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README.md

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# InfiniteBandedMatrices.jl
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A Julia repository for linear algebra with infinite banded and block-banded matrices
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This currently implements the infinite-dimensional QL decomposition for perturbations of Toeplitz operators. Here is an example:
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```
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The infinite-dimensional QL decomposition is a subtly thing: its defined when the operator has non-positive Fredholm index, and if the Fredholm index is not zero, it may not be unique. For the Bull head matrix `A`, here are plots of `ql(A-λ*I).L[1,1]` alongside the image of the symbol `A`, which depicts the essential spectrum of `A` and where the Fredholm index changes. Note we have two plots as the regions with negative Fredholm index have multiple QL decompositions. Where the Fredholm index is positive, the QL decomposition doesn't exist and is depected in black.
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<img src=https://github.com/JuliaApproximation/InfiniteBandedMatrices.jl/raw/master/images/ql1.png width=500 height=400>
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<img src=https://github.com/JuliaApproximation/InfiniteBandedMatrices.jl/raw/master/images/ql2.png width=500 height=400>
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<img src=https://github.com/JuliaMatrices/InfiniteBandedMatrices.jl/raw/master/images/ql1.png width=500 height=400>
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<img src=https://github.com/JuliaMatrices/InfiniteBandedMatrices.jl/raw/master/images/ql2.png width=500 height=400>
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