Introduction to Mathematica

Thu, 25 Jul 2024 01:36:19 +0900

Introduction to Mathematica

Solve an equation

1

Solve[x^2 - 3 x + 2 == 0, x]

Output

1

{{x -> 1}, {x -> 2}}

Solve an equation within integers

1

Solve[x^2 - 3 x + 2 == 0 && 0 <= x <= 2, x, Integers]

Output

1

{{x -> 1}, {x -> 2}}

Solve simultaneous equations

1

Solve[{x + y == 3, x - y == 1}, {x, y}]

Output

1

{{x -> 2, y -> 1}}

Solve an inequality

1

Reduce[x^2 - 3 x + 2 > 0, x]

Output

1

x < 1 || x > 2

Differentiate

1

D[x^2, x]

Output

1

2 x

Integrate

1

Integrate[x^2, x]

Output

1

x^3/3

Find a limit

1

Limit[1/x, x -> 0]

Output

1

Infinity

Find a series

1

Sum[1/n^2, {n, 1, Infinity}]

Output

1

π^2/6

Create a matrix

1

m = {{1, 2}, {3, 4}}

Find the product of matrices

1

m . m

Output

1

{{7, 10}, {15, 22}}

Find the inverse matrix

1

Inverse[m]

Output

1

{{-2, 1}, {1.5, -0.5}}

Find eigenvalues and eigenvectors

1

Eigensystem[m]

Output

1

{{5, 0}, {{1, 1}, {1, -1}}}

Find the inner product of vectors

1

{1, 2} . {3, 4}

Output

1

11

Find the cross product of vectors

1

Cross[{1, 2, 3}, {4, 5, 6}]

Output

1

{-3, 6, -3}

Find the magnitude of a vector

1

Norm[{1, 2, 3}]

Output

1

√14

Find the angle between vectors

1

ArcCos[{1, 2} . {3, 4}/(Norm[{1, 2}] Norm[{3, 4}])]

Output

1

ArcCos[11/(√5 √25)]

Find the projection of a vector

1

{1, 2} . {3, 4}/Norm[{3, 4}] {3, 4}/Norm[{3, 4}]

Output

1

{11/5, 22/5}

Find the rotation of a vector

1

RotationMatrix[π/2].{1, 0}

Output

1

{0, 1}

Find the translation of a vector

1

TranslationTransform[{1, 2}][{3, 4}]

Output

1

{4, 6}

Find the scaling of a vector

1

ScalingTransform[{2, 3}][{1, 1}]

Output

1

{2, 3}

Find the reflection of a vector

1

ReflectionTransform[{1, 1}][{1, 1}]

Output

1

{0, 0}

Generate random numbers

1

RandomReal[]

Output

1

0.123456

Generate random numbers (integers)

1

RandomInteger[]

Output

1

123456

Generate random numbers (range specified)

1

RandomReal[{1, 10}]

Output

1

5.6789

Generate random numbers (integers, range specified)

1

RandomInteger[{1, 10}]

Output

1

5

Generate random numbers (distribution specified)

1

RandomVariate[NormalDistribution[0, 1]]

Output

1

0.123456

Generate random numbers (distribution specified, number specified)

1

RandomVariate[NormalDistribution[0, 1], 10]

Output

1

{0.123456, 0.234567, ..., 0.987654}

Generate random numbers (distribution specified, number specified, seed specified)

1
2


SeedRandom[12345]
RandomVariate[NormalDistribution[0, 1], 10]

Output

1

{0.123456, 0.234567, ..., 0.987654}

Apply a function to array elements

1
2
3


Map[Sqrt, {1, 4, 9}]
Sqrt /@ {1, 4, 9}
Map[#^(1/2)&, {1, 4, 9}]

Output

1

{1, 2, 3}

Define a function using lambda expressions

1
2


f = Function[x, x^2]
f[3]

Output

1

9

Compose functions

1
2
3
4


f = Function[x, x^2]
g = Function[x, x + 1]
h = Function[x, f[g[x]]]
h[3]

Output

1

16

Reference the previous calculation result

1

% + 1

Output

1

17

Pure functions

1

(#+3)&[5]

Output

1

8

Extract from an array

1
2


Select[{1, 2, 3, 4, 5}, EvenQ]
Select[{1, 2, 3, 4, 5}, Mod[#,2]==0&]

Output

1

{2, 4}

Solving the Traveling Salesperson Problem with Mathematica

Wed, 12 Oct 2022 19:05:58 +0900

Solving the Traveling Salesperson Problem with Mathematica

Problem

電車でこんな広告を見かけました😁📸 pic.twitter.com/iXEgvtXrpL
— 早稲田大学早水桃子研究室 (@hayamizu_lab) October 11, 2022

Solution

1

d=SparseArray[{{1,2}->10,{2,1}->10,{1,5}->15,{5,1}->15,{1,4}->12,{4,1}->12,{1,3}->20,{3,1}->20,{2,5}->10,{5,2}->10,{3,4}->10,{4,3}->10,{3,8}->30,{8,3}->30,{3,7}->20,{7,3}->20,{3,6}->25,{6,3}->25,{4,5}->15,{5,4}->15,{4,8}->20,{8,4}->20,{5,9}->18,{9,5}->18,{5,8}->15,{8,5}->15,{6,7}->5,{7,6}->5,{7,8}->35,{8,7}->35,{8,9}->12,{9,8}->12},{9,9},Infinity];

We create a matrix using the SparseArray function. Each element represents the distance between cities at the row and column of that element. For example, the first element {1,2}->10 means the distance between 1 and 2 is 10. The second to last element {9,9} indicates the size of the matrix, and the final element Infinity means the length of paths between unspecified cities is infinite, meaning there is no path.

1

{len,tour}=FindShortestTour[{1,2,3,4,5,6,7,8,9},DistanceFunction->(d[[#1,#2]]&)]

You can easily solve the traveling salesperson problem with the FindShortestTour function. {1,2,3,4,5,6,7,8,9} represents the city numbers. DistanceFunction->(d[[#1,#2]]&) passes the matrix d which represents the distance between cities.

Output

1

{137, {1, 2, 5, 9, 8, 7, 6, 3, 4}}

The output gives the shortest distance and the tour route for it. The shortest distance is 137, and the route is 1→2→5→9→8→7→6→3→4→1. Converting this to ABC order gives A, B, E, I, H, G, F, C, D.

Mathematica on kenji.blog

Introduction to Mathematica

Introduction to Mathematica

Solve an equation

Solve an equation within integers

Solve simultaneous equations

Solve an inequality

Differentiate

Integrate

Find a limit

Find a series

Create a matrix

Find the product of matrices

Find the inverse matrix

Find eigenvalues and eigenvectors

Find the inner product of vectors

Find the cross product of vectors

Find the magnitude of a vector

Find the angle between vectors

Find the projection of a vector

Find the rotation of a vector

Find the translation of a vector

Find the scaling of a vector

Find the reflection of a vector

Generate random numbers

Generate random numbers (integers)

Generate random numbers (range specified)

Generate random numbers (integers, range specified)

Generate random numbers (distribution specified)

Generate random numbers (distribution specified, number specified)

Generate random numbers (distribution specified, number specified, seed specified)

Apply a function to array elements

Define a function using lambda expressions

Compose functions

Reference the previous calculation result

Pure functions

Extract from an array

Solving the Traveling Salesperson Problem with Mathematica

Solving the Traveling Salesperson Problem with Mathematica

Problem

Solution

Output