diff --git a/dev/index.html b/dev/index.html index 6127ab71..6c3fa2de 100644 --- a/dev/index.html +++ b/dev/index.html @@ -140,4 +140,4 @@ autoback_hesvec(f,x,v)

J*v and H*v Operators

The following produce matrix-free operators which are used for calculating Jacobian-vector and Hessian-vector products where the differentiation takes place at the vector u:

JacVec(f,x::AbstractArray;autodiff=true)
 HesVec(f,x::AbstractArray;autodiff=true)
-HesVecGrad(g,x::AbstractArray;autodiff=false)

These all have the same interface, where J*v utilizes the out-of-place Jacobian-vector or Hessian-vector function, whereas mul!(res,J,v) utilizes the appropriate in-place versions. To update the location of differentiation in the operator, simply mutate the vector u: J.u .= ....

+HesVecGrad(g,x::AbstractArray;autodiff=false)

These all have the same interface, where J*v utilizes the out-of-place Jacobian-vector or Hessian-vector function, whereas mul!(res,J,v) utilizes the appropriate in-place versions. To update the location of differentiation in the operator, simply mutate the vector u: J.u .= ....

diff --git a/dev/search/index.html b/dev/search/index.html index ade9373e..25766d18 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · SparseDiffTools.jl

Loading search...

    +Search · SparseDiffTools.jl

    Loading search...

      diff --git a/dev/sparsedifftools/index.html b/dev/sparsedifftools/index.html index 777c5449..42870992 100644 --- a/dev/sparsedifftools/index.html +++ b/dev/sparsedifftools/index.html @@ -5,18 +5,18 @@ - `alg`: The algorithm used for computing the matrix colors - `epsilon`: For Finite Differencing based Jacobian Approximations, any number smaller than `epsilon` is considered to be zero. If `nothing` is specified, then this value - is calculated as `100 * eps(eltype(x))`source
      SparseDiffTools.JacPrototypeSparsityDetection — Type
      JacPrototypeSparsityDetection(; jac_prototype, alg = GreedyD1Color())

      Use a pre-specified jac_prototype to compute the matrix colors of the Jacobian.

      Keyword Arguments

      - `jac_prototype`: The prototype Jacobian used for computing the matrix colors
-- `alg`: The algorithm used for computing the matrix colors

      See Also: SymbolicsSparsityDetection, PrecomputedJacobianColorvec

      source
      SparseDiffTools.PrecomputedJacobianColorvec — Type
      PrecomputedJacobianColorvec(jac_prototype, row_colorvec, col_colorvec)

      Use a pre-specified colorvec which can be directly used for sparse differentiation. Based on whether a reverse mode or forward mode or finite differences is used, the corresponding row_colorvec or col_colorvec is used. Atmost one of them can be set to nothing.

      Arguments

      - `jac_prototype`: The prototype Jacobian used for computing structural nonzeros
+  is calculated as `100 * eps(eltype(x))`
      source
      SparseDiffTools.JacPrototypeSparsityDetection — Type
      JacPrototypeSparsityDetection(; jac_prototype, alg = GreedyD1Color())

      Use a pre-specified jac_prototype to compute the matrix colors of the Jacobian.

      Keyword Arguments

      - `jac_prototype`: The prototype Jacobian used for computing the matrix colors
+- `alg`: The algorithm used for computing the matrix colors

      See Also: SymbolicsSparsityDetection, PrecomputedJacobianColorvec

      source
      SparseDiffTools.PrecomputedJacobianColorvec — Type
      PrecomputedJacobianColorvec(jac_prototype, row_colorvec, col_colorvec)

      Use a pre-specified colorvec which can be directly used for sparse differentiation. Based on whether a reverse mode or forward mode or finite differences is used, the corresponding row_colorvec or col_colorvec is used. Atmost one of them can be set to nothing.

      Arguments

      - `jac_prototype`: The prototype Jacobian used for computing structural nonzeros
 - `row_colorvec`: The row colorvec of the Jacobian
-- `col_colorvec`: The column colorvec of the Jacobian

      See Also: SymbolicsSparsityDetection, JacPrototypeSparsityDetection

      source
      SparseDiffTools.PrecomputedJacobianColorvec — Method
      PrecomputedJacobianColorvec(; jac_prototype, partition_by_rows::Bool = false,
+- `col_colorvec`: The column colorvec of the Jacobian

      See Also: SymbolicsSparsityDetection, JacPrototypeSparsityDetection

      source
      SparseDiffTools.PrecomputedJacobianColorvec — Method
      PrecomputedJacobianColorvec(; jac_prototype, partition_by_rows::Bool = false,
     colorvec = missing, row_colorvec = missing, col_colorvec = missing)

      Use a pre-specified colorvec which can be directly used for sparse differentiation. Based on whether a reverse mode or forward mode or finite differences is used, the corresponding row_colorvec or col_colorvec is used. Atmost one of them can be set to nothing.

      Keyword Arguments

      - `jac_prototype`: The prototype Jacobian used for computing structural nonzeros
 - `partition_by_rows`: Whether to partition the Jacobian by rows or columns (row
   partitioning is used for reverse mode AD)
 - `colorvec`: The colorvec of the Jacobian. If `partition_by_rows` is `true` then this
   is the row colorvec, otherwise it is the column colorvec
 - `row_colorvec`: The row colorvec of the Jacobian
-- `col_colorvec`: The column colorvec of the Jacobian

      See Also: SymbolicsSparsityDetection, JacPrototypeSparsityDetection

      source
      SparseDiffTools.SymbolicsSparsityDetection — Type
      SymbolicsSparsityDetection(; alg = GreedyD1Color())

      Use Symbolics to compute the sparsity pattern of the Jacobian. This requires Symbolics.jl to be explicitly loaded.

      Keyword Arguments

      - `alg`: The algorithm used for computing the matrix colors

      See Also: JacPrototypeSparsityDetection, PrecomputedJacobianColorvec

      source
      ArrayInterface.matrix_colors — Function
      matrix_colors(A, alg::ColoringAlgorithm = GreedyD1Color();
-    partition_by_rows::Bool = false)

      Return the colorvec vector for the matrix A using the chosen coloring algorithm. If a known analytical solution exists, that is used instead. The coloring defaults to a greedy distance-1 coloring.

      Note that if A isa SparseMatrixCSC, the sparsity pattern is defined by structural nonzeroes, ie includes explicitly stored zeros.

      If ArrayInterface.fast_matrix_colors(A) is true, then uses ArrayInterface.matrix_colors(A) to compute the matrix colors.

      source
      SparseDiffTools.JacVec — Function
      JacVec(f, u, [p, t]; fu = nothing, autodiff = AutoForwardDiff(), tag = DeivVecTag(),
+- `col_colorvec`: The column colorvec of the Jacobian

      See Also: SymbolicsSparsityDetection, JacPrototypeSparsityDetection

      source
      SparseDiffTools.SymbolicsSparsityDetection — Type
      SymbolicsSparsityDetection(; alg = GreedyD1Color())

      Use Symbolics to compute the sparsity pattern of the Jacobian. This requires Symbolics.jl to be explicitly loaded.

      Keyword Arguments

      - `alg`: The algorithm used for computing the matrix colors

      See Also: JacPrototypeSparsityDetection, PrecomputedJacobianColorvec

      source
      ArrayInterface.matrix_colors — Function
      matrix_colors(A, alg::ColoringAlgorithm = GreedyD1Color();
+    partition_by_rows::Bool = false)

      Return the colorvec vector for the matrix A using the chosen coloring algorithm. If a known analytical solution exists, that is used instead. The coloring defaults to a greedy distance-1 coloring.

      Note that if A isa SparseMatrixCSC, the sparsity pattern is defined by structural nonzeroes, ie includes explicitly stored zeros.

      If ArrayInterface.fast_matrix_colors(A) is true, then uses ArrayInterface.matrix_colors(A) to compute the matrix colors.

      source
      SparseDiffTools.JacVec — Function
      JacVec(f, u, [p, t]; fu = nothing, autodiff = AutoForwardDiff(), tag = DeivVecTag(),
     kwargs...)

      Returns SciMLOperators.FunctionOperator which computes jacobian-vector product df/du * v.

      Note

      For non-square jacobians with inplace f, fu must be specified, else JacVec assumes a square jacobian.

      L = JacVec(f, u)
 
 L * v         # = df/du * v
@@ -27,7 +27,7 @@
 f(u, p)     # p !== nothing
 f(u)        # Otherwise

      For In Place Functions:

      f(du, u, p, t)  # t !== nothing
 f(du, u, p)     # p !== nothing
-f(du, u)        # Otherwise
      source
      SparseDiffTools.VecJac — Function
      VecJac(f, u, [p, t]; fu = nothing, autodiff = AutoFiniteDiff())

      Returns SciMLOperators.FunctionOperator which computes vector-jacobian product (df/du)ᵀ * v.

      Note

      For non-square jacobians with inplace f, fu must be specified, else VecJac assumes a square jacobian.

      L = VecJac(f, u)
+f(du, u)        # Otherwise
      source
      SparseDiffTools.VecJac — Function
      VecJac(f, u, [p, t]; fu = nothing, autodiff = AutoFiniteDiff())

      Returns SciMLOperators.FunctionOperator which computes vector-jacobian product (df/du)ᵀ * v.

      Note

      For non-square jacobians with inplace f, fu must be specified, else VecJac assumes a square jacobian.

      L = VecJac(f, u)
 
 L * v         # = (df/du)ᵀ * v
 mul!(w, L, v) # = (df/du)ᵀ * v
@@ -37,27 +37,27 @@
 f(u, p)     # p !== nothing
 f(u)        # Otherwise

      For In Place Functions:

      f(du, u, p, t)  # t !== nothing
 f(du, u, p)     # p !== nothing
-f(du, u)        # Otherwise
      source
      SparseDiffTools._cols_by_rows — Method
      _cols_by_rows(rows_index,cols_index)

      Returns a vector of rows where each row contains a vector of its column indices.

      source
      SparseDiffTools._rows_by_cols — Method
      _rows_by_cols(rows_index,cols_index)

      Returns a vector of columns where each column contains a vector of its row indices.

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraphs, ::AcyclicColoring)

      Returns a coloring vector following the acyclic coloring rules (1) the coloring corresponds to a distance-1 coloring, and (2) vertices in every cycle of the graph are assigned at least three distinct colors. This variant of coloring is called acyclic since every subgraph induced by vertices assigned any two colors is a collection of trees—and hence is acyclic.

      Reference: Gebremedhin AH, Manne F, Pothen A. New Acyclic and Star Coloring Algorithms with Application to Computing Hessians

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraph, ::BacktrackingColor)

      Return a tight, distance-1 coloring of graph g using the minimum number of colors possible (i.e. the chromatic number of graph, χ(g))

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraph, ::GreedyStar1Color)

      Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

      A star coloring is a special type of distance - 1 coloring, For a coloring to be called a star coloring, it must satisfy two conditions:

      1. every pair of adjacent vertices receives distinct colors

      (a distance-1 coloring)

      1. For any vertex v, any color that leads to a two-colored path

      involving v and three other vertices is impermissible for v. In other words, every path on four vertices uses at least three colors.

      Reference: Gebremedhin AH, Manne F, Pothen A. What color is your Jacobian? Graph coloring for computing derivatives. SIAM review. 2005;47(4):629-705.

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraph, ::GreedyStar2Color)

      Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

      A star coloring is a special type of distance - 1 coloring, For a coloring to be called a star coloring, it must satisfy two conditions:

      1. every pair of adjacent vertices receives distinct colors

      (a distance-1 coloring)

      1. For any vertex v, any color that leads to a two-colored path

      involving v and three other vertices is impermissible for v. In other words, every path on four vertices uses at least three colors.

      Reference: Gebremedhin AH, Manne F, Pothen A. What color is your Jacobian? Graph coloring for computing derivatives. SIAM review. 2005;47(4):629-705.

      TODO: add text explaining the difference between star1 and star2

      source
      SparseDiffTools.color_graph — Method
      color_graph(G::VSafeGraph,::ContractionColor)

      Find a coloring of the graph g such that no two vertices connected by an edge have the same color.

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::VSafeGraph, alg::GreedyD1Color)

      Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

      source
      SparseDiffTools.contract! — Method
      contract!(g, y, x)

      Contract the vertex y to x, both of which belong to graph G, that is delete vertex y and join x with the neighbors of y if they are not already connected with an edge.

      source
      SparseDiffTools.find — Method
      find(w::Integer, x::Integer, g::Graphs.AbstractGraph,
-    two_colored_forest::DisjointSets{<:Integer})

      Returns the root of the disjoint set to which the edge connecting vertices w and x in the graph g belongs to

      source
      SparseDiffTools.find_edge_index — Method
      find_edge(g::Graphs.AbstractGraph, v::Integer, w::Integer)

      Returns an integer equivalent to the index of the edge connecting the vertices v and w in the graph g

      source
      SparseDiffTools.free_colors — Method
      free_colors(x::Integer,
+f(du, u)        # Otherwise
      source
      SparseDiffTools._cols_by_rows — Method
      _cols_by_rows(rows_index,cols_index)

      Returns a vector of rows where each row contains a vector of its column indices.

      source
      SparseDiffTools._rows_by_cols — Method
      _rows_by_cols(rows_index,cols_index)

      Returns a vector of columns where each column contains a vector of its row indices.

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraphs, ::AcyclicColoring)

      Returns a coloring vector following the acyclic coloring rules (1) the coloring corresponds to a distance-1 coloring, and (2) vertices in every cycle of the graph are assigned at least three distinct colors. This variant of coloring is called acyclic since every subgraph induced by vertices assigned any two colors is a collection of trees—and hence is acyclic.

      Reference: Gebremedhin AH, Manne F, Pothen A. New Acyclic and Star Coloring Algorithms with Application to Computing Hessians

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraph, ::BacktrackingColor)

      Return a tight, distance-1 coloring of graph g using the minimum number of colors possible (i.e. the chromatic number of graph, χ(g))

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraph, ::GreedyStar1Color)

      Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

      A star coloring is a special type of distance - 1 coloring, For a coloring to be called a star coloring, it must satisfy two conditions:

      1. every pair of adjacent vertices receives distinct colors

      (a distance-1 coloring)

      1. For any vertex v, any color that leads to a two-colored path

      involving v and three other vertices is impermissible for v. In other words, every path on four vertices uses at least three colors.

      Reference: Gebremedhin AH, Manne F, Pothen A. What color is your Jacobian? Graph coloring for computing derivatives. SIAM review. 2005;47(4):629-705.

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::Graphs.AbstractGraph, ::GreedyStar2Color)

      Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

      A star coloring is a special type of distance - 1 coloring, For a coloring to be called a star coloring, it must satisfy two conditions:

      1. every pair of adjacent vertices receives distinct colors

      (a distance-1 coloring)

      1. For any vertex v, any color that leads to a two-colored path

      involving v and three other vertices is impermissible for v. In other words, every path on four vertices uses at least three colors.

      Reference: Gebremedhin AH, Manne F, Pothen A. What color is your Jacobian? Graph coloring for computing derivatives. SIAM review. 2005;47(4):629-705.

      TODO: add text explaining the difference between star1 and star2

      source
      SparseDiffTools.color_graph — Method
      color_graph(G::VSafeGraph,::ContractionColor)

      Find a coloring of the graph g such that no two vertices connected by an edge have the same color.

      source
      SparseDiffTools.color_graph — Method
      color_graph(g::VSafeGraph, alg::GreedyD1Color)

      Find a coloring of a given input graph such that no two vertices connected by an edge have the same color using greedy approach. The number of colors used may be equal or greater than the chromatic number χ(G) of the graph.

      source
      SparseDiffTools.contract! — Method
      contract!(g, y, x)

      Contract the vertex y to x, both of which belong to graph G, that is delete vertex y and join x with the neighbors of y if they are not already connected with an edge.

      source
      SparseDiffTools.find — Method
      find(w::Integer, x::Integer, g::Graphs.AbstractGraph,
+    two_colored_forest::DisjointSets{<:Integer})

      Returns the root of the disjoint set to which the edge connecting vertices w and x in the graph g belongs to

      source
      SparseDiffTools.find_edge_index — Method
      find_edge(g::Graphs.AbstractGraph, v::Integer, w::Integer)

      Returns an integer equivalent to the index of the edge connecting the vertices v and w in the graph g

      source
      SparseDiffTools.free_colors — Method
      free_colors(x::Integer,
             A::AbstractVector{<:Integer},
             colors::AbstractVector{<:Integer},
             F::Array{Integer,1},
             g::Graphs.AbstractGraph,
-            opt::Integer)

      Returns set of free colors of x which are less than optimal chromatic number (opt)

      Arguments:

      x: Vertex who's set of free colors is to be calculated A: List of vertices of graph g sorted in non-increasing order of degree colors: colors[i] stores the number of distinct colors used in the coloring of vertices A[0], A[1]... A[i-1] F: F[i] stores the color of vertex i g: Graph to be colored opt: Current optimal number of colors to be used in the coloring of graph g

      source
      SparseDiffTools.grow_star! — Method
      grow_star!(two_colored_forest::DisjointSets{<:Integer},
+            opt::Integer)

      Returns set of free colors of x which are less than optimal chromatic number (opt)

      Arguments:

      x: Vertex who's set of free colors is to be calculated A: List of vertices of graph g sorted in non-increasing order of degree colors: colors[i] stores the number of distinct colors used in the coloring of vertices A[0], A[1]... A[i-1] F: F[i] stores the color of vertex i g: Graph to be colored opt: Current optimal number of colors to be used in the coloring of graph g

      source
      SparseDiffTools.grow_star! — Method
      grow_star!(two_colored_forest::DisjointSets{<:Integer},
     first_neighbor::AbstractVector{<:Tuple{Integer, Integer}}, v::Integer, w::Integer,
-    g::Graphs.AbstractGraph, color::AbstractVector{<:Integer})

      Grow a 2-colored star after assigning a new color to the previously uncolored vertex v, by comparing it with the adjacent vertex w. Disjoint set is used to store stars in sets, which are identified through key edges present in g.

      source
      SparseDiffTools.init_jacobian — Function
      init_jacobian(cache::AbstractMaybeSparseJacobianCache)

      Initialize the Jacobian based on the cache. Uses sparse jacobians if possible.

      Note

      This function doesn't alias the provided jacobian prototype. It always initializes a fresh jacobian that can be mutated without any side effects.

      source
      SparseDiffTools.init_jacobian — Method
      init_jacobian(cache::AbstractMaybeSparseJacobianCache;
-    preserve_immutable::Val = Val(false))

      Initialize the Jacobian based on the cache. Uses sparse jacobians if possible.

      If preserve_immutable is true, then the Jacobian returned might be immutable, this is relevant if the inputs are immutable like StaticArrays.

      source
      SparseDiffTools.insert_new_tree! — Method
      insert_new_tree!(two_colored_forest::DisjointSets{<:Integer}, v::Integer,
-    w::Integer, g::Graphs.AbstractGraph

      creates a new singleton set in the disjoint set 'twocoloredforest' consisting of the edge connecting v and w in the graph g

      source
      SparseDiffTools.least_index — Method
      least_index(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, opt::Integer)

      Returns least index i such that color of vertex A[i] is equal to opt (optimal chromatic number)

      source
      SparseDiffTools.length_common_neighbor — Method
      length_common_neighbor(g, z, x)

      Find the number of vertices that share an edge with both the vertices z and x belonging to the graph g.

      source
      SparseDiffTools.matrix2graph — Function
      matrix2graph(sparse_matrix, [partition_by_rows::Bool=true])

      A utility function to generate a graph from input sparse matrix, columns are represented with vertices and 2 vertices are connected with an edge only if the two columns are mutually orthogonal.

      Note that the sparsity pattern is defined by structural nonzeroes, ie includes explicitly stored zeros.

      source
      SparseDiffTools.max_degree_vertex — Method
      max_degree_vertex(G, nn)

      Find the vertex in the group nn of vertices belonging to the graph G which has the highest degree.

      source
      SparseDiffTools.max_degree_vertex — Method
      max_degree_vertex(G)

      Find the vertex in graph with highest degree.

      source
      SparseDiffTools.merge_trees! — Method
      merge_trees!(two_colored_forest::DisjointSets{<:Integer}, v::Integer, w::Integer,
-    x::Integer, g::Graphs.AbstractGraph)

      Subroutine to merge trees present in the disjoint set which have a common edge.

      source
      SparseDiffTools.min_index — Method
      min_index(forbidden_colors::AbstractVector{<:Integer}, v::Integer)

      Returns min{i > 0 such that forbidden_colors[i] != v}

      source
      SparseDiffTools.non_neighbors — Method
      non_neighbors(G, x)

      Find the set of vertices belonging to the graph G which do not share an edge with the vertex x.

      source
      SparseDiffTools.prevent_cycle! — Method
      prevent_cycle!(first_visit_to_tree::AbstractVector{<:Tuple{Integer, Integer}},
+    g::Graphs.AbstractGraph, color::AbstractVector{<:Integer})

      Grow a 2-colored star after assigning a new color to the previously uncolored vertex v, by comparing it with the adjacent vertex w. Disjoint set is used to store stars in sets, which are identified through key edges present in g.

      source
      SparseDiffTools.init_jacobian — Function
      init_jacobian(cache::AbstractMaybeSparseJacobianCache)

      Initialize the Jacobian based on the cache. Uses sparse jacobians if possible.

      Note

      This function doesn't alias the provided jacobian prototype. It always initializes a fresh jacobian that can be mutated without any side effects.

      source
      SparseDiffTools.init_jacobian — Method
      init_jacobian(cache::AbstractMaybeSparseJacobianCache;
+    preserve_immutable::Val = Val(false))

      Initialize the Jacobian based on the cache. Uses sparse jacobians if possible.

      If preserve_immutable is true, then the Jacobian returned might be immutable, this is relevant if the inputs are immutable like StaticArrays.

      source
      SparseDiffTools.insert_new_tree! — Method
      insert_new_tree!(two_colored_forest::DisjointSets{<:Integer}, v::Integer,
+    w::Integer, g::Graphs.AbstractGraph

      creates a new singleton set in the disjoint set 'twocoloredforest' consisting of the edge connecting v and w in the graph g

      source
      SparseDiffTools.least_index — Method
      least_index(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, opt::Integer)

      Returns least index i such that color of vertex A[i] is equal to opt (optimal chromatic number)

      source
      SparseDiffTools.length_common_neighbor — Method
      length_common_neighbor(g, z, x)

      Find the number of vertices that share an edge with both the vertices z and x belonging to the graph g.

      source
      SparseDiffTools.matrix2graph — Function
      matrix2graph(sparse_matrix, [partition_by_rows::Bool=true])

      A utility function to generate a graph from input sparse matrix, columns are represented with vertices and 2 vertices are connected with an edge only if the two columns are mutually orthogonal.

      Note that the sparsity pattern is defined by structural nonzeroes, ie includes explicitly stored zeros.

      source
      SparseDiffTools.max_degree_vertex — Method
      max_degree_vertex(G, nn)

      Find the vertex in the group nn of vertices belonging to the graph G which has the highest degree.

      source
      SparseDiffTools.max_degree_vertex — Method
      max_degree_vertex(G)

      Find the vertex in graph with highest degree.

      source
      SparseDiffTools.merge_trees! — Method
      merge_trees!(two_colored_forest::DisjointSets{<:Integer}, v::Integer, w::Integer,
+    x::Integer, g::Graphs.AbstractGraph)

      Subroutine to merge trees present in the disjoint set which have a common edge.

      source
      SparseDiffTools.min_index — Method
      min_index(forbidden_colors::AbstractVector{<:Integer}, v::Integer)

      Returns min{i > 0 such that forbidden_colors[i] != v}

      source
      SparseDiffTools.non_neighbors — Method
      non_neighbors(G, x)

      Find the set of vertices belonging to the graph G which do not share an edge with the vertex x.

      source
      SparseDiffTools.prevent_cycle! — Method
      prevent_cycle!(first_visit_to_tree::AbstractVector{<:Tuple{Integer, Integer}},
     forbidden_colors::AbstractVector{<:Integer}, v::Integer, w::Integer, x::Integer,
     g::Graphs.AbstractGraph, two_colored_forest::DisjointSets{<:Integer},
-    color::AbstractVector{<:Integer})

      Subroutine to avoid generation of 2-colored cycle due to coloring of vertex v, which is adjacent to vertices w and x in graph g. Disjoint set is used to store the induced 2-colored subgraphs/trees where the id of set is an integer representing an edge of graph 'g'

      source
      SparseDiffTools.remove_higher_colors — Method
      remove_higher_colors(U::AbstractVector{<:Integer}, opt::Integer)

      Remove all the colors which are greater than or equal to the opt (optimal chromatic number) from the set of colors U

      source
      SparseDiffTools.sort_by_degree — Method
      sort_by_degree(g::Graphs.AbstractGraph)

      Returns a list of the vertices of graph g sorted in non-increasing order of their degrees

      source
      SparseDiffTools.sparse_jacobian! — Function
      sparse_jacobian!(J::AbstractMatrix, ad, cache::AbstractMaybeSparseJacobianCache, f, x)
+    color::AbstractVector{<:Integer})

      Subroutine to avoid generation of 2-colored cycle due to coloring of vertex v, which is adjacent to vertices w and x in graph g. Disjoint set is used to store the induced 2-colored subgraphs/trees where the id of set is an integer representing an edge of graph 'g'

      source
      SparseDiffTools.remove_higher_colors — Method
      remove_higher_colors(U::AbstractVector{<:Integer}, opt::Integer)

      Remove all the colors which are greater than or equal to the opt (optimal chromatic number) from the set of colors U

      source
      SparseDiffTools.sort_by_degree — Method
      sort_by_degree(g::Graphs.AbstractGraph)

      Returns a list of the vertices of graph g sorted in non-increasing order of their degrees

      source
      SparseDiffTools.sparse_jacobian! — Function
      sparse_jacobian!(J::AbstractMatrix, ad, cache::AbstractMaybeSparseJacobianCache, f, x)
 sparse_jacobian!(J::AbstractMatrix, ad, cache::AbstractMaybeSparseJacobianCache, f!, fx,
-    x)

      Inplace update the matrix J with the Jacobian of f at x using the AD backend ad.

      cache is the cache object returned by sparse_jacobian_cache.

      source
      SparseDiffTools.sparse_jacobian! — Method
      sparse_jacobian!(J::AbstractMatrix, ad::AbstractADType, sd::AbstractSparsityDetection,
+    x)

      Inplace update the matrix J with the Jacobian of f at x using the AD backend ad.

      cache is the cache object returned by sparse_jacobian_cache.

      source
      SparseDiffTools.sparse_jacobian! — Method
      sparse_jacobian!(J::AbstractMatrix, ad::AbstractADType, sd::AbstractSparsityDetection,
     f, x; fx=nothing)
 sparse_jacobian!(J::AbstractMatrix, ad::AbstractADType, sd::AbstractSparsityDetection,
-    f!, fx, x)

      Sequentially calls sparse_jacobian_cache and sparse_jacobian! to compute the Jacobian of f at x. Use this if the jacobian for f is computed exactly once. In all other cases, use sparse_jacobian_cache once to generate the cache and use sparse_jacobian! with the same cache to compute the jacobian.

      source
      SparseDiffTools.sparse_jacobian — Method
      sparse_jacobian(ad::AbstractADType, cache::AbstractMaybeSparseJacobianCache, f, x)
-sparse_jacobian(ad::AbstractADType, cache::AbstractMaybeSparseJacobianCache, f!, fx, x)

      Use the sparsity detection cache for computing the sparse Jacobian. This allocates a new Jacobian at every function call.

      If x is a StaticArray, then this function tries to use a non-allocating implementation for the jacobian computation. This is possible only for a limited backends currently.

      source
      SparseDiffTools.sparse_jacobian — Method
      sparse_jacobian(ad::AbstractADType, sd::AbstractMaybeSparsityDetection, f, x; fx=nothing)
-sparse_jacobian(ad::AbstractADType, sd::AbstractMaybeSparsityDetection, f!, fx, x)

      Sequentially calls sparse_jacobian_cache and sparse_jacobian! to compute the Jacobian of f at x. Use this if the jacobian for f is computed exactly once. In all other cases, use sparse_jacobian_cache once to generate the cache and use sparse_jacobian! with the same cache to compute the jacobian.

      If x is a StaticArray, then this function tries to use a non-allocating implementation for the jacobian computation. This is possible only for a limited backends currently.

      source
      SparseDiffTools.sparse_jacobian_cache — Method
      sparse_jacobian_cache(ad::AbstractADType, sd::AbstractSparsityDetection, f, x;
+    f!, fx, x)

      Sequentially calls sparse_jacobian_cache and sparse_jacobian! to compute the Jacobian of f at x. Use this if the jacobian for f is computed exactly once. In all other cases, use sparse_jacobian_cache once to generate the cache and use sparse_jacobian! with the same cache to compute the jacobian.

      source
      SparseDiffTools.sparse_jacobian — Method
      sparse_jacobian(ad::AbstractADType, cache::AbstractMaybeSparseJacobianCache, f, x)
+sparse_jacobian(ad::AbstractADType, cache::AbstractMaybeSparseJacobianCache, f!, fx, x)

      Use the sparsity detection cache for computing the sparse Jacobian. This allocates a new Jacobian at every function call.

      If x is a StaticArray, then this function tries to use a non-allocating implementation for the jacobian computation. This is possible only for a limited backends currently.

      source
      SparseDiffTools.sparse_jacobian — Method
      sparse_jacobian(ad::AbstractADType, sd::AbstractMaybeSparsityDetection, f, x; fx=nothing)
+sparse_jacobian(ad::AbstractADType, sd::AbstractMaybeSparsityDetection, f!, fx, x)

      Sequentially calls sparse_jacobian_cache and sparse_jacobian! to compute the Jacobian of f at x. Use this if the jacobian for f is computed exactly once. In all other cases, use sparse_jacobian_cache once to generate the cache and use sparse_jacobian! with the same cache to compute the jacobian.

      If x is a StaticArray, then this function tries to use a non-allocating implementation for the jacobian computation. This is possible only for a limited backends currently.

      source
      SparseDiffTools.sparse_jacobian_cache — Method
      sparse_jacobian_cache(ad::AbstractADType, sd::AbstractSparsityDetection, f, x;
     fx=nothing)
-sparse_jacobian_cache(ad::AbstractADType, sd::AbstractSparsityDetection, f!, fx, x)

      Takes the underlying AD backend ad, sparsity detection algorithm sd, function f, and input x and returns a cache object that can be used to compute the Jacobian.

      If fx is not specified, it will be computed by calling f(x).

      Returns

      A cache for computing the Jacobian of type AbstractMaybeSparseJacobianCache.

      source
      SparseDiffTools.uncolor_all! — Method
      uncolor_all(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, start::Integer)

      Uncolors all vertices A[i] where i is greater than or equal to start

      source
      SparseDiffTools.uncolored_vertex_of_maximal_degree — Method
      uncolored_vertex_of_maximal_degree(A::AbstractVector{<:Integer},F::AbstractVector{<:Integer})

      Returns an uncolored vertex from the partially colored graph which has the highest degree

      source
      SparseDiffTools.vertex_degree — Method
      vertex_degree(g, z)

      Find the degree of the vertex z which belongs to the graph g.

      source
      +sparse_jacobian_cache(ad::AbstractADType, sd::AbstractSparsityDetection, f!, fx, x)

      Takes the underlying AD backend ad, sparsity detection algorithm sd, function f, and input x and returns a cache object that can be used to compute the Jacobian.

      If fx is not specified, it will be computed by calling f(x).

      Returns

      A cache for computing the Jacobian of type AbstractMaybeSparseJacobianCache.

      source
      SparseDiffTools.uncolor_all! — Method
      uncolor_all(F::AbstractVector{<:Integer}, A::AbstractVector{<:Integer}, start::Integer)

      Uncolors all vertices A[i] where i is greater than or equal to start

      source
      SparseDiffTools.uncolored_vertex_of_maximal_degree — Method
      uncolored_vertex_of_maximal_degree(A::AbstractVector{<:Integer},F::AbstractVector{<:Integer})

      Returns an uncolored vertex from the partially colored graph which has the highest degree

      source
      SparseDiffTools.vertex_degree — Method
      vertex_degree(g, z)

      Find the degree of the vertex z which belongs to the graph g.

      source