Examples¶
Demo¶
Getting Started¶
To see all the matrices in the collection, type
julia> mdinfo()
Currently loaded Matrices
–––––––––––––––––––––––––––
builtin(#)
––––––––––– ––––––––––– ––––––––––– –––––––––––– ––––––––––– ––––––––––––– ––––––––––––
1 baart 10 deriv2 19 gravity 28 kms 37 parter 46 rohess 55 ursell
2 binomial 11 dingdong 20 grcar 29 lehmer 38 pascal 47 rosser 56 vand
3 blur 12 erdrey 21 hadamard 30 lotkin 39 pei 48 sampling 57 wathen
4 cauchy 13 fiedler 22 hankel 31 magic 40 phillips 49 shaw 58 wilkinson
5 chebspec 14 forsythe 23 heat 32 minij 41 poisson 50 smallworld 59 wing
6 chow 15 foxgood 24 hilb 33 moler 42 prolate 51 spikes
7 circul 16 frank 25 invhilb 34 neumann 43 randcorr 52 toeplitz
8 clement 17 gilbert 26 invol 35 oscillate 44 rando 53 tridiag
9 companion 18 golub 27 kahan 36 parallax 45 randsvd 54 triw
user(#)
–––––––
Groups
––––––– ––––– ––––– ––––––– –––––– ––––––– –––––––––
all local eigen illcond posdef regprob symmetric
builtin user graph inverse random sparse
Suite Sparse of
–––––––––––– ––––
2772 2833
MatrixMarket of
–––––––––––– –––
488 498
We can generate a Hilbert matrix of size 4 by typing
matrixdepot("hilb", 4)
4x4 Array{Float64,2}:
1.0 0.5 0.333333 0.25
0.5 0.333333 0.25 0.2
0.333333 0.25 0.2 0.166667
0.25 0.2 0.166667 0.142857
and generate a circul matrix of size 5 by
matrixdepot("circul", 5)
5x5 Array{Float64,2}:
1.0 2.0 3.0 4.0 5.0
5.0 1.0 2.0 3.0 4.0
4.0 5.0 1.0 2.0 3.0
3.0 4.0 5.0 1.0 2.0
2.0 3.0 4.0 5.0 1.0
We can type the matrix name to get help.
mdinfo("hilb")
Hilbert matrix
≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡
The Hilbert matrix is a very ill conditioned matrix. It is symmetric
positive definite and totally positive.
Input options:
• [type,] dim: the dimension of the matrix;
• [type,] row_dim, col_dim: the row and column dimensions.
Groups: ["inverse", "ill-cond", "symmetric", "pos-def"]
Reference: M. D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math.
Monthly, 90 (1983), pp. 301-312.
N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for
Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002; sec. 28.1
mdinfo("hadamard")
Hadamard matrix
≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡
The Hadamard matrix is a square matrix whose entries are 1 or -1. It was
named after Jacques Hadamard. The rows of a Hadamard matrix are orthogonal.
Input options:
• [type,] n: the dimension of the matrix, n is a power of 2.
Groups: ["inverse", "orthogonal", "eigen"]
Reference: S. W. Golomb and L. D. Baumert, The search for Hadamard matrices,
Amer. Math. Monthly, 70 (1963) pp. 12-17
From the information given, we can create a 4-by-6 rectangular Hilbert matrix by
matrixdepot("hilb", 4, 6)
4x6 Array{Float64,2}:
1.0 0.5 0.333333 0.25 0.2 0.166667
0.5 0.333333 0.25 0.2 0.166667 0.142857
0.333333 0.25 0.2 0.166667 0.142857 0.125
0.25 0.2 0.166667 0.142857 0.125 0.111111
We can also specify the data type
matrixdepot("hilb", Float16, 5, 3)
5x3 Array{Float16,2}:
1.0 0.5 0.33325
0.5 0.33325 0.25
0.33325 0.25 0.19995
0.25 0.19995 0.16663
0.19995 0.16663 0.14282
Matrices can be accessed by groups.
mdlist(:symmetric)
19-element Array{ASCIIString,1}:
"hilb"
"cauchy"
"circul"
"dingdong"
"invhilb"
"moler"
"pascal"
"pei"
"clement"
"fiedler"
"minij"
"tridiag"
"lehmer"
"randcorr"
"poisson"
"wilkinson"
"randsvd"
"kms"
"wathen"
mdlist(:symmetric & :illcond)
7-element Array{ASCIIString,1}:
"hilb"
"cauchy"
"invhilb"
"moler"
"pascal"
"pei"
"tridiag"
mdlist(:inverse & :illcond & :symmetric)
7-element Array{ASCIIString,1}:
"hilb"
"cauchy"
"invhilb"
"moler"
"pascal"
"pei"
"tridiag"
User Defined Groups¶
We can add new groups to MatrixDepot. Since each group in Matrix Depot is a list of strings, you can simply do, for example,
spd = mdlist(:symmetric & :posdef)
10-element Array{ASCIIString,1}:
"hilb"
"cauchy"
"circul"
"invhilb"
"moler"
"pascal"
"pei"
"minij"
"tridiag"
"lehmer"
myprop = ["lehmer", "cauchy", "hilb"]
3-element Array{ASCIIString,1}:
"lehmer"
"cauchy"
"hilb"
Then use it in your tests like
for matrix in myprop
A = matrixdepot(matrix, 6)
L, U, p = lu(A) #LU factorization
err = norm(A[p,:] - L*U, 1) # 1-norm error
println("1-norm error for $matrix matrix is ", err)
end
1-norm error for lehmer matrix is 1.1102230246251565e-16
1-norm error for cauchy matrix is 5.551115123125783e-17
1-norm error for hilb matrix is 2.7755575615628914e-17
To add a group of matrices permanently for future use, we put the macro
@addgroup at the beginning.
@addgroup myfav = ["lehmer", "cauchy", "hilb"]
@addgroup test_for_paper2 = ["tridiag", "sampling", "wing"]
You can see the changes immediately:
mdinfo()
Currently loaded Matrices
–––––––––––––––––––––––––––
builtin(#)
––––––––––– ––––––––––– ––––––––––– –––––––––––– ––––––––––– ––––––––––––– ––––––––––––
1 baart 10 deriv2 19 gravity 28 kms 37 parter 46 rohess 55 ursell
2 binomial 11 dingdong 20 grcar 29 lehmer 38 pascal 47 rosser 56 vand
3 blur 12 erdrey 21 hadamard 30 lotkin 39 pei 48 sampling 57 wathen
4 cauchy 13 fiedler 22 hankel 31 magic 40 phillips 49 shaw 58 wilkinson
5 chebspec 14 forsythe 23 heat 32 minij 41 poisson 50 smallworld 59 wing
6 chow 15 foxgood 24 hilb 33 moler 42 prolate 51 spikes
7 circul 16 frank 25 invhilb 34 neumann 43 randcorr 52 toeplitz
8 clement 17 gilbert 26 invol 35 oscillate 44 rando 53 tridiag
9 companion 18 golub 27 kahan 36 parallax 45 randsvd 54 triw
user(#)
–––––––
Groups
––––––– ––––– ––––– ––––––– –––––– ––––––– ––––––––– –––––––––––––––
all local eigen illcond posdef regprob symmetric test_for_paper2
builtin user graph inverse random sparse myfav
Suite Sparse of
–––––––––––– ––––
2772 2833
MatrixMarket of
–––––––––––– –––
488 498
Notice new defined groups have been included. We can use them as
mdlist(:myfav)
3-element Array{ASCIIString,1}:
"lehmer"
"cauchy"
"hilb"
We can remove a group using the macro @rmgroup. As before, we
need to reload Julia to see the changes.
@rmgroup myfav
listgroups()
14-element Array{Symbol,1}:
:all
:builtin
:local
:user
:eigen
:graph
:illcond
:inverse
:posdef
:random
:regprob
:sparse
:symmetric
More Examples¶
An interesting test matrix is magic square. It can be generated as
M = matrixdepot("magic", Int, 5)
5x5 Array{Int64,2}:
17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9
sum(M, dims=1)
1x5 Array{Int64,2}:
65 65 65 65 65
sum(M, dims=2)
5x1 Array{Int64,2}:
65
65
65
65
65
sum(diag(M))
65
p = [5:-1:1]
sum(diag(M[:,p]))
65
Pascal Matrix can be generated as
P = matrixdepot("pascal", Int, 6)
6x6 Array{Int64,2}:
1 1 1 1 1 1
1 2 3 4 5 6
1 3 6 10 15 21
1 4 10 20 35 56
1 5 15 35 70 126
1 6 21 56 126 252
Notice the Cholesky factor of the Pascal matrix has Pascal’s triangle rows.
cholesky(P)
6x6 UpperTriangular{Float64,Array{Float64,2}}:
1.0 1.0 1.0 1.0 1.0 1.0
0.0 1.0 2.0 3.0 4.0 5.0
0.0 0.0 1.0 3.0 6.0 10.0
0.0 0.0 0.0 1.0 4.0 10.0
0.0 0.0 0.0 0.0 1.0 5.0
0.0 0.0 0.0 0.0 0.0 1.0