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HP 15c Scientific Numerical Integration, f(x), f(x), f(x), f(x), label, The basic rules for using fare:, The quantity I can be interpreted

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Section 14

Numerical Integration

Many problems in mathematics, science, and engineering require calculating the definite integral of a function. If the function is denoted by f(x) and the interval of integration is a to b, the integral can be expressed mathematically as

I = ∫ba f (x)dx.

The quantity I can be interpreted

geometrically as the area of a region bounded by the graph of f(x), the x-axis, and the limits x = a and x = b.*

When an integral is difficult or impossible to evaluate by analytical methods, it can be calculated using numerical techniques. Usually, this can be done only with a fairly complicated computer program. With your HP-15C, however, you can easily do numerical integration using the f (integrate) key.†

Using f

The basic rules for using fare:

1.In Program mode, key in a subroutine that evaluates the function f(x) that you want to integrate. This subroutine must begin with a label instruction (´b label) and end up with a value for f(x) in the X- register.

*Provided that f(x) is nonnegative throughout the interval of integration.

†The f function does not use the imaginary stack. Refer to the HP-15C Advanced Functions Handbook for information about using fin Complex mode.

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Contents

HP-15C Owner’s Handbook Legal Notice Introduction Contents Part I: HP-15CFundamentals Section 5: The Display and Continuous Memory Part II: HP-15CProgramming Section 6: Programming Basics Section 6: Programming Basics Section 7: Program Editing Section 8: Program Branching and Controls Section 9: Subroutines Part III: HP-15CAdvanced Functions Section 14: Numerical Integration Appendix A: Error Conditions Appendix C: Memory Allocation Appendix D: A Detailed Look at Appendix E: A Detailed Look at f Appendix F: Batteries Function Summary and Index Programming Summary and Index Subject Index The HP-15C: A Problem Solver To Compute Keystrokes Display 531,441.0000 Example: Writing the Program Loading the Program KeystrokesDisplay 001-42,21,11 PRGM Running the Program Page Page Getting Started "^ • G f > i O m ´ | P I l F T s ? t H b ´ CLEAR u CLEAR u (change sign) Keying in Exponents The “CLEAR” Keys Clearing Sequence Effect Display Clearing: `and − 12,345 1,234 912,349 Keystrokes Display Two-NumberFunctions and Terminating Digit Entry Chain Calculations [(5.4 × 0.8) ÷ (12.5 − 0.72 )] Numeric Functions |K“|‘ General Functions Reciprocal Factorial and Gamma Square Root Trigonometric Operations Trigonometric Modes RAD GRAD Trigonometric Functions Degrees/Radians Conversions Logarithmic Functions Natural Logarithm Natural Antilogarithm Common Logarithm Common Antilogarithm The Power Function To Calculate Percentages Percent 30 Section 2: Numeric Functions For example, to find the sales tax at 3% and total cost of a $15.76 item: Enters the base number (the price) 3 |k Calculates 3% of $15.76 (the tax) Rectangular Conversion The Automatic Memory Stack LAST X, and Data Storage Stack Lift No Stack Lift or Drop Stack Drop x + y Page The LAST X Register and K before execution of a numeric operation (LAST X) 12.9 + Oops! The wrong divisor 36 Section 3: The Memory Stack, LAST X, and Data Storage *287.0000 Reverses the function that produced the wrong answer 13.9 + The correct answer Calculator Functions and the Stack Order of Entry and the vKey Nested Calculations Arithmetic Calculations With Constants LAST Page Loading the Stack with a Constant 1,000 42 Section 3: The Memory Stack, LAST X, and Data Storage *1,150.0000 Population at the end of day 1,322.5000 Day 1,520.8750 Turn the calculator off. Next day, turn it back on again For storage arithmetic Recall Arithmetic For recall arithmetic x 8.33(4 − 5.2) ÷ [(8.33 − 7.46)0.32] 4.3 (3.15 − 2.75) − (1.71)(2.01) Page Statistics Functions 48 Section 4: Statistics Functions How many different four-cardhands can be dealt from a deck of 52 cards Fifty-two(y) cards dealt four (x) at a time 270,725.0000 statistics registers Register Contents 1,415.00 -value drops to four 52 Section 4: Statistics Functions Correcting Accumulated Statistics x, y Mean Example: Average kg of nitrogen, x, for all cases ®6.40 Average tons of rice, y, for all cases Standard Deviation Linear Regression Linear Estimation and Correlation Coefficient Linear Estimation Correlation Coefficient The The Display and Continuous Memory Scientific Notation Display (scientific) undisplayed With the previous number still in the display: ´i6 Mantissa Display Round-OffError Annunciators Digit Separators 12,345.67 12,345.6700 Error Display Overflow Low-PowerIndication Status Resetting Continuous Memory Page Page Programming Basics Location in Program Memory Program Begin Recording a Program one Executing a Program running 300.51300.51 ´A Restarting a Program User Mode Direct entry Program Memory Example Radius, r Height, h Base Area Volume Surface Area 007-44,40,1 011-44,40 019– 020–44,40,3 021– 43 ¦254.4690 Program Instructions Instruction Coding Instruction Code Keycode 25: second row, fifth key Memory Configuration 76 Section 6: Programming Basics 60 ´m% R60 and below allocated to data storage; five (R61 to R65) remain for programming 1 ´m% 19 ´m% lm% Displays the current highest data register The m and W (memory status) functions are described in detail in appendix C Unexpected Program Stops Pressing Any Key Error Stops Abbreviated Key Sequences User Mode USER Polynomial Expressions and Horner's Method 001-42,21,12 12,691.0000 Nonprogrammable Functions Page Program Editing The Back Step Instruction Deleting Program Lines Inserting Program Lines Deletions: 84 Section 7: Program Editing Program mode. (Assumes position is at line 000.) t“020 020-44,40,3 011-44,40,2 007-44,40 Single-StepOperations Single-Step Program Execution 001−42,21,11 Â002− 44 0 t0 2.5000 Result 003− 004− Insertions and Deletions Initializing Calculator Status ´b.1 001-42,21,.1 002-42,7 Interest Page Program Branching and Controls conditional test Tn 1.Direct: |£and | 2.Indirect: |Tn Test Page Example: Branching and Looping 010-45,20,1 013-43,30,9 016-44,40,0 Example: Flags 002-43,5 004-42,21,15 005-43,4 006-42,21 016-45,10,1 10,698.3049 (Repeat stack entries.) 10,645.0795 Set to Run mode. Monthly payment Payment periods (4 years × 12 months) Looping Conditional Branching Tests The System Flags: Flags 8 and Flag Page Subroutines Subroutine Limits MAIN PROGRAM 001- ´b9 002- |R 003- O0 004- ® xample: Nesting The Subroutine Return Nested Subroutines The Index Register and Loop Control Direct Versus Indirect Data Storage With the Index Register Indirect Program Control With the Index Register Program Loop Control Index Register Storage and Recall Direct Indirect Index Register Arithmetic Exchanging the X-Register Indirect Branching With To Labels To Line numbers Indirect Flag Control With Indirect Display Format Control With Loop Control With Counters: Iand e nnnnnx x x y y Start count at zero Count by twos Count up to nnnnn For Iterations Examples: Register Operations Storing and Recalling Exchanging the X-Register Storage Register Arithmetic Example: Loop Control with e −−011- 42 012-42,5 15 “ O Example: Display Format Control Value in R value in display To display fixed point notation for all possible decimal places on the HP-15C: ´B 9.000000000 8.00000000 7.0000000 6.000000 5.00000 4.0000 3.000 2.00 Display at ´©instruction Display when program halts Index Register Contents 116 Section 10: The Index Register and Loop Control Iand e must be specified as a cannot be zero the value for Page Page Page Calculating With Complex Numbers real RAD GRAD Deactivating Complex Mode Entering Complex Numbers Page Page Stack Lift in Complex Mode Manipulating the Real and Imaginary Stacks real exchange imaginary Clearing a Complex Number Clearing the Real X-Register Clearing the Imaginary Entering Complex Numbers with −. The clearing functions − and (−)(0.0000) Entering a Real Number Entering a Pure Imaginary Number 130 Section 11: Calculating With Complex Numbers Storing and Recalling Complex Numbers X-register only; a + ib ¤xNo∕@'a:; Two-NumberFunctions +-*÷y ®) ( v K Example: Complex Arithmetic Complex Results from Real Numbers and without disturbing the stack contents normally would result in an Error 0. complex value arc sin 2.404 can be |F8 ´% (hold) Page +3.1434 |: Page Page Calculating With Matrices ´>12.0000 1,1 X = A-1B running C – C 1,1 Dimensioning a Matrix ´ m Key the number of rows 2.Key the number of columns (x) into the X-register 3.Press ´ m followed by a letter key, A through E Displaying Matrix Dimensions Keystrokes l>B Changing Matrix Dimensions Storing and Recalling All Elements in Order null A 1,1 Checking and Changing Matrix Elements Individually Using R and R 2,3 Using the Stack Storing a Number in All Elements of a Matrix Matrix Descriptors 148 Section 12: Calculating with Matrices LU factorization The Result Matrix result matrix maximum Copying a Matrix One-MatrixOperations 150 Section 12: Calculating with Matrices One-MatrixOperations: Sign Change, Inverse, Transpose, Norms, Determinant Result in Effect on Matrix Scalar Operations Elements of Result Matrix Operation Matrix in Y-Register Scalar in Y-Register Scalar in X-Register Arithmetic Operations Calculates Y + Y C = B - A 154 Section 12: Calculating with Matrices Calculates B - A and stores values in redimensioned result The result is Matrix Multiplication C = AT B Keystrokes Display l> A A 17⎦ 156 Section 12: Calculating with Matrices Solving the Equation AX = B The ÷function is useful for solving matrix equations of the form AX = B where A is the coefficient matrix, B is Week Total Weight (kg) Total Value Solution: AD = B 158 Section 12: Calculating with Matrices 274OB 233OB 331OB 120.32 OB 112.96 OB 151.36 OB ´<Á Stores b11 Cabbage (kg) Broccoli (kg) Calculating the Residual R–YX B – AC Using Matrices in LU Form then Z can be represented in the calculator by Into ⎡4 + Z = ⎢ ⎣1 + The Complex Transformations Between ZP and Z Inverting a Complex Matrix Multiplying Complex Matrices Keystrokes l>A l>B C4 C 1,1 –2.8500 –10 –4.0000 –111.0000 1.0000 –113.8000 –101.0000 –11 –1.0500 –10 Solving the Complex Equation AX = B AX = B –200.0000 Transforms AP into Ã Designates matrix C as Calculates XP and stores in C Transforms XP into XC Page Using a Matrix Element With Register Operations O*l O{+, -, *, ÷} l{+, -, *, ÷} Using Matrix Descriptors in the Index Register If the Index register contains a matrix descriptor: Conditional Tests on Matrix Descriptors Page Page Results Page Keystroke(s) Results Finding the Roots of an Equation Page 182 Section 13: Finding the Roots of an Equation ´b0 001–42,21,0 Begin with binstruction Subroutine assumes stack –10 –2.0000 G G r Keystrokes |¥ 000– 001–42,21,11 009– 010– 011– 012– 013– 001–42,21,1 Page Page 001–42,21,3 Page Page Page Page Numerical Integration |¥ 000– Page 001-42,21,1 Page Find Si(2) Key in the following subroutine to evaluate the function f(x) = (sin x) / x ´b.2 001–42,21 Begin subroutine with a b 200 Section 14: Numerical Integration no more You'll recall that the HP-15Cprovides three types of display formatting: ´ • V i V approximates ´i2 3.1400 Set display format to 1.300 Integral approximated in ® 1.8 - Uncertainty of 202 Section 14: Numerical Integration Set display format to Roll down stack until upper limit appears in X-register Integral approximated in ´CLEAR u Page Error Conditions Error 1: Improper Matrix Operation Error 2: Improper Statistics Operation Error 3: Improper Register Number or Matrix Element Error 4: Improper Line Number or Label Call Error 5: Subroutine Level Too Deep Error 6: Improper Flag Number Error 7: Recursive _or f Pr Error (Power Error) Pr Error Stack Lift and the LAST X Register Disabling Operations Stack Lift Imaginary Enabling Operations Appendix B: Stack Lift and the LAST X Register Stack enabled when the next number is keyed or recalled into the display Neutral Operations 212 Appendix B: Stack Lift and the LAST X Register The following operations save x in the LAST X register: -x H\ k +[ H] ∆ *\ h : Memory Allocation Total allocatable memory: Memory Status (W) To view the current memory configuration of the calculator, press | W(memory), holding W to retain the display.* The display will be four numbers dduu pp-b dd= the number of the highest-numbered register in the data storage 216 Appendix C: Memory Allocation Place , the available for programming) will be (65 – 2.Press ´m% Automatic Program Memory Reallocation Two-ByteProgram Instructions Function Registers Needed ´ V, ´ } "8 is executed until you dimension it A Detailed Look at Page Page Page 001–42,21,12 010–43,30,7 012–43,30,0 226 Appendix D: A Detailed Look at Execute _again: ´vB Value of modified f(t) at root Page 001–42,21,2 019–42,21,9 022– 1,000.0000 Page ´b.0 001–42,21,.0 10.0000 Error 8 455.335 48,026,721.85 1.0000 455.4335 Error 8 48,026,721.85 1.0000 Error 8 2.1213 2.1471 0.3788 2.1213 Error 8 2.1213 1.0000 –20 ´_.0 –20 –16 Another x-value.Previous value. Same function value x = a 001-42,21,2 023– 024– 025– 026– 027– 028– 029– 030– 031– 032– –208.4999 –1.0929 –07 035– 037– 038– Page Counting Iterations Specifying a Tolerance A Detailed Look at f by just one Page Keystrokes ´i3 –03 7.786 –03 1.448 –04 Page Page –04 =0.5×10−n+m( x) relative absolute Page Page Page 001-42,21 “002- 1 Page Page Page Page Page Page Page Batteries Page 1.L 2.C 3.H Function Summary and Index (page 19) (page 22) Display Control (page 58) (page 59) (page 19) Hyperbolic (page 28) Index Register Control (page 29) Mathematics (page 25) (page 24) (page 180) 152-155) (pages 152- 155) to ZP (page164) to XT (page 150) X (page 154) Percentage Probability (page 47) Stack Manipulation (page 21) (page21) Statistics (page 49) (page 52) Trigonometry 26) (page 26) 26) (page 26) Programming Summary and Index (page 101) (page 68) G(page 101) (page 92) (page 91) Subject Index Page D E F G H I K L M Page N O P 66 Q R 184-186 S Page T U V W X Y Z Product Regulatory & Environment Information Canadian Notice Avis Canadien European Union Regulatory Notice Japanese Notice Korean Notice Chemical Substances Perchlorate Material - special handling may apply