diff --git a/src/apiutils.jl b/src/apiutils.jl
index 5a2a1df9..39608d6c 100644
--- a/src/apiutils.jl
+++ b/src/apiutils.jl
@@ -40,32 +40,68 @@ end
     return Expr(:tuple, [:(single_seed(Partials{N,V}, Val{$i}())) for i in 1:N]...)
 end
 
+# Only seed indices that are structurally non-zero
+structural_eachindex(x::AbstractArray) = structural_eachindex(x, x)
+function structural_eachindex(x::AbstractArray, y::AbstractArray)
+    require_one_based_indexing(x, y)
+    eachindex(x, y)
+end
+function structural_eachindex(x::UpperTriangular, y::AbstractArray)
+    require_one_based_indexing(x, y)
+    if size(x) != size(y)
+        throw(DimensionMismatch())
+    end
+    n = size(x, 1)
+    return (CartesianIndex(i, j) for j in 1:n for i in 1:j)
+end
+function structural_eachindex(x::LowerTriangular, y::AbstractArray)
+    require_one_based_indexing(x, y)
+    if size(x) != size(y)
+        throw(DimensionMismatch())
+    end
+    n = size(x, 1)
+    return (CartesianIndex(i, j) for j in 1:n for i in j:n)
+end
+function structural_eachindex(x::Diagonal, y::AbstractArray)
+    require_one_based_indexing(x, y)
+    if size(x) != size(y)
+        throw(DimensionMismatch())
+    end
+    return diagind(x)
+end
+
 function seed!(duals::AbstractArray{Dual{T,V,N}}, x,
                seed::Partials{N,V} = zero(Partials{N,V})) where {T,V,N}
-    duals .= Dual{T,V,N}.(x, Ref(seed))
+    for idx in structural_eachindex(duals, x)
+        duals[idx] = Dual{T,V,N}(x[idx], seed)
+    end
     return duals
 end
 
 function seed!(duals::AbstractArray{Dual{T,V,N}}, x,
                seeds::NTuple{N,Partials{N,V}}) where {T,V,N}
-    dual_inds = 1:N
-    duals[dual_inds] .= Dual{T,V,N}.(view(x,dual_inds), seeds)
+    for (i, idx) in zip(1:N, structural_eachindex(duals, x))
+        duals[idx] = Dual{T,V,N}(x[idx], seeds[i])
+    end
     return duals
 end
 
 function seed!(duals::AbstractArray{Dual{T,V,N}}, x, index,
                seed::Partials{N,V} = zero(Partials{N,V})) where {T,V,N}
     offset = index - 1
-    dual_inds = (1:N) .+ offset
-    duals[dual_inds] .= Dual{T,V,N}.(view(x, dual_inds), Ref(seed))
+    idxs = Iterators.drop(structural_eachindex(duals, x), offset)
+    for idx in idxs
+        duals[idx] = Dual{T,V,N}(x[idx], seed)
+    end
     return duals
 end
 
 function seed!(duals::AbstractArray{Dual{T,V,N}}, x, index,
                seeds::NTuple{N,Partials{N,V}}, chunksize = N) where {T,V,N}
     offset = index - 1
-    seed_inds = 1:chunksize
-    dual_inds = seed_inds .+ offset
-    duals[dual_inds] .= Dual{T,V,N}.(view(x, dual_inds), getindex.(Ref(seeds), seed_inds))
+    idxs = Iterators.drop(structural_eachindex(duals, x), offset)
+    for (i, idx) in zip(1:chunksize, idxs)
+        duals[idx] = Dual{T,V,N}(x[idx], seeds[i])
+    end
     return duals
 end
diff --git a/src/gradient.jl b/src/gradient.jl
index 6d4cc791..98bdb091 100644
--- a/src/gradient.jl
+++ b/src/gradient.jl
@@ -16,7 +16,7 @@ Set `check` to `Val{false}()` to disable tag checking. This can lead to perturba
 function gradient(f::F, x::AbstractArray, cfg::GradientConfig{T} = GradientConfig(f, x), ::Val{CHK}=Val{true}()) where {F, T, CHK}
     require_one_based_indexing(x)
     CHK && checktag(T, f, x)
-    if chunksize(cfg) == length(x)
+    if chunksize(cfg) == structural_length(x)
         return vector_mode_gradient(f, x, cfg)
     else
         return chunk_mode_gradient(f, x, cfg)
@@ -35,7 +35,7 @@ This method assumes that `isa(f(x), Real)`.
 function gradient!(result::Union{AbstractArray,DiffResult}, f::F, x::AbstractArray, cfg::GradientConfig{T} = GradientConfig(f, x), ::Val{CHK}=Val{true}()) where {T, CHK, F}
     result isa DiffResult ? require_one_based_indexing(x) : require_one_based_indexing(result, x)
     CHK && checktag(T, f, x)
-    if chunksize(cfg) == length(x)
+    if chunksize(cfg) == structural_length(x)
         vector_mode_gradient!(result, f, x, cfg)
     else
         chunk_mode_gradient!(result, f, x, cfg)
@@ -63,12 +63,19 @@ function extract_gradient!(::Type{T}, result::DiffResult, dual::Dual) where {T}
 end
 
 extract_gradient!(::Type{T}, result::AbstractArray, y::Real) where {T} = fill!(result, zero(y))
-extract_gradient!(::Type{T}, result::AbstractArray, dual::Dual) where {T}= copyto!(result, partials(T, dual))
+function extract_gradient!(::Type{T}, result::AbstractArray, dual::Dual) where {T}
+    idxs = structural_eachindex(result)
+    for (i, idx) in zip(1:npartials(dual), idxs)
+        result[idx] = partials(T, dual, i)
+    end
+    return result
+end
 
 function extract_gradient_chunk!(::Type{T}, result, dual, index, chunksize) where {T}
     offset = index - 1
-    for i in 1:chunksize
-        result[i + offset] = partials(T, dual, i)
+    idxs = Iterators.drop(structural_eachindex(result), offset)
+    for (i, idx) in zip(1:chunksize, idxs)
+        result[idx] = partials(T, dual, i)
     end
     return result
 end
@@ -106,10 +113,10 @@ end
 
 function chunk_mode_gradient_expr(result_definition::Expr)
     return quote
-        @assert length(x) >= N "chunk size cannot be greater than length(x) ($(N) > $(length(x)))"
+        @assert structural_length(x) >= N "chunk size cannot be greater than ForwardDiff.structural_length(x) ($(N) > $(structural_length(x)))"
 
         # precalculate loop bounds
-        xlen = length(x)
+        xlen = structural_length(x)
         remainder = xlen % N
         lastchunksize = ifelse(remainder == 0, N, remainder)
         lastchunkindex = xlen - lastchunksize + 1
diff --git a/src/jacobian.jl b/src/jacobian.jl
index e7ca96f5..d654d38b 100644
--- a/src/jacobian.jl
+++ b/src/jacobian.jl
@@ -18,7 +18,7 @@ Set `check` to `Val{false}()` to disable tag checking. This can lead to perturba
 function jacobian(f::F, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f, x), ::Val{CHK}=Val{true}()) where {F,T,CHK}
     require_one_based_indexing(x)
     CHK && checktag(T, f, x)
-    if chunksize(cfg) == length(x)
+    if chunksize(cfg) == structural_length(x)
         return vector_mode_jacobian(f, x, cfg)
     else
         return chunk_mode_jacobian(f, x, cfg)
@@ -36,7 +36,7 @@ Set `check` to `Val{false}()` to disable tag checking. This can lead to perturba
 function jacobian(f!::F, y::AbstractArray, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {F,T, CHK}
     require_one_based_indexing(y, x)
     CHK && checktag(T, f!, x)
-    if chunksize(cfg) == length(x)
+    if chunksize(cfg) == structural_length(x)
         return vector_mode_jacobian(f!, y, x, cfg)
     else
         return chunk_mode_jacobian(f!, y, x, cfg)
@@ -57,7 +57,7 @@ Set `check` to `Val{false}()` to disable tag checking. This can lead to perturba
 function jacobian!(result::Union{AbstractArray,DiffResult}, f::F, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f, x), ::Val{CHK}=Val{true}()) where {F,T, CHK}
     result isa DiffResult ? require_one_based_indexing(x) : require_one_based_indexing(result, x)
     CHK && checktag(T, f, x)
-    if chunksize(cfg) == length(x)
+    if chunksize(cfg) == structural_length(x)
         vector_mode_jacobian!(result, f, x, cfg)
     else
         chunk_mode_jacobian!(result, f, x, cfg)
@@ -78,7 +78,7 @@ Set `check` to `Val{false}()` to disable tag checking. This can lead to perturba
 function jacobian!(result::Union{AbstractArray,DiffResult}, f!::F, y::AbstractArray, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {F,T,CHK}
     result isa DiffResult ? require_one_based_indexing(y, x) : require_one_based_indexing(result, y, x)
     CHK && checktag(T, f!, x)
-    if chunksize(cfg) == length(x)
+    if chunksize(cfg) == structural_length(x)
         vector_mode_jacobian!(result, f!, y, x, cfg)
     else
         chunk_mode_jacobian!(result, f!, y, x, cfg)
@@ -169,10 +169,10 @@ const JACOBIAN_ERROR = DimensionMismatch("jacobian(f, x) expects that f(x) is an
 function jacobian_chunk_mode_expr(work_array_definition::Expr, compute_ydual::Expr,
                                   result_definition::Expr, y_definition::Expr)
     return quote
-        @assert length(x) >= N "chunk size cannot be greater than length(x) ($(N) > $(length(x)))"
+        @assert structural_length(x) >= N "chunk size cannot be greater than ForwardDiff.structural_length(x) ($(N) > $(structural_length(x)))"
 
         # precalculate loop bounds
-        xlen = length(x)
+        xlen = structural_length(x)
         remainder = xlen % N
         lastchunksize = ifelse(remainder == 0, N, remainder)
         lastchunkindex = xlen - lastchunksize + 1
diff --git a/src/prelude.jl b/src/prelude.jl
index ed4782e9..9e037afa 100644
--- a/src/prelude.jl
+++ b/src/prelude.jl
@@ -12,8 +12,15 @@ function Chunk(input_length::Integer, threshold::Integer = DEFAULT_CHUNK_THRESHO
     Base.@nif 12 d->(N == d) d->(Chunk{d}()) d->(Chunk{N}())
 end
 
+structural_length(x::AbstractArray) = length(x)
+function structural_length(x::Union{LowerTriangular,UpperTriangular})
+    n = size(x, 1)
+    return (n * (n + 1)) >> 1
+end
+structural_length(x::Diagonal) = size(x, 1)
+
 function Chunk(x::AbstractArray, threshold::Integer = DEFAULT_CHUNK_THRESHOLD)
-    return Chunk(length(x), threshold)
+    return Chunk(structural_length(x), threshold)
 end
 
 # Constrained to `N <= threshold`, minimize (in order of priority):
diff --git a/test/GradientTest.jl b/test/GradientTest.jl
index a386c479..4f46c167 100644
--- a/test/GradientTest.jl
+++ b/test/GradientTest.jl
@@ -226,4 +226,33 @@ end
     @test dx ≈ sum(a * b)
 end
 
+# issue #738
+@testset "LowerTriangular, UpperTriangular and Diagonal" begin
+    for n in (3, 10, 20)
+        M = rand(n, n)
+        for T in (LowerTriangular, UpperTriangular, Diagonal)
+            @test ForwardDiff.gradient(sum, T(randn(n, n))) == T(ones(n, n))
+            @test ForwardDiff.gradient(x -> dot(M, x), T(randn(n, n))) == T(M)
+
+            # Check number of function evaluations and chunk sizes
+            fevals = Ref(0)
+            npartials = Ref(0)
+            y = ForwardDiff.gradient(T(randn(n, n))) do x
+                fevals[] += 1
+                npartials[] += ForwardDiff.npartials(eltype(x))
+                return sum(x)
+            end
+            if npartials[] <= ForwardDiff.DEFAULT_CHUNK_THRESHOLD
+                # Vector mode (single evaluation)
+                @test fevals[] == 1
+                @test npartials[] == sum(y)
+            else
+                # Chunk mode (multiple evaluations)
+                @test fevals[] > 1
+                @test sum(y) <= npartials[] < sum(y) + fevals[]
+            end
+        end
+    end
+end
+
 end # module