I created a sparse matrix for a given size n
mat = SparseArray[{Band[{1, 2}] -> U10, Band[{2, 1}] -> V10}, {n*n, n*n}]
For[j = 0, j < n - 1, j++, mat += SparseArray[{Band[{1, 1 + (n*(j + 1))}]->U01, Band[{1 + (n*(j + 1)), 1}] -> V01}, {n*n, n*n}]]
now I would like to replace the entries U10,V10,U01 and V01 by some 4x4 matrices such that the original structure is preserved
Thanks for any help
a SparseArray can not hold arrays as elements. You need to convert to a normal array, eg.
mat=Normal#mat /. {U10->{{1, 2},{3,4}},U01->{{1, 0},{0, 1}},... }
If you think to convert that back to a sparse array:
SparseArray[mat]
you get the error:
SparseArray::rect: Rectangular array or list of rules expected at position 1
because your array is not rectangular.
Related
The following was my question given by my teacher,
Generate a sequence of N = 1000 independent observations of random variable with distribution: (c) Exponential with parameter λ = 1 , by
inversion method.
Present graphically obtained sequences(except for those generated in point e) i.e. e.g. (a) i. plot in the coordinates (no. obs.,
value of the obs) ii. plot in the coordinates (obs no n, obs. no n +
i) for i = 1, 2, 3. iii. plot so called covariance function for some
values. i.e. and averages:
I have written the following code,
(*****************************************************************)
(*Task 01(c) and 02(a)*)
(*****************************************************************)
n = 1000;
taskC = Table[-Log[RandomReal[]], {n}];
ListPlot[taskC, AxesLabel->{"No. obs", "value of the obs"}]
i = 1;
ListPlot[Table[
{taskC[[k]], taskC[[k+i]]},
{k, 1, n-i,1}],
AxesLabel->{"obs.no.n", "obs.no.n+1"}]
i++;
ListPlot[Table[
{taskC[[k]], taskC[[k+i]]},
{k, 1, n-i,1}],
AxesLabel-> {"obs.no.n", "obs.no.n+2"}]
i++;
ListPlot[Table[
{taskC[[k]], taskC[[k+i]]},
{k,1,n-i,1}],
AxesLabel->{"obs.no.n", "obs.no.n+3"}]
avg = (1/n)*Sum[taskC[[i]], {i,n}];
ListPlot[Table[1/(n-tau) * Sum[(taskC[[i]]-avg)*(taskC[[i+tau]] - avg), n], {tau, 1,100}],
Joined->True,
AxesLabel->"Covariance Function"]
He has commented,
The plots of co-variance functions should start from 0-shift. Note
that for larger than 0 shifts you are estimating co-variance between
independent observations which is zero, while for 0 shift you are
estimating variance of observation which is large. Thus the contrast
between these two cases is a clear indication that the observations
are uncorrelated.
What did I do wrong?
How can I correct my code?
Zero-shift means calculating the covariance for tau = 0, which is simply the variance.
Labeled[ListPlot[Table[{tau,
1/(n - tau)*Sum[(taskC[[i]] - avg)*(taskC[[i + tau]] - avg), {i, n - tau}]},
{tau, 0, 5}], Filling -> Axis, FillingStyle -> Thick, PlotRange -> All,
Frame -> True, PlotRangePadding -> 0.2, AspectRatio -> 1],
{"Covariance Function K(n)", "n"}, {{Top, Left}, Bottom}]
Variance[taskC]
0.93484
Covariance[taskC, taskC]
0.93484
(* n = 1 *)
Covariance[Most[taskC], Rest[taskC]]
0.00926913
How can I select a subset of columns or rows from a matrix in SPSS?
Given the following example, I want to compute a matrix X2 containing the first two columns of X.
MATRIX.
COMPUTE
X = {1, 2, 2;
0, -1, 1;
1, 1, -2}.
* Compute new matrix X2 that contains the first two columns of X
MAGIC CODE ;)
END MATRIX.
What is the syntax for matrix subsetting operations in SPSS?
You can subset a matrix, so it would be simply COMPUTE XSub = X(:,1:2). Full example below.
MATRIX.
COMPUTE X = {1, 2, 2;
0, -1, 1;
1, 1, -2}.
COMPUTE XSub = X(:,1:2).
PRINT XSub.
END MATRIX.
To the add-on question in the comments, 1:n basically SPSS understands as a row vector of 1 2 3 .... n. You can create your own vector to subset the matrix though, such as {1,3} or {2,2} or {3,1} or whatever. The last example will return the 3rd column first and the first column second in the subsetted matrix. Example below:
MATRIX.
COMPUTE X = {1, 2, 2;
0, -1, 1;
1, 1, -2}.
COMPUTE XSub = X(:,{3,1}).
PRINT XSub.
END MATRIX.
Which prints out
Run MATRIX procedure:
XSUB
2 1
1 0
-2 1
------ END MATRIX -----
MATRIX.
COMPUTE X = {1, 2, 3; 4, 5, 6; 7, 8, 9}.
COMPUTE Y=MAKE(NROW(X),2,0).
LOOP i=1 to NROW(Y).
LOOP j=1 to NCOL(Y).
COMPUTE Y(i,j)=X(i,j).
END LOOP.
END LOOP.
PRINT X.
PRINT Y.
END MATRIX.
The following code gives me the first k eigenvalues of a certain big matrix. Because of the symmetries of the matrix, the eigenvalues are in pairs, one positive and the other negative, with the same absolute value. This is indeed the case if I run the code with the exact matrices, without using the sparse version. However when I make them sparse, the resulting eigenvalues appear to lose the sign information, as now the pairs can be both negative, or both positive, depending on the number I put on "nspins" (which controls the size of the matrix). The variable "sparse" controls whether I use sparse matrices or not.
This issue gives me considerable trouble. Can anybody tell me why the sparse version of the computation gives wrong signs, and how to fix it?
sparse = 1; (*Parameter that controls whether I will use sparse \
matrices, 0 means not sparse, 1 means sparse*)
(*Base matrices of my big matrix*)
ox = N[{{0, 1}, {1, 0}}];
oz = N[{{1, 0}, {0, -1}}];
id = N[{{1, 0}, {0, 1}}];
(*Transformation into sparse whether desired*)
If[sparse == 1,
ox = SparseArray[ox];
oz = SparseArray[oz];
id = SparseArray[id];
]
(*Dimension of the big matrix, must be even*)
nspins = 8;
(*Number of eigenvalues computed*)
neigenv = 4;
(*Algorithm to create big matrices*)
Do[
Do[
If[j == i, mata = ox; matc = oz;, mata = id; matc = id;];
If[j == 1,
o[1, i] = mata;
o[3, i] = matc;
,
o[1, i] = KroneckerProduct[o[1, i], mata];
o[3, i] = KroneckerProduct[o[3, i], matc];
];
, {j, 1, nspins}];
, {i, 1, nspins}];
(*Sum of big matrices*)
ham = Sum[o[1, i].o[1, i + 1], {i, 1, nspins - 1}] +
o[1, nspins].o[1, 1] + 0.5*Sum[o[3, i], {i, 1, nspins}];
(*Print the desired eigenvalues*)
Do[Print [Eigenvalues[ham, k][[k]]], {k, 1, neigenv}];
I'm working on a script in mathematica that will take simulate a string held at either end and plucked, by solving the wave equation via numerical methods. (http://en.wikipedia.org/wiki/Wave_equation#Investigation_by_numerical_methods)
n = 5; (*The number of discreet elements to be used*)
L = 1.0; (*The length of the string that is vibrating*)
a = 1.0/3.0; (*The distance from the left side that the string is \
plucked at*)
T = 1; (*The tension in the string*)
[Rho] = 1; (*The length density of the string*)
y0 = 0.1; (*The vertical distance of the string pluck*)
[CapitalDelta]x = L/n; (*The length of each discreet element*)
m = ([Rho]*L)/n;(*The mass of each individual node*)
c = Sqrt[T/[Rho]];(*The speed at which waves in the string propogate*)
I set all my variables
Y[t] = Array[f[t], {n - 1, 1}];
MatrixForm(*Creates a vector size n-1 by 1 of functions \
representing each node*)
I define my Vector of nodal position functions
K = MatrixForm[
SparseArray[{Band[{1, 1}] -> -2, Band[{2, 1}] -> 1,
Band[{1, 2}] -> 1}, {n - 1,
n - 1}]](*Creates a matrix size n by n governing the coupling \
between each node*)
I create the stiffness matrix relating all the nodal functions to one another
Y0 = MatrixForm[
Table[Piecewise[{{(((i*L)/n)*y0)/a,
0 < ((i*L)/n) < a}, {(-((i*L)/n)*y0)/(L - a) + (y0*L)/(L - a),
a < ((i*L)/n) < L}}], {i, 1, n - 1}]]
I define the initial positions of each node using a piecewise function
NDSolve[{Y''[t] == (c/[CapitalDelta]x)^2 Y[t].K, Y[0] == Y0,
Y'[0] == 0},
Y, {t, 0, 10}];(*Numerically solves the system of second order DE's*)
Finally, This should solve for the values of the individual nodes, but it returns an error:
"NDSolve::ndinnt : Initial condition [Y0 table] is not a number or a rectangular array"
So , it would seem that I don't have a firm grasp on how matrices work in mathematica. I would greatly appreciate it if anyone could help me get this last line of code to run properly.
Thank you,
Brad
I don't think you should use MatrixForm when defining the matrices. MatrixForm is used to format a list of list as a matrix, usually when you display it. Try removing it and see if it works.
Does anyone know why the following random distributions of matrices generate different plots? (This is code to generate a plot of the PDFs for first cells from a set of 10x10 matrices sampled using an inverse Wishart distribution; amazingly, the plots are different depending on the way one performs the matrix inverse - and it seems the right plots are obtained by Inverse[_], why?)
base code:
<< MultivariateStatistics`;
Module[{dist, p, k, data, samples, scale, graphics, distribution},
p = 10;
k = 13;
samples = 500;
dist = WishartDistribution[IdentityMatrix[p], k];
(* a samples x p x p array *)
data = Inverse[#] & /# RandomVariate[dist, samples];
(* distribution graphics *)
distribution[i_, j_] := Module[{fiber, f, mean, rangeAll, colorHue},
fiber = data[[All, i, j]];
dist = SmoothKernelDistribution[fiber];
f = PDF[dist];
Plot[f[z], {z, -2, 2},
PlotLabel -> ("Mean=" <> ToString[Mean[fiber]]),
PlotRange -> All]
];
Grid # Table[distribution[i, j], {i, 1, 3}, {j, 1, 5}]
]
code variant: above, change line
data = Inverse[#] & /# RandomVariate[dist, samples];
by this
data = #^(-1) & /# RandomVariate[dist, samples];
and you will see the plotted distributions are different.
Inverse computes a matrix inverse, i.e. if a is a square matrix, then Inverse[a].a will be the identity matrix.
a^(-1) is the same as 1/a, i.e. it gives you the reciprocal of each matrix element. The ^ operator gives powers element-wise. If you want a matrix power, use MatrixPower.