Category: C++ Computer Applications

Text and Exercise from “Boundary Value Problems Second Edition” by David L. Powers in Progress (c) Wednesday, April 17, 2024, James Pate Williams, Jr.

text-and-exercise-from-boundary-value-problems-Download

Solution of the One-Dimensional Heat Equation for a Rod Using Finite Differences by James Pate Williams, Jr. Created on Wednesday April 3, 2024

solution-of-a-one-dimensional-heat-equation-using-finite-differencesDownload

Undamped Mass-Spring Eigenvalue – Eigenvector Problem by James Pate Williams, Jr. (c) Monday April 1, 2024

We extend the results of the following website:

https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/3%3A_Systems_of_ODEs/3.6%3A_Second_order_systems_and_applications

The five masses in the problem have a maximum value of 8. The six springs have a maximum value of 4 for their Hooke’s coefficients. The first 5 by 5 matrix is the inverse mass matrix, the second matrix is the Hooke’s coefficient 5 by 5 matrix, the third 5 by 5 matrix is the product of the inverse mass matrix times the Hooke’s coefficient matrix. The final row vector is the eigenvalue vector.

New 1d Integration Results (c) March 24, 2024, by James Pate Williams, Jr.

I tested NUMAL’s integration function versus homegrown trapezoidal rule and Simpson’s rule. The second and third algorithms were closed (included both endpoints). There exist higher order Newton-Cotes integration formulas. I did not test the Gauss-Legendre and/or Gauss-Laguerre integration method(s). The trapezoidal rule can be improved if the derivative of the integrand is known and is easily calculated.

new-1d-integration-resultsDownload

Microsoft Visual Studio 2019 Community Version – Creating and Using a C Based Dynamic Link Library © March 22, 2024, by James Pate Williams, Jr.

Create a solution using the DLL C++ DLL template. Rename pch.cpp to pch.c and dllmain.cpp to dllmain.c. Add a header file for the DLL containing the following macro:

#define DLL_Export __declspec(dllexport)

Mark all the exportable functions in the DLL header file with the DLL_Export macro. Suppose the DLL is named CSortingDLL. Next create a console C++ project in the DLL solution to use the DLL. Open the Properties page of the console app. If you want to use the ASCII character set, select the Advanced Configuration Properties, and set the Character Set to Not Set. Now open the C/C++ General Property and under Additional Include Directories add ..\CSortingDLL\;. Do not include the previous period mark. Next open the Linker General Property page. Set Additional Library Directories to ..\CSortingDLL\$(IntDir);. The penultimate step is to set the Linker Input Additional Dependencies to CSortingDLL.lib;. The final step for the app is to add a reference to CSortingDLL.  Now you have a functioning DLL and an app using the DLL. If you have any questions, please contact me via email at jamespate@mac.com.  Also, Microsoft has a DLL walkthrough complete with source code. I will add my CSortingDLL header source code as a PDF document.

csortingdll_hDownload

Factorizations of Some Fibonacci Sequence Numbers, Lucas Sequence Numbers and Some Other Numbers Using Arjen K. Lenstra’s Free Large Integer Package and the Elliptic Curve Method (c) January 28, 2024, by James Pate Williams, Jr.

All of the following computations were performed on a late 2015 Dell XPS 8900 personal computer with a 64-bit Intel Core I7 processor @ 4.0GHz with 16GB of DDR2 RAM.

Factorization of Six Fibonacci Sequence Numbers:

 

Fibonacci 500

# digits 105

5 ^ 2 p # digits 1

15 c # digits 2

101 p # digits 3

401 p # digits 3

1661 c # digits 4

3001 p # digits 4

10291 c # digits 5

570601 p # digits 6

112128001 p # digits 9

1353439001 p # digits 10

28143378001 p # digits 11

5465167948001 p # digits 13

84817574770589638001 p # digits 20

158414167964045700001 p # digits 21

Runtime (s) = 1.206000



Fibonacci 505

# digits 106

5 p # digits 1

743519377 p # digits 9

44614641121 p # digits 11

770857978613 p # digits 12

960700389041 p # digits 12

12588421794766514566269164716286291055826556238643852856601641 p # digits 62

Runtime (s) = 1.959000



Fibonacci 510

# digits 107

2 ^ 3 p # digits 1

11 p # digits 2

61 p # digits 2

1021 p # digits 4

1597 p # digits 4

3469 p # digits 4

3571 p # digits 4

9521 p # digits 4

53551 p # digits 5

95881 p # digits 5

142445 c # digits 6

1158551 p # digits 7

3415914041 p # digits 10

20778644396941 p # digits 14

20862774425341 p # digits 14

81358225616651 c # digits 14

162716451241291 p # digits 15

Runtime (s) = 2.682000



Fibonacci 515

# digits 108

5 p # digits 1

519121 p # digits 6

5644193 p # digits 7

512119709 p # digits 9

84388938382141 p # digits 14

300367026458796424297447559250634818495937628065437243817852436228914621 p # digits 72

Runtime (s) = 7.861000



Fibonacci 520

# digits 109

131 p # digits 3

451 c # digits 3

521 p # digits 3

2081 p # digits 4

2161 p # digits 4

3121 p # digits 4

24571 p # digits 5

90481 p # digits 5

2519895 c # digits 7

21183761 p # digits 8

57089761 p # digits 8

102193207 p # digits 9

1932300241 p # digits 10

14736206161 p # digits 11

5836312049326721 p # digits 16

42426476041450801 p # digits 17

Runtime (s) = 5.155000



Fibonacci 525

# digits 110

2 p # digits 1

5 p # digits 1

421 p # digits 3

701 p # digits 3

3001 p # digits 4

3965 c # digits 4

4201 p # digits 4

141961 p # digits 6

2553601 p # digits 7

230686501 p # digits 9

8288823481 p # digits 10

82061511001 p # digits 11

19072991752501 c # digits 14

8481116649425701 p # digits 16

17231203730201189308301 p # digits 23

Runtime (s) = 2.026000

Factorization of Six Lucas Sequence Numbers



Lucas 340

113709744839525149336680459091826532688903186653162057995534262332121127

# digits 72

7 p # digits 1

2161 p # digits 4

5441 p # digits 4

897601 p # digits 6

23230657239121 p # digits 14

17276792316211992881 p # digits 20

3834936832404134644974961 p # digits 25

Runtime (s) = 109.103000



Lucas 345

# digits 73

2 ^ 2 p # digits 1

31 p # digits 2

461 p # digits 3

1151 p # digits 4

1529 c # digits 4

324301 p # digits 6

686551 p # digits 6

1485571 p # digits 7

4641631 p # digits 7

19965899801 c # digits 11

117169733521 p # digits 12

3490125311294161 p # digits 16

Runtime (s) = 0.032000



Lucas 350

13985374084677485786380981408251904922622980674054858121032362563653278123

# digits 74

3 p # digits 1

401 p # digits 3

2801 p # digits 4

11521 c # digits 5

28001 p # digits 5

570601 p # digits 6

12317523121 p # digits 11

248773766357061401 p # digits 18

7358192362316341243805801 p # digits 25

Runtime (s) = 21.047000



Lucas 355

69362907070206748494476200566565775354902428015845969798000696945226974645

# digits 74

5 p # digits 1

4261 p # digits 4

6673 p # digits 4

75309701 p # digits 8

309273161 p # digits 9

46165371073 p # digits 11

9207609261398081 p # digits 16

49279722643391864192801 p # digits 23

Runtime (s) = 40.726000



Lucas 360

769246427201094785080787978422393713094534885688979999504447628313150135520

# digits 75

2 ^ 5 p # digits 1

3 ^ 2 p # digits 1

23 p # digits 2

41 p # digits 2

105 c # digits 3

107 p # digits 3

241 p # digits 3

2161 p # digits 4

2521 p # digits 4

3439 c # digits 4

8641 p # digits 4

20641 p # digits 5

103681 p # digits 6

109441 p # digits 6

191306797 c # digits 9

10783342081 p # digits 11

13373763765986881 p # digits 17

Runtime (s) = 0.032000



Lucas 365

19076060504701386559675231910437330047906343529583769121365013189782992678011

# digits 77

11 p # digits 2

151549 p # digits 6

514651 p # digits 6

7015301 p # digits 7

8942501 p # digits 7

9157663121 p # digits 10

11899937029 p # digits 11

3252336525249736694804553589211 p # digits 31

The following two numbers were first factorized by J. M. Pollard on an 8-bit Phillips P2012 personal computer with 64 KB RAM and two 640 KB disc drives. The times required by Pollard were 41 and 47 hours.

2^144-3
22300745198530623141535718272648361505980413

# digits 44

492729991333 p # digits 12

45259565260477899162010980272761 p # digits 32

Runtime (s) = 0.086000

2^153+3
11417981541647679048466287755595961091061972995

# digits 47

5 p # digits 1

11 p # digits 2

600696432006490087537 p # digits 21

345598297796034189382757 p # digits 24

Runtime (s) = 0.676000

Partial factorization of the Twelfth Fermat Number 2^4096+1
# digits 1234

114689 p # digits 6

26017793 p # digits 8

63766529 p # digits 8

190274191361 p # digits 12

Runtime (s) = 1532.878000