sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P'

sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P'

sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P'

C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P' sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For

C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P' sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For

C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P' sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For

matrix-matrix mul�plica�on (#30372). • Sparse vector outer products are more performant and maintain sparsity in products of the form kron(u, v'), u * v', and u .* v' where u and v are sparse vectors or column sparse matrices differ from their dense counterparts in that the resul�ng matrix follows the same sparsity pa�ern as a given sparse matrix S, or that the resul�ng sparse matrix has density d, i.e. each matrix

matrix-matrix mul�plica�on (#30372). • Sparse vector outer products are more performant and maintain sparsity in products of the form kron(u, v'), u * v', and u .* v' where u and v are sparse vectors or column sparse matrices differ from their dense counterparts in that the resul�ng matrix follows the same sparsity pa�ern as a given sparse matrix S, or that the resul�ng sparse matrix has density d, i.e. each matrix

sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P'

sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each dimensions m x n with structural zeros at S[I[k], J[k]]. This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. sparse(I, J, zeros(length(I))). For C::Sparse, update::Cint) Update an LDLt or LLt Factorization F of A to a factorization of A ± C*C'. If sparsity preserving factorization is used, i.e. L*L' == P*A*P' then the new factor will be L*L' == P*A*P'