inv returns wrong result for this SymTridiagonal([1., 0., 0.], [0., 1.])

[atbug]

julia> using LinearAlgebra
julia> A = SymTridiagonal([1., 0., 0.], [0., 1.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
 1.0  0.0   ⋅
 0.0  0.0  1.0
  ⋅   1.0  0.0

julia> inv(A)
3×3 Array{Float64,2}:
 NaN  NaN  NaN
 NaN  NaN  NaN
 NaN  NaN  NaN

julia> inv(convert(Array, A))
3×3 Array{Float64,2}:
 1.0  -0.0  -0.0
 0.0  -0.0   1.0
 0.0   1.0   0.0

julia> versioninfo()
Julia Version 1.1.0
Commit 80516ca202 (2019-01-21 21:24 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: Intel(R) Core(TM) i5-7200U CPU @ 2.50GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-6.0.1 (ORCJIT, skylake)
Environment:
  JULIA_EDITOR = "C:\Users\wangc\AppData\Local\atom\app-1.34.0\atom.exe" -a
  JULIA_NUM_THREADS = 2

[runapp]

This issue is caused by #30861 by further check.
In detail, fatorize failed to check if the array was pos-def and uses LDLt decomposition rather than LU decomp, resulting nan elements.

[brenhinkeller]

Looks like we now get an error, instead of silently wrong results (progress?)

julia> A = SymTridiagonal([1., 0., 0.], [0., 1.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 1.0  0.0   ⋅
 0.0  0.0  1.0
  ⋅   1.0  0.0

julia> inv(A)
ERROR: ZeroPivotException: factorization encountered one or more zero pivots. Consider switching to a pivoted LU factorization.
Stacktrace:
 [1] ldlt!(S::SymTridiagonal{Float64, Vector{Float64}})
   @ LinearAlgebra /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/LinearAlgebra/src/ldlt.jl:122
 [2] ldlt(M::SymTridiagonal{Float64, Vector{Float64}}; shift::Bool)
   @ LinearAlgebra /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/LinearAlgebra/src/ldlt.jl:169
 [3] ldlt
   @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/LinearAlgebra/src/ldlt.jl:163 [inlined]
 [4] factorize
   @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/LinearAlgebra/src/ldlt.jl:172 [inlined]
 [5] inv(A::SymTridiagonal{Float64, Vector{Float64}})
   @ LinearAlgebra /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/LinearAlgebra/src/generic.jl:1039
 [6] top-level scope
   @ REPL[16]:1

julia> inv(convert(Array, A))
3×3 Matrix{Float64}:
 1.0  -0.0  0.0
 0.0  -0.0  1.0
 0.0   1.0  0.0

julia> versioninfo()
Julia Version 1.8.3
Commit 0434deb161e (2022-11-14 20:14 UTC)
Platform Info:
  OS: macOS (arm64-apple-darwin21.3.0)
  CPU: 10 × Apple M1 Max
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-13.0.1 (ORCJIT, apple-m1)
  Threads: 1 on 8 virtual cores