Skip to main content

list manipulation - How to create overlapped block matrices and simplify the result


This is sort of a duplicate of Creating overlapped block matrices, but the question wasn't fully answered since the OP never returned to clarify.


I am using Kuba's answer for FEM, but I have been unable to get it to work properly. The problem is that the block matrix isn't being simplified even when the arrays we are working with are known (not arbitrary). Below is sample code. I have used examples for a, b and c, but in my actual code the matrices are larger and much more complicated.


a = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
b = {{3, 2, 7}, {2, 0, 3}, {4, 1, 2}};
c = {{5, 1, 4}, {4, 5, 8}, {9, 1, 6}};

n = {3, 3};

arrays = Array[#, n] & /@ {a, b, c}; (*arrays to work with*)

app[a1_, a2_, overlap_: 1] := With[{dim = Dimensions@a1},
                                   ArrayPad[a1, {0, n[[1]] - overlap}] + 
                                   ArrayPad[a2, Transpose@{dim - overlap, {0, 0}}]];

Fold[app, First@arrays, Rest@arrays] // MatrixForm

Result


Notice how Mathematica returns Matrix[index] rather than simplifying it. How can I adjust the code and simplify this, and replace e.g. {{1,2,3},{4,5,6},{7,8,9}}[1,1] with 1 since that is row 1, column 1 of the matrix?




Answer



Here is a somewhat general routine for building block diagonal matrices with overlaps:


blockOverlap[matList_?ArrayQ, r : (_Integer | {__Integer}) : 1] := 
     With[{spopt = SystemOptions["SparseArrayOptions"]},
          Internal`WithLocalSettings[SetSystemOptions["SparseArrayOptions" ->
                                     {"TreatRepeatedEntries" -> 1}],
          SparseArray[Sort[Flatten[MapThread[
                      ArrayRules[SparseArray[Band[{#1, #1}] -> {#2}]] &,
                      {Accumulate[Prepend[(Length /@ Most[matList]) - r, 1]], 
                       matList}]]]],

          SetSystemOptions[spopt]]]

OP's example:


blockOverlap[{a, b, c}] // MatrixForm

$$\begin{pmatrix} 1 & 2 & 3 & 0 & 0 & 0 & 0 \\ 4 & 5 & 6 & 0 & 0 & 0 & 0 \\ 7 & 8 & 12 & 2 & 7 & 0 & 0 \\ 0 & 0 & 2 & 0 & 3 & 0 & 0 \\ 0 & 0 & 4 & 1 & 7 & 1 & 4 \\ 0 & 0 & 0 & 0 & 4 & 5 & 8 \\ 0 & 0 & 0 & 0 & 9 & 1 & 6 \\ \end{pmatrix}$$


A slightly more general example:


blockOverlap[{a, b, c}, {1, 2}] // MatrixForm

$$\begin{pmatrix} 1 & 2 & 3 & 0 & 0 & 0 \\ 4 & 5 & 6 & 0 & 0 & 0 \\ 7 & 8 & 12 & 2 & 7 & 0 \\ 0 & 0 & 2 & 5 & 4 & 4 \\ 0 & 0 & 4 & 5 & 7 & 8 \\ 0 & 0 & 0 & 9 & 1 & 6 \\ \end{pmatrix}$$



Comments

Post a Comment

Popular posts from this blog

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...
Post a Comment
Read more

dynamic - How can I make a clickable ArrayPlot that returns input?

I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...
Post a Comment
Read more

list manipulation - Selecting multiple columns from a matrix?

Sample data: data = { {{2013, 1, 1}, 24.13, 167.67, 231.82}, {{2013, 1, 2}, 32.15, 170.92, 225.99}, {{2013, 1, 3}, 35.43, 172.68, 221.67}, {{2013, 1, 4}, 36.73, 173.05, 218.32}, {{2013, 1, 5}, 58.19, 165.96, 197.05}, {{2013, 1, 6}, 69.99, 163.50, 187.52}, {{2013, 1, 7}, 71.37, 154.21, 175.58}, {{2013, 1, 8}, 72.51, 149.66, 163.25}}; I want a DateListPlot with three graphs, so for a matrix formed by columns 1 and 2, one for columns 1 and 3, and 1 for columns 1 and 4. At the moment I'm using this code: data2 = Transpose[{data[[All, 1]], data[[All, 2]]}]; data3 = Transpose[{data[[All, 1]], data[[All, 3]]}]; data4 = Transpose[{data[[All, 1]], data[[All, 4]]}]; DateListPlot[{data2, data3, data4}, Joined -> True, Filling -> {3 -> {1}}] but I have a hunch that this can be done more efficiently. I don't like the Transpose s in particular. Any ideas? edit (for extra credit) What if I need to multiply the second column by 2, which in my solution is simp...
Post a Comment
Read more