Incorrect conversion from band matrix to sparse matrix

[andreasvarga]

The following illustrates the issue:

julia> sparse(BandedMatrix(1=>1:3))
4×4 SparseMatrixCSC{Int64, Int64} with 4 stored entries:
 0  1  ⋅  ⋅
 ⋅  ⋅  2  ⋅
 ⋅  ⋅  ⋅  3
 ⋅  ⋅  ⋅  ⋅

The expected result is

julia> sparse(Matrix(BandedMatrix(1=>1:3)))
4×4 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
 ⋅  1  ⋅  ⋅
 ⋅  ⋅  2  ⋅
 ⋅  ⋅  ⋅  3
 ⋅  ⋅  ⋅  ⋅

Similarly:

julia> a = BandedMatrix(1=>1:3,-1=>3:5)
4×4 BandedMatrix{Int64} with bandwidths (1, 1):
 0  1  ⋅  ⋅
 3  0  2  ⋅
 ⋅  4  0  3
 ⋅  ⋅  5  0

julia> sparse(a)
4×4 SparseMatrixCSC{Int64, Int64} with 10 stored entries:
 0  1  ⋅  ⋅
 3  0  2  ⋅
 ⋅  4  0  3
 ⋅  ⋅  5  0

julia> sparse(Matrix(a))
4×4 SparseMatrixCSC{Int64, Int64} with 6 stored entries:
 ⋅  1  ⋅  ⋅
 3  ⋅  2  ⋅
 ⋅  4  ⋅  3
 ⋅  ⋅  5  ⋅

[jecs]

It's even worse when bands span more of the matrix. For example,

julia> A=BandedMatrix(-5=>1:3,-3=>1:5,0=>1:8,3=>1:5,5=>1:3)
8×8 BandedMatrix{Int64} with bandwidths (5, 5):
 1  0  0  1  0  1  ⋅  ⋅
 0  2  0  0  2  0  2  ⋅
 0  0  3  0  0  3  0  3
 1  0  0  4  0  0  4  0
 0  2  0  0  5  0  0  5
 1  0  3  0  0  6  0  0
 ⋅  2  0  4  0  0  7  0
 ⋅  ⋅  3  0  5  0  0  8

julia> sparse(A)
8×8 SparseMatrixCSC{Int64, Int64} with 58 stored entries:
 1  0  0  1  0  1  ⋅  ⋅
 0  2  0  0  2  0  2  ⋅
 0  0  3  0  0  3  0  3
 1  0  0  4  0  0  4  0
 0  2  0  0  5  0  0  5
 1  0  3  0  0  6  0  0
 ⋅  2  0  4  0  0  7  0
 ⋅  ⋅  3  0  5  0  0  8

julia> sparse(Matrix(A))
8×8 SparseMatrixCSC{Int64, Int64} with 24 stored entries:
 1  ⋅  ⋅  1  ⋅  1  ⋅  ⋅
 ⋅  2  ⋅  ⋅  2  ⋅  2  ⋅
 ⋅  ⋅  3  ⋅  ⋅  3  ⋅  3
 1  ⋅  ⋅  4  ⋅  ⋅  4  ⋅
 ⋅  2  ⋅  ⋅  5  ⋅  ⋅  5
 1  ⋅  3  ⋅  ⋅  6  ⋅  ⋅
 ⋅  2  ⋅  4  ⋅  ⋅  7  ⋅
 ⋅  ⋅  3  ⋅  5  ⋅  ⋅  8

[dlfivefifty]

A PR to correct this would be very much appreciated!