Specialize isbanded for StridedMatrix (#56487)

This improves performance, as the loops in `istriu` and `istril` may be
fused to improve cache-locality.
This also changes the quick-return behavior, and only returns after the
check over all the upper or lower bands for a column is complete.

```julia
julia> using LinearAlgebra

julia> A = zeros(2, 10_000);

julia> @Btime isdiag($A);
  32.682 μs (0 allocations: 0 bytes) # nightly v"1.12.0-DEV.1593"
  9.481 μs (0 allocations: 0 bytes) # this PR

julia> A = zeros(10_000, 2);

julia> @Btime isdiag($A);
  10.288 μs (0 allocations: 0 bytes)  # nightly 
  2.579 μs (0 allocations: 0 bytes) # this PR

julia> A = zeros(100, 100);

julia> @Btime isdiag($A);
  6.616 μs (0 allocations: 0 bytes) # nightly
  3.075 μs (0 allocations: 0 bytes) # this PR

julia> A = diagm(0=>1:100);  A[3,4] = 1;

julia> @Btime isdiag($A);
  2.759 μs (0 allocations: 0 bytes) # nightly
  85.371 ns (0 allocations: 0 bytes) # this PR
```

A similar change is added to `istriu`/`istril` as well, so that
```julia
julia> A = zeros(2, 10_000);

julia> @Btime istriu($A); # trivial
  7.358 ns (0 allocations: 0 bytes) # nightly
  13.779 ns (0 allocations: 0 bytes) # this PR

julia> @Btime istril($A);
  33.464 μs (0 allocations: 0 bytes) # nightly
  9.476 μs (0 allocations: 0 bytes) # this PR

julia> A = zeros(10_000, 2);

julia> @Btime istriu($A);
  10.020 μs (0 allocations: 0 bytes) # nightly
  2.620 μs (0 allocations: 0 bytes) # this PR

julia> @Btime istril($A); # trivial
  6.793 ns (0 allocations: 0 bytes) # nightly
  14.473 ns (0 allocations: 0 bytes) # this PR

julia> A = zeros(100, 100);

julia> @Btime istriu($A);
  3.435 μs (0 allocations: 0 bytes) # nightly
  1.637 μs (0 allocations: 0 bytes) # this PR

julia> @Btime istril($A);
  3.353 μs (0 allocations: 0 bytes) # nightly
  1.661 μs (0 allocations: 0 bytes) # this PR
```

---------

Co-authored-by: Daniel Karrasch <[email protected]>

stdlib/LinearAlgebra/src/generic.jl

@@ -1353,6 +1353,14 @@ end
 
 ishermitian(x::Number) = (x == conj(x))
 
+# helper function equivalent to `iszero(v)`, but potentially without the fast exit feature
+# of `all` if this improves performance
+_iszero(V) = iszero(V)
+# A Base.FastContiguousSubArray view of a StridedArray
+FastContiguousSubArrayStrided{T,N,P<:StridedArray,I<:Tuple{AbstractUnitRange, Vararg{Any}}} = Base.SubArray{T,N,P,I,true}
+# using mapreduce instead of all permits vectorization
+_iszero(V::FastContiguousSubArrayStrided) = mapreduce(iszero, &, V, init=true)
+
 """
     istriu(A::AbstractMatrix, k::Integer = 0) -> Bool
 
@@ -1384,20 +1392,9 @@ julia> istriu(c, -1)
 true
 ```
 """
-function istriu(A::AbstractMatrix, k::Integer = 0)
-    require_one_based_indexing(A)
-    return _istriu(A, k)
-end
+istriu(A::AbstractMatrix, k::Integer = 0) = _isbanded_impl(A, k, size(A,2)-1)
 istriu(x::Number) = true
 
-@inline function _istriu(A::AbstractMatrix, k)
-    m, n = size(A)
-    for j in 1:min(n, m + k - 1)
-        all(iszero, view(A, max(1, j - k + 1):m, j)) || return false
-    end
-    return true
-end
-
 """
     istril(A::AbstractMatrix, k::Integer = 0) -> Bool
 
@@ -1429,20 +1426,9 @@ julia> istril(c, 1)
 true
 ```
 """
-function istril(A::AbstractMatrix, k::Integer = 0)
-    require_one_based_indexing(A)
-    return _istril(A, k)
-end
+istril(A::AbstractMatrix, k::Integer = 0) = _isbanded_impl(A, -size(A,1)+1, k)
 istril(x::Number) = true
 
-@inline function _istril(A::AbstractMatrix, k)
-    m, n = size(A)
-    for j in max(1, k + 2):n
-        all(iszero, view(A, 1:min(j - k - 1, m), j)) || return false
-    end
-    return true
-end
-
 """
     isbanded(A::AbstractMatrix, kl::Integer, ku::Integer) -> Bool
 
@@ -1474,7 +1460,66 @@ julia> LinearAlgebra.isbanded(b, -1, 0)
 true
 ```
 """
-isbanded(A::AbstractMatrix, kl::Integer, ku::Integer) = istriu(A, kl) && istril(A, ku)
+isbanded(A::AbstractMatrix, kl::Integer, ku::Integer) = _isbanded(A, kl, ku)
+_isbanded(A::AbstractMatrix, kl::Integer, ku::Integer) = istriu(A, kl) && istril(A, ku)
+# Performance optimization for StridedMatrix by better utilizing cache locality
+# The istriu and istril loops are merged
+# the additional indirection allows us to reuse the isbanded loop within istriu/istril
+# without encountering cycles
+_isbanded(A::StridedMatrix, kl::Integer, ku::Integer) = _isbanded_impl(A, kl, ku)
+function _isbanded_impl(A, kl, ku)
+    Base.require_one_based_indexing(A)
+
+    #=
+    We split the column range into four possible groups, depending on the values of kl and ku.
+
+    The first is the bottom left triangle, where bands below kl must be zero,
+    but there are no bands above ku in that column.
+
+    The second is where there are both bands below kl and above ku in the column.
+    These are the middle columns typically.
+
+    The third is the top right, where there are bands above ku but no bands below kl
+    in the column.
+
+    The fourth is mainly relevant for wide matrices, where there is a block to the right
+    beyond ku, where the elements should all be zero. The reason we separate this from the
+    third group is that we may loop over all the rows using A[:, col] instead of A[rowrange, col],
+    which is usually faster.
+    =#
+
+    last_col_nonzeroblocks = size(A,1) + ku # fully zero rectangular block beyond this column
+    last_col_emptytoprows = ku + 1 # empty top rows before this column
+    last_col_nonemptybottomrows = size(A,1) + kl - 1 # empty bottom rows after this column
+
+    colrange_onlybottomrows = firstindex(A,2):min(last_col_nonemptybottomrows, last_col_emptytoprows)
+    colrange_topbottomrows = max(last_col_emptytoprows, last(colrange_onlybottomrows))+1:last_col_nonzeroblocks
+    colrange_onlytoprows_nonzero = last(colrange_topbottomrows)+1:last_col_nonzeroblocks
+    colrange_zero_block = last_col_nonzeroblocks+1:lastindex(A,2)
+
+    for col in intersect(axes(A,2), colrange_onlybottomrows) # only loop over the bottom rows
+        botrowinds = max(firstindex(A,1), col-kl+1):lastindex(A,1)
+        bottomrows = @view A[botrowinds, col]
+        _iszero(bottomrows) || return false
+    end
+    for col in intersect(axes(A,2), colrange_topbottomrows)
+        toprowinds = firstindex(A,1):min(col-ku-1, lastindex(A,1))
+        toprows = @view A[toprowinds, col]
+        _iszero(toprows) || return false
+        botrowinds = max(firstindex(A,1), col-kl+1):lastindex(A,1)
+        bottomrows = @view A[botrowinds, col]
+        _iszero(bottomrows) || return false
+    end
+    for col in intersect(axes(A,2), colrange_onlytoprows_nonzero)
+        toprowinds = firstindex(A,1):min(col-ku-1, lastindex(A,1))
+        toprows = @view A[toprowinds, col]
+        _iszero(toprows) || return false
+    end
+    for col in intersect(axes(A,2), colrange_zero_block)
+        _iszero(@view A[:, col]) || return false
+    end
+    return true
+end
 
 """
     isdiag(A) -> Bool

stdlib/LinearAlgebra/src/hessenberg.jl

@@ -77,6 +77,16 @@ Base.@constprop :aggressive function istriu(A::UpperHessenberg, k::Integer=0)
     k <= -1 && return true
     return _istriu(A, k)
 end
+# additional indirection to dispatch to optimized method for banded parents (defined in special.jl)
+@inline function _istriu(A::UpperHessenberg, k)
+    P = parent(A)
+    m = size(A, 1)
+    for j in firstindex(P,2):min(m + k - 1, lastindex(P,2))
+        Prows = @view P[max(begin, j - k + 1):min(j+1,end), j]
+        _iszero(Prows) || return false
+    end
+    return true
+end
 
 function Matrix{T}(H::UpperHessenberg) where T
     m,n = size(H)

stdlib/LinearAlgebra/src/special.jl

@@ -592,3 +592,4 @@ end
 # istriu/istril for triangular wrappers of structured matrices
 _istril(A::LowerTriangular{<:Any, <:BandedMatrix}, k) = istril(parent(A), k)
 _istriu(A::UpperTriangular{<:Any, <:BandedMatrix}, k) = istriu(parent(A), k)
+_istriu(A::UpperHessenberg{<:Any, <:BandedMatrix}, k) = istriu(parent(A), k)

stdlib/LinearAlgebra/src/triangular.jl

@@ -348,25 +348,29 @@ Base.@constprop :aggressive function istril(A::LowerTriangular, k::Integer=0)
     k >= 0 && return true
     return _istril(A, k)
 end
+# additional indirection to dispatch to optimized method for banded parents (defined in special.jl)
 @inline function _istril(A::LowerTriangular, k)
     P = parent(A)
     for j in max(firstindex(P,2), k + 2):lastindex(P,2)
-        all(iszero, @view(P[j:min(j - k - 1, end), j])) || return false
+        _iszero(@view P[max(j, begin):min(j - k - 1, end), j]) || return false
     end
     return true
 end
+
 Base.@constprop :aggressive function istriu(A::UpperTriangular, k::Integer=0)
     k <= 0 && return true
     return _istriu(A, k)
 end
+# additional indirection to dispatch to optimized method for banded parents (defined in special.jl)
 @inline function _istriu(A::UpperTriangular, k)
     P = parent(A)
     m = size(A, 1)
     for j in firstindex(P,2):min(m + k - 1, lastindex(P,2))
-        all(iszero, @view(P[max(begin, j - k + 1):j, j])) || return false
+        _iszero(@view P[max(begin, j - k + 1):min(j, end), j]) || return false
     end
     return true
 end
+
 istril(A::Adjoint, k::Integer=0) = istriu(A.parent, -k)
 istril(A::Transpose, k::Integer=0) = istriu(A.parent, -k)
 istriu(A::Adjoint, k::Integer=0) = istril(A.parent, -k)

stdlib/LinearAlgebra/test/generic.jl

@@ -3,6 +3,8 @@
 module TestGeneric
 
 using Test, LinearAlgebra, Random
+using Test: GenericArray
+using LinearAlgebra: isbanded
 
 const BASE_TEST_PATH = joinpath(Sys.BINDIR, "..", "share", "julia", "test")
 
@@ -511,56 +513,110 @@ end
 end
 
 @testset "generic functions for checking whether matrices have banded structure" begin
-    using LinearAlgebra: isbanded
     pentadiag = [1 2 3; 4 5 6; 7 8 9]
     tridiag   = [1 2 0; 4 5 6; 0 8 9]
+    tridiagG  = GenericArray([1 2 0; 4 5 6; 0 8 9])
+    Tridiag   = Tridiagonal(tridiag)
     ubidiag   = [1 2 0; 0 5 6; 0 0 9]
+    ubidiagG  = GenericArray([1 2 0; 0 5 6; 0 0 9])
+    uBidiag   = Bidiagonal(ubidiag, :U)
     lbidiag   = [1 0 0; 4 5 0; 0 8 9]
+    lbidiagG  = GenericArray([1 0 0; 4 5 0; 0 8 9])
+    lBidiag   = Bidiagonal(lbidiag, :L)
     adiag     = [1 0 0; 0 5 0; 0 0 9]
+    adiagG    = GenericArray([1 0 0; 0 5 0; 0 0 9])
+    aDiag     = Diagonal(adiag)
     @testset "istriu" begin
         @test !istriu(pentadiag)
         @test istriu(pentadiag, -2)
         @test !istriu(tridiag)
+        @test istriu(tridiag) == istriu(tridiagG) == istriu(Tridiag)
         @test istriu(tridiag, -1)
+        @test istriu(tridiag, -1) == istriu(tridiagG, -1) == istriu(Tridiag, -1)
         @test istriu(ubidiag)
+        @test istriu(ubidiag) == istriu(ubidiagG) == istriu(uBidiag)
         @test !istriu(ubidiag, 1)
+        @test istriu(ubidiag, 1) == istriu(ubidiagG, 1) == istriu(uBidiag, 1)
         @test !istriu(lbidiag)
+        @test istriu(lbidiag) == istriu(lbidiagG) == istriu(lBidiag)
         @test istriu(lbidiag, -1)
+        @test istriu(lbidiag, -1) == istriu(lbidiagG, -1) == istriu(lBidiag, -1)
         @test istriu(adiag)
+        @test istriu(adiag) == istriu(adiagG) == istriu(aDiag)
     end
     @testset "istril" begin
         @test !istril(pentadiag)
         @test istril(pentadiag, 2)
         @test !istril(tridiag)
+        @test istril(tridiag) == istril(tridiagG) == istril(Tridiag)
         @test istril(tridiag, 1)
+        @test istril(tridiag, 1) == istril(tridiagG, 1) == istril(Tridiag, 1)
         @test !istril(ubidiag)
+        @test istril(ubidiag) == istril(ubidiagG) == istril(ubidiagG)
         @test istril(ubidiag, 1)
+        @test istril(ubidiag, 1) == istril(ubidiagG, 1) == istril(uBidiag, 1)
         @test istril(lbidiag)
+        @test istril(lbidiag) == istril(lbidiagG) == istril(lBidiag)
         @test !istril(lbidiag, -1)
+        @test istril(lbidiag, -1) == istril(lbidiagG, -1) == istril(lBidiag, -1)
         @test istril(adiag)
+        @test istril(adiag) == istril(adiagG) == istril(aDiag)
     end
     @testset "isbanded" begin
         @test isbanded(pentadiag, -2, 2)
         @test !isbanded(pentadiag, -1, 2)
         @test !isbanded(pentadiag, -2, 1)
         @test isbanded(tridiag, -1, 1)
+        @test isbanded(tridiag, -1, 1) == isbanded(tridiagG, -1, 1) == isbanded(Tridiag, -1, 1)
         @test !isbanded(tridiag, 0, 1)
+        @test isbanded(tridiag, 0, 1) == isbanded(tridiagG, 0, 1) == isbanded(Tridiag, 0, 1)
         @test !isbanded(tridiag, -1, 0)
+        @test isbanded(tridiag, -1, 0) == isbanded(tridiagG, -1, 0) == isbanded(Tridiag, -1, 0)
         @test isbanded(ubidiag, 0, 1)
+        @test isbanded(ubidiag, 0, 1) == isbanded(ubidiagG, 0, 1) == isbanded(uBidiag, 0, 1)
         @test !isbanded(ubidiag, 1, 1)
+        @test isbanded(ubidiag, 1, 1) == isbanded(ubidiagG, 1, 1) == isbanded(uBidiag, 1, 1)
         @test !isbanded(ubidiag, 0, 0)
+        @test isbanded(ubidiag, 0, 0) == isbanded(ubidiagG, 0, 0) == isbanded(uBidiag, 0, 0)
         @test isbanded(lbidiag, -1, 0)
+        @test isbanded(lbidiag, -1, 0) == isbanded(lbidiagG, -1, 0) == isbanded(lBidiag, -1, 0)
         @test !isbanded(lbidiag, 0, 0)
+        @test isbanded(lbidiag, 0, 0) == isbanded(lbidiagG, 0, 0) == isbanded(lBidiag, 0, 0)
         @test !isbanded(lbidiag, -1, -1)
+        @test isbanded(lbidiag, -1, -1) == isbanded(lbidiagG, -1, -1) == isbanded(lBidiag, -1, -1)
         @test isbanded(adiag, 0, 0)
+        @test isbanded(adiag, 0, 0) == isbanded(adiagG, 0, 0) == isbanded(aDiag, 0, 0)
         @test !isbanded(adiag, -1, -1)
+        @test isbanded(adiag, -1, -1) == isbanded(adiagG, -1, -1) == isbanded(aDiag, -1, -1)
         @test !isbanded(adiag, 1, 1)
+        @test isbanded(adiag, 1, 1) == isbanded(adiagG, 1, 1) == isbanded(aDiag, 1, 1)
     end
     @testset "isdiag" begin
         @test !isdiag(tridiag)
+        @test isdiag(tridiag) == isdiag(tridiagG) == isdiag(Tridiag)
         @test !isdiag(ubidiag)
+        @test isdiag(ubidiag) == isdiag(ubidiagG) == isdiag(uBidiag)
         @test !isdiag(lbidiag)
+        @test isdiag(lbidiag) == isdiag(lbidiagG) == isdiag(lBidiag)
         @test isdiag(adiag)
+        @test isdiag(adiag) ==isdiag(adiagG) == isdiag(aDiag)
+    end
+end
+
+@testset "isbanded/istril/istriu with rectangular matrices" begin
+    @testset "$(size(A))" for A in [zeros(0,4), zeros(2,5), zeros(5,2), zeros(4,0)]
+        @testset for m in -(size(A,1)-1):(size(A,2)-1)
+            A .= 0
+            A[diagind(A, m)] .= 1
+            G = GenericArray(A)
+            @testset for (kl,ku) in Iterators.product(-6:6, -6:6)
+                @test isbanded(A, kl, ku) == isbanded(G, kl, ku) == isempty(A) || (m in (kl:ku))
+            end
+            @testset for k in -6:6
+                @test istriu(A,k) == istriu(G,k) == isempty(A) || (k <= m)
+                @test istril(A,k) == istril(G,k) == isempty(A) || (k >= m)
+            end
+        end
     end
 end
 

stdlib/LinearAlgebra/test/hessenberg.jl

@@ -279,4 +279,30 @@ end
     @test H.H == D
 end
 
+@testset "istriu/istril forwards to parent" begin
+    n = 10
+    @testset "$(nameof(typeof(M)))" for M in [Tridiagonal(rand(n-1), rand(n), rand(n-1)),
+                Tridiagonal(zeros(n-1), zeros(n), zeros(n-1)),
+                Diagonal(randn(n)),
+                Diagonal(zeros(n)),
+                ]
+        U = UpperHessenberg(M)
+        A = Array(U)
+        for k in -n:n
+            @test istriu(U, k) == istriu(A, k)
+            @test istril(U, k) == istril(A, k)
+        end
+    end
+    z = zeros(n,n)
+    P = Matrix{BigFloat}(undef, n, n)
+    copytrito!(P, z, 'U')
+    P[diagind(P,-1)] .= 0
+    U = UpperHessenberg(P)
+    A = Array(U)
+    @testset for k in -n:n
+        @test istriu(U, k) == istriu(A, k)
+        @test istril(U, k) == istril(A, k)
+    end
+end
+
 end # module TestHessenberg