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Round–Off Error

UPC - 082916001456

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Checksum and length: B956 75 You can subsequently delete line J0003 to save memory. Solve for X using initial guesses of 10-8 and -10-8. Keys: (In RPN mode) Display: Description: a8^IX 1^a8^ |W J X %/ .) Enters guesses. Selects program "J" as the function. Solves for X; displays the result. Round-Off Error The limited (12-digit) precision of the calculator can cause errors due to rounding off, which adversely affect the iterative solutions of SOLVE and integration. For example, [( x + 1) + 1015]2 - 1030 = 0 has no roots because f(x) is always greater than zero. However, given initial guesses of 1 and 2, SOLVE returns the answer 1.0000 due to round-off error. Round-off error can also cause SOLVE to fail to find a root. The equation x2 - 7 = 0 has a root at 7 . However, no 12-digit number exactly equals 7 , so the calculator can never make the function equal to zero. Furthermore, the function never changes sign SOLVE returns the message However, the final estimate of x (press b to see it) is the best possible 12-digit approximation of the root when the routine quits. More about Solving D-13

Section	Page
Basic Operation	15
Getting Started	17
Important Preliminaries	17
Turning the Calculator On and Off	17
Adjusting Display Contrast	17
Highlights of the Keyboard and Display	18
Shifted Keys	18
Alpha Keys	19
Cursor Keys	19
Silver Paint Keys	20
Backspacing and Clearing	20
Using Menus	23
Exiting Menus	25
RPN and ALG Keys	26
The Display and Annunciators	27
Keying in Numbers	30
Making Numbers Negative	30
Exponents of Ten	30
Understanding Digit Entry	31
Range of Numbers and OVERFLOW	32
Doing Arithmetic	32
One–Number Functions	33
Two–Number Functions	33
Controlling the Display Format	34
Periods and Commas in Numbers	34
Number of Decimal Places	35
SHOWing Full 12–Digit Precision	36
Fractions	37
Entering Fractions	37
Displaying Fractions	39
Messages	39
Calculator Memory	40
Checking Available Memory	40
Clearing All of Memory	40
RPN: The AutomaticMemory Stack	41
What the Stack Is	41
The X and Y–Registers are in the Display	42
Clearing the X–Register	42
Reviewing the Stack	43
Exchanging the X– and Y–Registers in the Stack	44
Arithmetic – How the Stack Does It	44
How ENTER Works	45
How CLEAR x Works	46
The LAST X Register	47
Correcting Mistakes with LAST X	48
Reusing Numbers with LAST X	49
Chain Calculations in RPN mode	51
Work from the Parentheses Out	51
Exercises	53
Order of Calculation	53
More Exercises	54
Storing Data into Variables	57
Storing and Recalling Numbers	58
Viewing a Variable without Recalling It	59
Reviewing Variables in the VAR Catalog	59
Clearing Variables	60
Arithmetic with Stored Variables	60
Storage Arithmetic	60
Recall Arithmetic	61
Exchanging x with Any Variable	62
The Variable \	63
Real–Number Functions	65
Exponential and Logarithmic Functions	65
Quotient and Remainder of Division	66
Power Functions	66
Trigonometry	67
Entering (	67
Setting the Angular Mode	68
Trigonometric Functions	68
Hyperbolic Functions	70
Percentage Functions	70
Physics Constants	72
Conversion Functions	73
Coordinate Conversions	74
Time Conversions	76
Angle Conversions	77
Unit Conversions	77
Probability Functions	78
Factorial	78
Gamma	78
Probability	78
Parts of Numbers	80
Names of Functions	81
Fractions	83
Entering Fractions	83
Fractions in the Display	84
Display Rules	84
Accuracy Indicators	85
Longer Fractions	86
Changing the Fraction Display	86
Setting the Maximum Denominator	87
Choosing a Fraction Format	87
Examples of Fraction Displays	88
Rounding Fractions	89
Fractions in Equations	90
Fractions in Programs	91
Entering and Evaluating Equations	93
How You Can Use Equations	93
Summary of Equation Operations	95
Entering Equations into the Equation List	96
Variables in Equations	96
Numbers in Equations	97
Functions in Equations	97
Parentheses in Equations	98
Displaying and Selecting Equations	98
Editing and Clearing Equations	99
Types of Equations	101
Evaluating Equations	101
Using ENTER for Evaluation	103
Using XEQ for Evaluation	104
Responding to Equation Prompts	104
The Syntax of Equations	105
Operator Precedence	105
Equation Functions	107
Syntax Errors	110
Verifying Equations	110
Solving Equations	111
Solving an Equation	111
Understanding and Controlling SOLVE	115
Verifying the Result	116
Interrupting a SOLVE Calculation	117
Choosing Initial Guesses for SOLVE	117
For More Information	121
Integrating Equations	123
Integrating Equations ( ( FN)	124
Accuracy of Integration	127
Specifying Accuracy	128
Interpreting Accuracy	128
For More Information	130
Operations withComplex Numbers	131
The Complex Stack	131
Complex Operations	132
Using Complex Numbers in Polar Notation	135
Base Conversions and Arithmetic	139
Arithmetic in Bases 2, 8, and 16	140
The Representation of Numbers	142
Negative Numbers	142
Range of Numbers	143
Windows for Long Binary Numbers	144
Statistical Operations	145
Entering Statistical Data	145
Entering One–Variable Data	146
Entering Two–Variable Data	146
Correcting Errors in Data Entry	146
Statistical Calculations	148
Mean	148
Sample Standard Deviation	150
Population Standard Deviation	150
Linear Regression	151
Limitations on Precision of Data	153
Summation Values and the Statistics Registers	154
Summation Statistics	154
The Statistics Registers in Calculator Memory	155
Access to the Statistics Registers	155
Programming	157
Simple Programming	159
Designing a Program	161
Selecting a Mode	161
Program Boundaries (LBL and RTN)	161
Using RPN, ALG and Equations in Programs	162
Data Input and Output	162
Entering a Program	163
Keys That Clear	164
Function Names in Programs	165
Running a Program	167
Executing a Program (XEQ)	167
Testing a Program	167
Entering and Displaying Data	169
Using INPUT for Entering Data	169
Using VIEW for Displaying Data	171
Using Equations to Display Messages	172
Displaying Information without Stopping	174
Stopping or Interrupting a Program	175
Programming a Stop or Pause (STOP, PSE)	175
Interrupting a Running Program	175
Error Stops	175
Editing a Program	176
Program Memory	177
Viewing Program Memory	177
Memory Usage	178
The Catalog of Programs (MEM)	178
Clearing One or More Programs	178
The Checksum	179
Nonprogrammable Functions	180
Programming with BASE	180
Selecting a Base Mode in a Program	180
Numbers Entered in Program Lines	181
Polynomial Expressions and Horner's Method	181
Programming Techniques	185
Routines in Programs	185
Calling Subroutines (XEQ, RTN)	186
Nested Subroutines	187
Branching (GTO)	188
A Programmed GTO Instruction	189
Using GTO from the Keyboard	189
Conditional Instructions	190
Tests of Comparison (x?y, x?0)	191
Flags	192
Loops	200
Conditional Loops (GTO)	201
Loops with Counters (DSE, ISG)	202
Indirectly Addressing Variables and Labels	204
The Variable \	204
The Indirect Address, (i)	205
Program Control with (i)	206
Equations with (i)	208
Solving and Integrating Programs	211
Solving a Program	211
Using SOLVE in a Program	216
Integrating a Program	217
Using Integration in a Program	219
Restrictions on Solving and Integrating	221
Mathematics Programs	223
Vector Operations	223
Solutions of Simultaneous Equations	234
Polynomial Root Finder	242
Coordinate Transformations	254
Statistics Programs	261
Curve Fitting	261
Normal and Inverse–Normal Distributions	271
Grouped Standard Deviation	277
Miscellaneous Programs and Equations	283
Time Value of Money	283
Prime Number Generator	288
Appendixes and Reference	293
Support, Batteries,and Service	295
Calculator Support	295
Answers to Common Questions	295
Environmental Limits	296
Changing the Batteries	296
Testing Calculator Operation	298
The Self–Test	299
Warranty	300
Service	301
Regulatory Information	303
User Memory and the Stack	305
Managing Calculator Memory	305
Resetting the Calculator	306
Clearing Memory	307
The Status of Stack Lift	308
Disabling Operations	308
Neutral Operations	308
The Status of the LAST X Register	310
ALG: Summary	311
About ALG	311
Doing Two–number Arithmetic in ALG	312
Simple Arithmetic	312
Power Functions	312
Percentage Calculations	313
Permutations and Combinations	314
Quotient and Remainder Of Division	314
Parentheses Calculations	315
Chain Calculations	315
Reviewing the Stack	316
Coordinate Conversions	317
Integrating an Equation	318
Operations with Complex Numbers	319
Arithmetic in Bases 2, 8, and 16	321
Entering Statistical Two–Variable Data	322
More about Solving	325
How SOLVE Finds a Root	325
Interpreting Results	327
When SOLVE Cannot Find a Root	332
Round–Off Error	337
Underflow	338
More about Integration	339
How the Integral Is Evaluated	339
Conditions That Could Cause Incorrect Results	340
Conditions That Prolong Calculation Time	345
Messages	349
Operation Index	353