[gragusa]

cholesky(Symmetric(A, :L)) (and Hermitian(A, :L)) throws a PosDefException for a positive definite BandedMatrix whose upper bandwidth is positive. Only bandwidths (l, 0) work:

using BandedMatrices, LinearAlgebra, Random
Random.seed!(1)
B = brand(8, 8, 2, 2); B = B + B' + 10I    # symmetric positive definite, bandwidths (2, 2)
cholesky(Symmetric(B, :U))                  # works
cholesky(Symmetric(B, :L))                  # throws PosDefException

Cause

The LowerTriangular method of banded_chol! passes bandeddata(A) to pbtrf!('L', ...). LAPACK's lower band format expects the diagonal in row 1, but a BandedMatrix with upper bandwidth u stores it in row u + 1. So LAPACK reads the top superdiagonal as the diagonal. The 'L' branch of _ldiv! passes the same storage to pbtrs!.

Fix

Both calls now pass a view of the band storage from the diagonal row down (_lower_bandeddata). The leading dimension is unchanged.

Tests

A new test factors Symmetric/Hermitian(A, :L) with bandwidths (2, 1), (2, 2) and (2, 3) for Float64, ComplexF64 and BigFloat, and checks the factor and a solve against dense results. Without this fix, the test errors; with it, it passes, and the full test suite does too. (Note: The existing :L tests use bandwidths (1, 0) where the two layouts agree, and tests did not catch the bug.

(The test compares Matrix(Ac.L) instead of Ac.L because of an unrelated issue: for BigFloat, Ac.L ≈ ... throws an UndefRefError, since the copy cholesky makes leaves the upper band storage undefined.)

Fixes #141.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 90.84%. Comparing base (70c629c) to head (e7e2645).
⚠️ Report is 3 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #512      +/-   ##
==========================================
+ Coverage   90.69%   90.84%   +0.15%     
==========================================
  Files          27       27              
  Lines        3784     3814      +30     
==========================================
+ Hits         3432     3465      +33     
+ Misses        352      349       -3     

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