Finding the Roots of an Equation, Desired root | HP 15c Scientific

	Section 13: Finding the Roots of an Equation 183
Keystrokes	Display
´_ 0	5.0000	The desired root.

After the routine finds and displays the root, you can ensure that the displayed number is indeed a root of f(x) = 0 by checking the stack. You have seen that the display (X-register) contains the desired root. The Y-register contains a previous estimate of the root, which should be very close to the displayed root. The Z-register contains the value of your function evaluated at the displayed root.

Keystrokes	Display
)	5.0000	A previous estimate of the
		root.
)	0.0000	Value of the function at the
		root showing that f(x) = 0.

Quadratic equations, such as the one you are solving, can have two roots. If you specify two new initial estimates, you can check for a second root. Try estimates of 0 and -10 to look for a negative root.

Keystrokes	Display
0 v	0.0000
10 “	–10
´ _ 0	–2.0000
)	–2.0000
)	0.0000

Initial estimates.

The second root.

A previous estimate of the root.

Value of f(x) at second root.

Image 183

HP 15c Scientific manual Finding the Roots of an Equation, Desired root

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 E a Detailed Look at f Contents Appendix a Error ConditionsAppendix C Memory Allocation Appendix D a Detailed Look at Subject Index Contents Appendix F BatteriesFunction Summary and Index Programming Summary and Index Quick Look at HP-15C Problem Solver To Compute Keystrokes Display Manual Solutions Keystrokes Display Programmed Solutions 300.51 KeystrokesDisplay001-42,21,11 002 003 004 005 006 007 008 009 8313 HP-15C a Problem Solver Part l HP-15C Fundamentals Section Power On and OffGetting Started Keyboard Operation I O m ´ P I l F T s ? t H b Prefix KeysChanging Signs Keying in Exponents Clear Keys Display Clearing ` and − Digit entry not terminatedClears only the last digit 6532 CalculationsOne-Number Functions Two-Number Functions 78.0000 17 +26.0000 22.0000 5000 13.0000 Number Alteration Functions Numeric Functions General Functions One-Number Functions Trigonometric Operations Time and Angle ConversionsPressing Calculates 40.5000 Degrees/Radians Conversions7069 Radians Hyperbolic Functions Logarithmic Functions To Calculate Keystrokes Display Power FunctionTwo-Number Functions Percentages Polar Conversion. Pressing Polar and Rectangular Coordinate ConversionsEnters the base number the price Calculates 3% of $15.76 the tax Keystrokes Display Always displayed Automatic Memory Stack Last X, and Data StorageAutomatic Memory Stack Stack Manipulation Automatic Memory Stack Registers Stack Manipulation Functions Memory Stack, Last X, and Data StorageLost Lost 12.9000 Last X Register and K287.0000 22.2481 Calculator Functions and the Stack 13.9 +20.6475 +15 X15 Order of Entry and the v Key 69.0000 Nested Calculations7 + 65.0000 Arithmetic Calculations With Constants 5 ‛15 Keys Keystrokes Display Growth factor 1000000 520.8750 Storage Register OperationsStoring and Recalling Numbers 322.5000 Storage and Recall Arithmetic Clearing Data Storage Registers For recall arithmetic For storage arithmetic 15.0000 ProblemsOverflow and Underflow 24 l-0 Memory Stack, Last X, and Data Storage Statistics Functions Probability Calculations60.0000 3422 Random Number Generator270,725.0000 5764 Registers Accumulating Statistics Register Contents Σy2 20.00 40.00 60.00 80.00 Kg per hectareMetric tons per Hectare, y 20 z 61v 40 z 7.21 60 z 7.78 80 z l 20 w 20 z Correcting Accumulated Statistics Mean Standard Deviation40.00 Application Linear Regression31.62 Standard deviation about the mean nitrogen Linear Estimation and Correlation Coefficient Statistics Functions 70 ´j Other Applications Display Continuous Memory Display ControlFixed Decimal Display 234567 Scientific Notation DisplayEngineering Notation Display 234568 Annunciators Round-Off ErrorSpecial Displays Mantissa Display 12.345.6700 Error DisplayDigit Separators 12,345.67 Low-Power Indication Continuous MemoryStatus Resetting Continuous Memory Page Part ll HP-15C Programming Loading a Program Programming BasicsMechanics Creating 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 010 002004 005 007-44,40 Or G a Further Information Program InstructionsInstruction Coding Keycode 25 second row, fifth key Memory Configuration 60 ´ m% Initial Memory Configuration 19.0000 Program Boundaries´ m % 19 ´ m% Unexpected Program Stops Abbreviated Key Sequences´bA ´b3 End of memory LOG % User ModePolynomial Expressions and Horners Method ¤ @ y ∕ 12,691.0000 Nonprogrammable Functions001-42,21,12 002 003 004 005 006 007 008 009 0000 Problems Moving to a Line in Program Memory Program Editing Examples Deleting Program LinesInserting Program Lines Or use Â Single-Step Operations Result Line PositionÂhold Release 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 016-44,40 010-45,20013-43,30 014 Formula is Example Flags 006-42,21 002-43004-42,21,15 005-43, 4 10,698.3049 Go to250.0000 48.0000 Conditional Branching Looping System Flags Flags 8 Program Branching and Controls ´b.1 SubroutinesGo To Subroutine and Return Subroutine Execution Subroutine Limits 004 000 001- ´b9002- R 003- O0 ´b.5 ´ b.4 Nested Subroutines Subroutine Return Index Register Loop Control V and % Keys106 Index Register and Loop Control Indirect Program Control With the Index RegisterProgram Loop Control Index Register Storage and Recall 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 12.3456 Examples Register OperationsIterations Storing and Recalling Keystrokes Display Example Loop Control with e Exchanging the X-RegisterStorage Register Arithmetic 013- 22 Loop control number in R2−− 011- 42 012-42, 5 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 120 Complex Stack and Complex ModeCalculating With Complex Numbers Creating the Complex Stack 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 L 2 ´ Operations With Complex NumbersStoring and Recalling Complex Numbers ´ O + * ÷ y ¤x N o ∕ @ a 0491 20007000 0428 ´ % hold Release1.5708 Polar and Rectangular Coordinate ConversionsComplex Results from Real Numbers 5708 Cos θ + i sin θ = re iθ Polar + ib = ∠ θ 8452 2981+ 3.1434 4721 352.0000872.0000 2361 For Further Information 138 Calculating With Matrices = A-1B Keystrokes Display Deactivates Complex Mode 2496 Matrix DimensionsRunning 11.2887 Number Rows Columns Dimensioning a Matrix Keystrokes l B Display Displaying Matrix DimensionsChanging Matrix Dimensions ´mA 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 8600 24 OA2400 86 OA 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 1543 03721311 0437 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 005 006 007 ´b0001-42,21 002 003 Desired root Finding the Roots of an Equation 003 004 Keystrokes ¥´ bA 000 001-42,21,11 200 t 5000 1 e tBrings another t-value Into X-register 000 001-42,21 002 003 004 005 When No Root Is Found Error Choosing Initial Estimates Label 6 x + 8 003 004 005 007X + 8 008 009 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 205 Error ConditionsError 0 Improper Mathematics Operation Appendix a Error 2 Improper Statistics Operation Error 1 Improper Matrix Operation Error 6 Improper Flag Number Error 3 Improper Register Number or Matrix ElementError 4 Improper Line Number or Label Call Error 5 Subroutine Level Too Deep Pr Error Power Error Appendix B Stack Lift Last X RegisterDigit Entry Termination Stack Lift Enabling Operations Disabling Operations Nnn Clear u ¥ Stack Stack Enabled. disabled 53.1301 No stack LiftNeutral Operations Appendix B Stack Lift and the Last X Register Keys \ k + H ∆ \ h ÷ À P* q r c ‘ / N z ∕ P\ o j Last X Register Registers Memory AllocationMemory Space Appendix C 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 Together Memory Requirements for the Advanced FunctionsTwo-Byte Program Instructions If executed Appendix C Memory Allocation 220 Detailed Look atHow Works Appendix D 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 End subroutine Test for x rangeBranch for x ≥ 3x 45x 2 + Possible root 000.0000Initial estimates 1358 Appendix D a Detailed Look at 013 014 015 016 ´ b.0 001-42,21,.0 002 003 004 005Bring x-value into X-register 007 008 009 010 10 v ´ ‛ 20 017 018 Finding Several Roots Error 0000 1250 5626 002 003 004 005 006 007 Fx = xx a3 = 6667 Deflated function value Same initial estimatesSecond root Stores root for deflation Deflation for third root Limiting the Estimation Time For Advanced Information Counting IterationsSpecifying a Tolerance 240 Detailed Look at fHow f Works Appendix E X = π1 0π cos4θ − x sinθ dθ Accuracy, Uncertainty, and Calculation Time ´ i ´ f 0000 1416 ´ f ´ Clear u hold Keystrokes Display Return approximation to´ Clear u Hold Keystrokes ´ i Display 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 Uncertainty Keys lower limit intoKeys upper limit into Approximation to integral Appendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Batteries Low-Power IndicationInstalling New Batteries Batteries Appendix F Batteries 2.C 3.H Verifying Proper Operation Self-Tests Digit Entry Function Summary and IndexComplex Functions Conversions Mantissa. Pressing Display ControlIndex Register Control Logarithmic Exponential Functions Mathematics Matrix Functions146 Number Alteration To ZP page164To XT Clear u PercentageProbability Stack Manipulation 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