HP 15c Scientific Calculator User Manual

HEWLETT-PACKARD HP-15C ADVANCED FUNCTIONS HANDBOOK

10 Section 1: Using _ Effectively 10 If c isn't a root, but f(c) is closer to zero than f(b), then b is relabeled as a, c is relabeled as b, an

100 Section 4: Using Matrix Operations 100 matrix A + ΔA represented by the LU decomposition. It can be zero only if the product's magnitude b

Section 4: Using Matrix Operations 101 One-Step Residual Correction The following program solves the system of equations AX = B for X, then perfo

102 Section 4: Using Matrix Operations 102 Example: Use the residual correction program to calculate the inverse of matrix A for .1748571024721633

Section 4: Using Matrix Operations 103 Let )()()()()()()(and,)()()()(,12211112121xxxxxxxFxxxxfxpppppppF

104 Section 4: Using Matrix Operations 104 21)1()1(2)2/exp()2/()(2xxmmadwwwf x. Here x2 > xl > 0, a and n are known (n > 1), and m =

Section 4: Using Matrix Operations 105 Keystrokes Display ´UO4 023- 44 4 Stores )(1kxin R4. ´UlA ´U 024u 45 11 Skips next line for last elem

106 Section 4: Using Matrix Operations 106 Keystrokes Display l2 059 45 2 3 060- 3 - 061- 30 2 062- 2 ÷ 063- 10 C

Section 4: Using Matrix Operations 107 Now run the program. For example, choose the values n = 11 and a = 0.05. The suggested initial guesses are x1

108 Section 4: Using Matrix Operations 108 Solving a Large System of Complex Equations Example: Find the output voltage at a radian frequency of ω =

Section 4: Using Matrix Operations 109 Keystrokes Display ´mA 8.0000 Dimensions matrix A to 4 × 8. ´>1 8.0000 ´U 8.0000 Activates User mode

Section 1: Using _ Effectively 11 If _ hasn't found a sign change and a sample value c doesn't yield a function value with diminished mag

110 Section 4: Using Matrix Operations 110 Keystrokes Display 1v8 8 ´mC 8.0000 Redimensions matrix C to 1 × 8. l< C 1 8 ´>4 C

Section 4: Using Matrix Operations 111 Least-Squares Using Normal Equations The unconstrained least-squares problem is known in statistical literatu

112 Section 4: Using Matrix Operations 112 and  .)(||ˆ||E22 pnFbXy For b ≠ 0. When the simpler model y = r is correct, both of these expectatio

Section 4: Using Matrix Operations 113 The analysis of variance (ANOVA) table below partitions the total sum of squares (Tot SS) into the regression

114 Section 4: Using Matrix Operations 114 You will need the original dependent variable data for each regression. If there is not enough room to s

Section 4: Using Matrix Operations 115 Keystrokes Display ´>6 018-42,16, 6 Calculates residuals of fit in B. ´>8 019-42,16, 8 |x 020- 4

116 Section 4: Using Matrix Operations 116 4. Press ´> 1 to set registers R0 and R1. 5. Press ´U to activate User mode. 6. For each observation

Section 4: Using Matrix Operations 117 ´>1 1.0000 ´U 1.0000 1OA 1.0000 Enters independent variable data. 3.9OA 3.9000 3.5OA 3.5000 ⋮

118 Section 4: Using Matrix Operations 118 lÁ 6.963636364 b1 estimate. ´U 6.963636364 Deactivates User mode. ´•4 6.9636 The Reg SS for the PPI va

Section 4: Using Matrix Operations 119 )rows1(),row1(rows)(ˆpnpq000gUV and Û is an upper-triangular matrix. If this factorization resul

12 Section 1: Using _ Effectively 12 In general, every equation is one of an infinite family of equivalent equations with the same real roots. And s

120 Section 4: Using Matrix Operations 120 )row1(),row1(rows)(*pq*000gUA** Where U* is also an upper-triangular matrix. You can o

Section 4: Using Matrix Operations 121 Keystrokes Display lmA 018-45,23,11 Recalls dimensions p + 2 and p + 1. ® 019- 34 O2 020- 44 2 S

122 Section 4: Using Matrix Operations 122 Keystrokes Display l0 053- 45 0 Recalls k (row). |£ 054- 43 10 Tests k ≤ l. t2 055- 22 2 Loops

Section 4: Using Matrix Operations 123 Keystrokes Display + 089- 40 l0 090- 45 0 ´mA 091-42,23,11 Dimensions matrix A as (p + 2) × (p

124 Section 4: Using Matrix Operations 124 8. Optionally, press C|x to calculate and display the residual sum of squares q2 and to calculate the cu

Section 4: Using Matrix Operations 125 These estimates agree (to within 3 in the ninth significant digit) with the results of the pre

126 Section 4: Using Matrix Operations 126 An orthogonal change of variables x = Qz, which is equivalent to rotating the coordinate axes, chang

Section 4: Using Matrix Operations 127 Keystrokes Display |¥ Program mode. ´CLEARM 000 ´bA 001-42,21,11 l>A 002-45,16,11 O>B 003-44,1

128 Section 4: Using Matrix Operations 128 Keystrokes Display l0 037- 45 0 v 038- 36 l|A 039-45,43,11 Recalls aii. l1 040- 45 1 v

Section 4: Using Matrix Operations 129 Keystrokes Display ´>5 073-42,16, 5 Calculates C = BTA. l>B 074-45,16,12 ´<A 075-42,26,11 * 0

Section 1: Using _ Effectively 13 This function equals zero at no more than n real values of x, called zeros of the polynomial. A limit to the numbe

130 Section 4: Using Matrix Operations 130 Keystrokes Display ⋮ 3OA 3.0000 Enters a32. 4OA 4.0000 Enters a33. A 0.8660 Calculates ratio－too

Section 4: Using Matrix Operations 131 The value used for λk need not be exact; the calculated eigenvector is determined accurately in spite of s

132 Section 4: Using Matrix Operations 132 Keystrokes Display l>C 024-45,16,13 O>Á 025-44,16,14 Stores z(n) in D. O< 026- 44 26 l>

Section 4: Using Matrix Operations 133 3. Dimension and enter the elements into matrix A using ´mA, ´>1, and OA. 4. Key in the eigenvalue a

134 Section 4: Using Matrix Operations 134 Keystrokes Display lC -0.5000 lC 1.0000 Eigenvector forλ2. lC -0.5000 6.8730C 0.7371 Uses λ3=6.8730 (a

Section 4: Using Matrix Operations 135 Labels used: E and 8. Registers used: no additional registers. Matrices used: A (from previous program) and E

136 Section 4: Using Matrix Operations 136 maximum.afindingfor)(minimumafindingfor)(xxsff Once the direction is determined from the gra

Section 4: Using Matrix Operations 137 N Determines the maximum number of iterations that the program will attempt in each of two procedures: the

138 Section 4: Using Matrix Operations 138 Keystrokes Display |n 008- 43 32 ´b7 009-42,21, 7 Line search routine. l4 010- 45 4 l÷6 011-45

Section 4: Using Matrix Operations 139 Keystrokes Display register. t6 046- 22 6 Branch for continuation. ´b5 047-42,21, 5 Interval reduction

14 Section 1: Using _ Effectively 14 First convert 5° 18'S to decimal degrees (press 5.18”|À), obtaining −5.3000 (using •4 display

140 Section 4: Using Matrix Operations 140 Keystrokes Display ´>5 083-42,16, 5 Calculates slope as (f)Ts. 1 084- 1 v 085- 36 l|B

Section 4: Using Matrix Operations 141 Labels used: A, B, and 2 through 8. Registers used: R2 through R9, R.0, R.1, and Index register. Matrices use

142 Section 4: Using Matrix Operations 142 Example: Use the optimization program to find the dimensions of the box of largest volume with the sum of

Section 4: Using Matrix Operations 143 Keystrokes Display l.3 144- 45 .3 l+.3 145-45,40,.3 - 146- 30 Replaces ei with lei − 2wh, the gr

144 Section 4: Using Matrix Operations 144 Keystrokes Display 9.253 03 Volume at this iteration. ¦ 3.480 01 Gradient. 1.121 03 Slop

145 Appendix: Accuracy of Numerical Calculations Misconceptions About Errors Error is not sin, nor is it always a mistake. Numerical error is merely

146 Appendix: Accuracy of Numerical Calculations 146 where payment = $0.01 = one penny per second, i = 0.1125 = 11.25 percent per annum interest ra

Appendix: Accuracy of Numerical Calculations 147 needed to line up the bridge girders before she can rivet them together. Both see the same discrepa

148 Appendix: Accuracy of Numerical Calculations 148 0 < x < 1 above. When R(x) is squared 50 times to produce F(x) = S(R(x)), the result is cl

Appendix: Accuracy of Numerical Calculations 149 Regarding the first misconception, example 1 would behave in the same perverse way if it suffered o

Section 1: Using _ Effectively 15 In Run mode, key in the five coefficients: Keystrokes Display |¥ Run mode. 4.2725 “8” 4.2725 -08 O4 4.2725

150 Appendix: Accuracy of Numerical Calculations 150 A Hierarchy of Errors Some errors are easier to explain and to tolerate than others.

Appendix: Accuracy of Numerical Calculations 151 significantly smaller than one unit in the 10th significant digit of the result, include ∆

152 Appendix: Accuracy of Numerical Calculations 152 She calculated $376,877.67 on her HP-15C, but the bank's total was $333,783.35, and this la

Appendix: Accuracy of Numerical Calculations 153 Keystrokes Display - 011- 30 Calculates u − 1 when u≠1. ÷ 012- 10 Calculates x/(u − 1)

154 Appendix: Accuracy of Numerical Calculations 154 numbers. In other words, every complex function f in Level 1C will produce a calculated com

Appendix: Accuracy of Numerical Calculations 155 with [ (1014 $) = 0.7990550814, although the true sin (1014 $) = −0.78387… The wrong sign is an err

156 Appendix: Accuracy of Numerical Calculations 156 [(2x) = −0.00001100815000 2[(x) \(x) = −0.00001100815000. Note the close agreem

Appendix: Accuracy of Numerical Calculations 157 It all seems like much ado about very little. After a blizzard of formulas and examples, we conclud

158 Appendix: Accuracy of Numerical Calculations 158 Some transformations f are stable in the presence of input noise; they keep Δy rel

Appendix: Accuracy of Numerical Calculations 159 introduced to explain diverse noise sources actually distributed throughout F. Some of the noise ap

16 Section 1: Using _ Effectively 16 Keystrokes Display − 278.4497 Last estimate tried. ) 276.7942 A previous estimate. ) 7.8948 Nonzero value of

160 Appendix: Accuracy of Numerical Calculations 160 Example 6: The Smaller Root of a Quadratic. The two roots x and y of the quadrati

Appendix: Accuracy of Numerical Calculations 161 when we wish to calculate f(x), the best we could hope to get is f(x + Δx), but we actually get F(x

162 Appendix: Accuracy of Numerical Calculations 162 all calculated accurately because N is in Level 1. What might happen if N were in Level 2 instea

Appendix: Accuracy of Numerical Calculations 163 Example 7: The Angle in a Triangle. The cosine law for triangles says r2 = p2 + q2 – 2pq cos θ for

164 Appendix: Accuracy of Numerical Calculations 164 Disparate Results from Three Programs FA, FB, FC Case 1 Case 2 Case 3 p 1. 9.999999996

Appendix: Accuracy of Numerical Calculations 165 Keystrokes Display * 007- 20 ® 008- 34 |K 009- 43 36 |x 010- 43 11 + 011-

166 Appendix: Accuracy of Numerical Calculations 166 Keystrokes Display ) 046- 33 |£ 047- 43 10 ® 048- 34 O1 049- 44 1 O+0 0

Appendix: Accuracy of Numerical Calculations 167 Keystrokes Display ¤ 085- 11 * 086- 20 l0 087- 45 0 |: 088- 43 1 |~ 089-

168 Appendix: Accuracy of Numerical Calculations 168 constitute the edge lengths of a feasible triangle, so FC might produce an error message when it

Appendix: Accuracy of Numerical Calculations 169 (A + ΔA)-1 is at least about as far from A-1 in norm as the calculated ⁄ (A). The figure below illu

Section 1: Using _ Effectively 17 For some systems of equations, expressed as f1(x1, …, xn) = 0 ⋮ fn(x1, …, xn) = 0 it is possible through algebraic

170 Appendix: Accuracy of Numerical Calculations 170 000,5200003.000,5000002.0004503.000,50000,5004503.000,5000050000020 ,.X and 

Appendix: Accuracy of Numerical Calculations 171 Of course, none of these horrible things could happen if X were not so nearly singul

172 Appendix: Accuracy of Numerical Calculations 172 Example 6 Continued. The program listed below solves the real quadratic equation c − 2 bz + az2

Appendix: Accuracy of Numerical Calculations 173 Keystrokes Display ÷ 021- 10 |K 022- 43 36 |( 023- 43 33 ÷ 024- 10 |n 025-

174 Appendix: Accuracy of Numerical Calculations 174 The results of several cases are summarized below. Case 1 Case 2 Case 3 Case 4 c 3 4 1 654,321

Appendix: Accuracy of Numerical Calculations 175 Subroutine "B" below, which uses such a trick,* is a very short program that guarantees n

176 Appendix: Accuracy of Numerical Calculations 176 Keystrokes Display |( 032- 43 33 O8 033- 44 8 l7 034- 45 7 O9 035- 44 9 |a

Appendix: Accuracy of Numerical Calculations 177 Keystrokes Display l÷V 071-45,10,25 v 072- 36 ” 073- 16 l0 074- 45 0 lV 075-

178 Index Page numbers in bold type indicate primary references; page numbers in regular type indicate secondary references. A Absolute error · 146,

Index 179 Complementary normal distribution function · 51–55 Complex components, accurate · 64 Complex equations, solving large system · 107–110 Com

18 Section 1: Using _ Effectively 18 A final consideration for this example is to choose the initial estimates that would be appropriat

180 Index 180 Electrostatic field · 50 Endpoint, f sampling at · 41, 48 Equations complex, solving large system · 107–110 equivalent · 11–12 solving

Index 181 Extended precision · 41 Extremes of function · 18–24 F F actorization, orthogonal · 95–98, 118–25 F ratio · 112–18 Field Intensity · 18–24

182 Index 182 backward error analysis · 168–171 IRR · 34–39 Iterative refinement · 88, 100–102 J Jordon canonical form · 131 L Large system of comple

Index 183 Multivalued functions, complex · 59–62 N n th roots of complex number · 59, 67–69 Nearly singular matrix · 88, 93 Net present value · 34–3

184 Index 184 Preconditioning a system · 91–93 Present value · 24–39 Principal branch · 59–62 Principal value · 59–62 Q Quadratic equation, roots · 1

Index 185 Sampling, f · 41, 48, 63 Scaling a matrix · 88–91 Scaling a system · 90–91 Secant method · 9 Sign change · 10 Sign symmetry · 151, 156 Sin

186 Index 186 V Variables, transforming · 47–48 W Weighted least-squares · 93–94, 97–98, 120 Weighted normal equations · 94 Y Yield · 35 Z Zero of po

Section 1: Using _ Effectively 19 Keystrokes Display |¥ Program mode. ´CLEAR M 000- ´ b0 001-42,21, 0 \ 002- 24 l0 003- 45 0 * 00

Legal Notice This manual and any examples contained herein are provided “as is” and are subject to change without notice. Hewlett-Packa

20 Section 1: Using _ Effectively 20 The relative field intensity is maximum at an angle of 90° (perpendicular to the tower). To find the minimum,

Section 1: Using _ Effectively 21 The accuracy of this approximation depends upon the increment Δ and the nature of the function. Small

22 Section 1: Using _ Effectively 22 Keystrokes Display ® 029- 34 [ 030- 23 ÷ 031- 10 l2 032- 45 2 * 033- 20 |n 03

Section 1: Using _ Effectively 23  If a root of g(x) is found, either the number e is not beyond the extreme value of f(x) or else _ has found a d

24 Section 1: Using _ Effectively 24 Based on these samples, try using an extreme estimate of −0.25 and initial _ estimates (in radians

Section 1: Using _ Effectively 25 n The number of compounding periods. (For example, a 30 year loan with monthly payments has n = 12 x

26 Section 1: Using _ Effectively 26 compounding period—months or years, for example. Vertical arrows represent exchanges of money, following the con

Section 1: Using _ Effectively 27  If a problem has a specified interest rate of 0, the program generates an Error 0 display (or Er

28 Section 1: Using _ Effectively 28 Keystrokes Display v 029- 36 “ 030- 26 ” 031- 16 3 032- 3 |"1 033-43, 5, 1 C

Section 1: Using _ Effectively 29 Keystrokes Display ¦ 066- 31 G1 067- 32 1 Calculates FV. l+3 068-45,40, 3 l÷7 069-45,10, 7 ” 070-

HEWLETT PACKARD HP-15C Advanced Functions Handbook August 1982 00015-90011 Printed in U.S.A. © Hewlett-Packard Company 1982

30 Section 1: Using _ Effectively 30 Keystrokes Display l+3 103-45,40, 3 _ subroutine continues. ´b2 104-42,21, 2 l5 105- 45 5 l*7 106-45,2

Section 1: Using _ Effectively 31 Example: You place $155 in a savings account paying 5¾% compounded monthly. What sum of money can you withdraw at

32 Section 1: Using _ Effectively 32 Keystrokes Display ´CLEARQ Clears financial variables 30v12*A 360.00 Enters n = 30 × 12. 13v12÷B 1.08 Enters

Section 1: Using _ Effectively 33 Keystrokes Display ´CLEARQ Clears financial variables 360A 360.00 Enters n = 360. 14v12÷B 1.17 Enters i = 14/1

34 Section 1: Using _ Effectively 34 Keystrokes Display ´CLEARQ Clears financial variables. |F0 Specifies beginning of period payments. 5v12*A

Section 1: Using _ Effectively 35 flows must occur at equal intervals; if no cash flow occurs for several time periods, enter 0 for the cash flow am

36 Section 1: Using _ Effectively 36 Keystrokes Display “ 010- 26 ” 011- 16 3 012- 3 ´_2 013-42,10, 2 t1 014- 22 1 t0

Section 1: Using _ Effectively 37 Keystrokes Display - 048- 30 l÷2 049-45,10, 2 l*3 050-45,20, 3 t5 051- 22 5 ´b4 052-42,21, 4 ®

38 Section 1: Using _ Effectively 38  To calculate NPV, enter periodic interest rate i in percent and press A. Repeat for as many interest rates as

Section 1: Using _ Effectively 39 Since the NPV is negative, the investment does not achieve the desired 9% yield. Calculate the IRR. Keystrokes Dis

4 Contents Contents ...4 Introduction ...

40 Section 2: Working with f The HP-15C gives you the ability to perform numerical integration using f. This section shows you how to use f effectivel

Section 2: Working with f 41 The HP-15C doesn't prevent you from declaring that f(x) is far more accurate than it really is. You can specify

42 Section 2: Working with f 42 Functions Related to Physical Situations Functions like cos(4 - sin) are pure mathematical functions. In this conte

Section 2: Working with f 43 mathematical operations—may not be accurate to all 10 digits that can be displayed. Note that round-off error affect

44 Section 2: Working with f 44 4. To get the integral over the entire interval of integration, add together the approximations and th

Section 2: Working with f 45 Keystrokes Display 200v 2.000 00 Try a smaller value of x. vv 2.000 02 G1 2.768 -85 Calculator doesn’t

46 Section 2: Working with f 46 Keystrokes Display ® 1.841 -04 Uncertainty of approximation. )) 1.000 01 Roll down stack until upper limit

Section 2: Working with f 47 Transformation of Variables In many problems where the function changes very slowly over most of a very wide interval o

48 Section 2: Working with f 48 The approximation agrees with the value calculated In the previous problem for the same integral. Evaluating Diff

Section 2: Working with f 49 Although the branch for u=1 adds four steps to your subroutine, integration near x = 0 becomes more accu

Contents 5 Applications ... 65

50 Section 2: Working with f 50 tutduuedxxf1tan02tan0)tan1()(2, which is calculated readily even with t as large as 1010. Using the same substi

Section 2: Working with f 51 01.0/  acr However, this integral is nearly improper because q and r are both so nearly zero. But by using an integra

52 Section 2: Working with f 52 The program has the following characteristics:  The display format specifies the accuracy of the integrand in the s

Section 2: Working with f 53 Keystrokes Display ´b4 030-42,21, 4 Subroutine for erf(x) or erfc(x). |"1 031-43, 5, 1 Clears flag 1. O1 032-

54 Section 2: Working with f 54 Keystrokes Display - 067- 30 + 068- 40 Adjusts integral for sign of x and function. ” 069- 16 |n

Section 2: Working with f 55 Keystrokes Display 1.234´A 8.914 -01 P(1.234). .5´E 5.205 -01 erf(0.5). Example: For a Normally distribute

56 Section 3: Calculating in Complex Mode Physically important problems involving real data are often solved by performing relatively simple calculat

Section 3: Calculating in Complex Mode 57 Keystrokes Display 6 002- 6 OV 003- 44 25 Stores counter in Index register. ® 004- 34

58 Section 3: Calculating in Complex Mode 58 Keystrokes Display 195v371÷ 0.5256 O4 0.5256 Stores a4. 1.011523068O5 1.0115 Stores a5. 1.517473649

Section 3: Calculating in Complex Mode 59 Definitions of Math Functions The lists that follow define the operation of the HP-15C in Com

6

60 Section 3: Calculating in Complex Mode 60 The illustrations that follow show the principal branches of the inverse relations. The left-hand graph

Section 3: Calculating in Complex Mode 61

62 Section 3: Calculating in Complex Mode 62 The principal branches in the last four graphs above are obtained from the equations shown, but don&apo

Section 3: Calculating in Complex Mode 63 Using _ and f in Complex Mode The _ and f functions use algorithms that sample your function at values

64 Section 3: Calculating in Complex Mode 64 To require approximations with accurate components is to demand more than keeping relative er

Section 3: Calculating in Complex Mode 65 then think of Z as 0.0000001234567890 × 10−3 + i(2.222222222 × 10−3). then the accurate digits are 0.000

66 Section 3: Calculating in Complex Mode 66 Keystrokes Display |n 011- 43 32 ´bE 012-42,21,15 “Recall” program. O0 013- 44 0 R0 = k. |` 0

Section 3: Calculating in Complex Mode 67 Calculating the nth Roots of a Complex Number This program calculates the nth roots of a complex number.

68 Section 3: Calculating in Complex Mode 68 Keystrokes Display ´V 029- 42 25 Forms complex z0. * 030- 20 Calculates z0eik360 / n, root num

Section 3: Calculating in Complex Mode 69 Keystrokes Display ¦ 0.9980 Calculates z1 (real part). ´% (hold) 0.0628 Imaginary part of z1. 50O V

7 Introduction The HP-15C provides several advanced capabilities never before combined so conveniently in a handheld calculator:  Finding the roots

70 Section 3: Calculating in Complex Mode 70 negative values of Re(z). This would slow convergence considerably unless the first guess z0 were extrem

Section 3: Calculating in Complex Mode 71 Keystrokes Display O0 016- 44 0 ® 017- 34 ´V 018- 42 25 Forms complex A(n). |n 019- 4

72 Section 3: Calculating in Complex Mode 72 Keystrokes Display 8 054- 8 ® 055- 34 ÷ 056- 10 + 057- 40 |a 058- 43 16

Section 3: Calculating in Complex Mode 73 Since all roots have negative real parts, the system is stable, but the margin of stability (the smalles

74 Section 3: Calculating in Complex Mode 74 Keystrokes Display v 012- 36 1 013- 1 ´f0 014-42,20, 0 Calculates Im(I) and Im(ΔI). O2

Section 3: Calculating in Complex Mode 75 To use this program: 1. Enter your function subroutine labeled "B" into program memory. 2. P

76 Section 3: Calculating in Complex Mode 76 Keystrokes Display ' 054- 12 Calculates eiz. ® 055- 34 ÷ 056- 10 Calculates f(z

Section 3: Calculating in Complex Mode 77 The program listed below is set up to compute the values of yk from evenly spaced values of xk. You mus

78 Section 3: Calculating in Complex Mode 78 Keystrokes Display O+V 027-44,40,25 Increments counter k in Index register. l4 028- 45 4 Recalls h.

Section 3: Calculating in Complex Mode 79 Keystrokes Display l5 059- 45 5 Recalls c. - 060- 30 Calculates Im(P(zk)) – c. |n 061- 43

8

80 Section 3: Calculating in Complex Mode 80 Keystrokes Display |n 067- 43 32 Determine the streamline using z0 = −2 + 0.1 i, step size h = 0.5,

Section 3: Calculating in Complex Mode 81 Example: For the same potential as the previous example, P(z) = 1/z + z, compute the velo

82 Section 4: Using Matrix Operations Matrix algebra is a powerful tool. It allows you to more easily formulate and solve many complicated problems,

Section 4: Using Matrix Operations 83 Row interchanges can also reduce rounding errors that can occur during the calculation of the decomposition. T

84 Section 4: Using Matrix Operations 84 ILL-Conditioned Matrices and the Condition Number In order to discuss errors in matrix calculations, it&apos

Section 4: Using Matrix Operations 85 that X and B are nonzero vectors satisfying AX = B for some square matrix A. Suppose A is perturbed by ΔA and

86 Section 4: Using Matrix Operations 86  ASAA  min)(1 K and  SAA1min1, where the minimum is taken over all singular matrices S. That is, if

Section 4: Using Matrix Operations 87 In the left diagram, ||ΔA|| < 1/||A−1||. If ||ΔA|| << 1 / ||ΔA−1|| (or K(A) ||ΔA||/||A|| << 1

88 Section 4: Using Matrix Operations 88  )10log(logcarrieddigitsofnumberdigits decimalcorrectofnumbern1AA where n is the dimen

Section 4: Using Matrix Operations 89 For example, let matrix A be 1121112110340A. The HP-15C correctly calculates A−1 to 10-digit acc

9 Section 1: Using _ Effectively The _ algorithm provides an effective method for finding a root of an equation. This section describes the numeric

90 Section 4: Using Matrix Operations 90 Multiplying the calculated inverse and the original matrix verifies that the calculated inverse is poor. The

Section 4: Using Matrix Operations 91 If (LER) −1 is verifiably poor, you can repeat the scaling, using LER in place of E and using new scaling matr

92 Section 4: Using Matrix Operations 92 11111111111111111111111116.2014and6.20146.20146.20146.20146.20141-EX. Subs

Section 4: Using Matrix Operations 93 Note that rTE was scaled by 107 so that each row of E and A has roughly the same norm as every other. Using th

94 Section 4: Using Matrix Operations 94 niiiFrw1222Wr where W is a diagonal n × n matrix with positive diagonal elements w1, w2, ... , wn. Then )

Section 4: Using Matrix Operations 95 and .97.999,903.000,10yXT However, when rounded to 10 digits, ,1010101010101010XXT which is t

96 Section 4: Using Matrix Operations 96 Any n × p matrix X can be factored as X = QTU, where Q is an n × n orthogonal matrix characterized by QT = Q

Section 4: Using Matrix Operations 97 Only the first p + 1 rows (and columns) of V need to be retained. (Note that Q here is not the same as that m

98 Section 4: Using Matrix Operations 98 columns in the original data and refactor the weighted constraint equations. Repeat this procedu

Section 4: Using Matrix Operations 99 101003313333333333.01LU, which is nonsingular. The singular matrix B can't be dist