dx + E Polynomial Expressions and Horners Method13-26Simple Programming A    A X     13-28Simple Programming Programming Techniques Programming TechniquesRoutines in Programs Calling Subroutines XEQ, RTN 14-2Programming Techniques Nested Subroutines A2 + b2 + c2 + d2 A2 + B2 Branching GTO14-4Programming Techniques Programmed GTO Instruction Using GTO from the Keyboard 14-6Programming Techniques Conditional Instructions Test Menus Tests of Comparison x?y, x?0 14-8Programming Techniques Program Lines Description RPN mode Meanings of Flags Flags 14-10Programming Techniques Flag Fraction-Control FlagsClear Programming Techniques 14-12Programming Techniques Flags Menu … takes the natural log of the X-input S  value   value  14-14Programming Techniques Number 14-16Programming Techniques LoopsF      Conditional Loops GTO 14-18Programming Techniques Loops with Counters DSE, ISG Programming Techniques 14-20Programming Techniques Indirectly Addressing Variables and LabelsVariables I and J Indirect Address, I and J 14-22Programming Techniques If I/J contains Then I/J will address Unnamed indirect variables Program Control with I/JEquations with I/J 14-24Programming Techniques Solving and Integrating Programs Solving and Integrating ProgramsSolving a Program V= N x R x T 15-2Solving and Integrating Programs P  value   value Keys Display Description ALG modeProgram Lines Description ALG mode G    H    LP   label  variable Using Solve in a Program15-6Solving and Integrating Programs Integrating a Program Program Lines In RPN mode 15-8Solving and Integrating Programs Sin  S   X  label   variable Using Integration in a Program15-10Solving and Integrating Programs Restrictions on Solving and Integrating 15-12Solving and Integrating Programs Statistics Programs Statistics ProgramsCurve Fitting 16-2Statistics Programs = B + MIn Program Listing 16-4Statistics Programs    16-6Statistics Programs Determines if D001 or B001 should be run 16-8Statistics Programs   S           Logarithmic Exponential PowerNormal and Inverse-Normal Distributions  16-12Statistics Programs Qx = 0.5 − σ 12π xx e−x −x ⎟σ 2 ⎟2dx This routine initializes the normal distribution program 16-14Statistics Programs Program Lines Program Lines Program S 16-16Statistics Programs To the inverse D  ∑ x f Grouped Standard Deviation16-18Statistics Programs   16-20Statistics Programs Register Summation Σ f Summation Σ x i f Index variable used to indirectly address the correctStatistics register  U       16-24Statistics Programs TVM equation is Miscellaneous Programs and EquationsTime Value of Money Miscellaneous Programs and Equations NÕ P4 I I P  value KeysDisplay In RPN mode    17-4Miscellaneous Programs and Equations    17-6Miscellaneous Programs and Equations I    Prime Number Generator 17-8Miscellaneous Programs and Equations Tests to see whether all possible factors have been tried 17-10Miscellaneous Programs and Equations   Cross Product in Vectors 17-12Miscellaneous Programs and Equations z 17-14Miscellaneous Programs and Equations Appendixes and Reference Page Answers to Common Questions Support, Batteries, and ServiceSupport, Batteries, and Service A-1 Calculator Support Environmental Limits 2Support, Batteries, and Service Changing the Batteries Support, Batteries, and Service A-3 4Support, Batteries, and Service Testing Calculator Operation Self-Test Support, Batteries, and Service A-5 6Support, Batteries, and Service Support, Batteries, and Service A-7 Warranty Customer Support 8Support, Batteries, and Service Emea Support, Batteries, and Service A-9 10Support, Batteries, and Service Invent Support, Batteries, and Service A-11 12Support, Batteries, and Service Yyyy+ Support, Batteries, and Service A-13 14Support, Batteries, and Service User Memory and the Stack B-1 User Memory and the StackManaging Calculator Memory 2User Memory and the Stack Resetting the Calculator User Memory and the Stack B-3 Clearing Memory 4User Memory and the Stack DefaultStatus of Stack Lift Category User Memory and the Stack B-5 Disabling OperationsNeutral Operations 6User Memory and the Stack Status of the Last X Registerˆ ˆ Accessing Stack Register Contents 8User Memory and the Stack ALG Summary C-1 ALG SummaryAbout ALG , Simple Arithmetic Doing Two argument Arithmetic in ALG Percentage Calculations Power Functions xÕ Parentheses CalculationsPermutations and Combinations Quotient and Remainder Of Division  4 Õ  Hyperbolic functions Trigonometric Functions ALG Summary C-7 Reviewing the Stack 8ALG Summary Integrating an EquationOperations with Complex Numbers ALG Summary C-9 6 6Õ4 6 8Ë4 66  Entering Statistical Two-Variable Data 12ALG Summary z ΣZ6 ALG Summary C-13   ˆ Õ ˆ 14ALG Summary More about Solving D-1 More about SolvingHow Solve Finds a Root Function Whose Roots Can Be Found 2More about Solving More about Solving D-3 Interpreting Results 4More about Solving X  X X  X  XX X  6More about Solving XÕ  More about Solving D-7 X  X4 X Õ  8More about Solving When Solve Cannot Find a Root X   More about Solving D-9  X  10More about Solving X    X  More about Solving D-11 X4 X ÕÕ 12More about Solving X   More about Solving D-13 Round-Off Error X   14More about Solving More about Integration More about IntegrationHow the Integral Is Evaluated 2More about Integration Conditions That Could Cause Incorrect Results ∞ xe−xdx 4More about Integration   More about Integration 6More about Integration   Conditions That Prolong Calculation Time 8More about Integration More about Integration 10More about Integration Messages F-1 Messages Or program. This might take a while 2Messages Messages F-3    Solve include EQN and PGM modecannot find 4Messages Messages F-5 FN each uses a level, they can also generate this error 6Messages Operation Index G-1 Operation IndexName Keys and Description 2Operation Index ×Ø Operation Index G-3 4Operation Index =  Denominator Operation Index G-5 6Operation Index  Fractional part Operation Index G-7Natural exponential label 8Operation Index   Integer part ISG variable variable Operation Index G-9 Input variable variable Natural logarithm 10Operation Index LBL label label Degrees to radians. Returns 2π/360 Operation Index G-11RCL variable  Pause 12Operation Index Operation Index G-13 XEQ label label 14Operation IndexHyperbolic Tangent View variable variable Operation Index G-15 ÕÕ≠ Õ≤ 11-4 16Operation Index=y? 14-7 Menu ≠0? Operation Index G-17 Power 18Operation Index Index-1 Index Index-2 EQN List TOP 6-7, F-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Angle between two vectors Index-11 Index-12