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For each source row, use a temporary dense array to accumulate contributions from nonzero pairs. For every stored $A_{ik}$, traverse stored entries in row $k$ of the other sparse matrix.During accumulation, acc(j) is the sum of the already processed terms contributing to position $(i,j)$. Missing entries multiply to zero, so ignoring them loses no contribution under the finite arithmetic convention of part (e). At completion,Descending j and prepending produce strictly increasing stored indices. Omitting zero also handles cancellation. The temporary array uses $O(N)$ auxiliary space beyond the output. The time complexity isat worst $O(N^3)$, but potentially much smaller when the matrices are sparse. The $N^2$ term covers clearing and scanning the row accumulators. Finite-precision rounding can affect the numerical result, as in any Double matrix multiplication.

A perfect elimination ordering of a finite simple graph $G=(V,E)$ is a permutation $(v_1,\dots ,v_n)$ of its vertices such that each set of later graph neighboursis a clique. Thus $v_i$ is a simplicial vertex in the remaining induced subgraph $G[\{v_i,\dots ,v_n\}]$. Empty and one-vertex sets satisfy the clique condition automatically. The order here is the order of deletion: $v_1$ goes first.

The characterization is powerful: a finite simple graph is a chordal graph if and only if it has a perfect elimination ordering. For one proof outline, the earliest vertex of an induced cycle of length at least four would have two later graph neighbours on that graph cycle that are not adjacent, contradicting the clique condition. Conversely, every nonempty chordal graph has a simplicial vertex, and every induced subgraph of a chordal graph is a chordal graph; repeatedly deleting such a vertex supplies the ordering.^[ref]

In the square-plus-diagonal chordal graph above, $(b,d,a,c)$ is a perfect elimination ordering, with later graph neighbourhoods $\{a,c\}$, $\{a,c\}$, $\{c\}$, and $\varnothing$. Starting instead with $a$ fails: its graph neighbours $b,c,d$ do not form a clique, because $bd$ is absent. Thus even a chordal graph can have a poor elimination ordering.

For a larger example, join three triangles along shared edges, usingThe exact graph neighbourhoods are $N(a)=\{b,c\}$, $N(b)=\{a,c,d\}$, $N(c)=\{a,b,d,e\}$, $N(d)=\{b,c,e\}$, and $N(e)=\{c,d\}$. The ordering $(a,b,c,d,e)$ has later graph neighbourhoods $\{b,c\}$, $\{c,d\}$, $\{d,e\}$, $\{e\}$, and $\varnothing$, all cliques; this certifies that the graph is a chordal graph without enumerating its graph cycles. Its maximal cliques $\{a,b,c\}$, $\{b,c,d\}$, $\{c,d,e\}$ form a junction tree in that order, with intersections $\{b,c\}$ and $\{c,d\}$.

To find an ordering, maximum cardinality search starts with all scores zero, repeatedly selects an unselected vertex of largest score, and increases each unselected graph neighbour's score by one. Reverse the selection order to obtain a candidate perfect elimination ordering. For the three-triangle graph, one valid selection sequence is $(e,d,c,b,a)$, with scores at selection $0,1,2,2,2$; reversing it gives the displayed deletion order. On a chordal graph every tie choice works. On an arbitrary graph, test the resulting later graph neighbourhoods for the clique condition to recognize chordal graphs. Suitable adjacency lists, score buckets and a linear-time verification method give $O(|V|+|E|)$ recognition.^[ref]

The deletion rule also explains sparse Gaussian elimination. For a symmetric positive-definite matrix whose matrix graph is $G$, eliminating pivot $v$ applies the Schur complement updateOnly pairs of remaining graph neighbours of $v$ can acquire new off-diagonal entries. A perfect elimination ordering already has all those edges, so Cholesky decomposition creates no symbolic fill-in. Positive definiteness ensures nonzero pivots in exact arithmetic. Special numerical cancellation can suppress predicted entries, and floating-point rounding affects their values.^[ref] In the three-triangle graph, deleting $c$ first instead connects every pair in $\{a,b,d,e\}$ and adds exactly $ad,ae,be$. The graph was already a chordal graph; the extra fill-in comes from the ordering.

One further reward is optimal graph coloring. Process a perfect elimination ordering backwards and give each vertex the smallest positive color absent from its already colored graph neighbours. Those graph neighbours form a clique, so this greedy algorithm uses at most the clique number $\omega (G)$ colors; any clique of that size requires at least that many. In the three-triangle graph, coloring $(e,d,c,b,a)$ gives colors $(1,2,3,1,2)$, so both the chromatic number and the clique number are three. Every maximal clique also occurs among $\{v_i\}\cup N_i^{+}$: take its earliest vertex and use maximality. The same ordering therefore supports graph coloring, maximal clique extraction and sparse computation.

Historically, Donald J. Rose, Robert E. Tarjan and George S. Lueker's 1976 paper connected graph elimination to symmetric sparse matrices and gave a search-based $O(|V|+|E|)$ algorithm for finding a perfect elimination ordering when one exists. Its abstract distinguishes a minimal elimination ordering from a minimum one: removing avoidable fill-in does not by itself minimize the total number of added edges.^[ref]

A piecewise-linear hat function is a continuous piecewise linear function that selects one node of a finite element mesh: it is one there and zero at every other node. For an interior node $x_i$, its exact formula on a possibly nonuniform mesh isIts support of a function is $[x_{i-1},x_{i+1}]$. It rises on the left interval and falls on the right, producing its hat-shaped graph. The interior hat functions form a basis of the continuous $P_1$ conforming finite element space with homogeneous Dirichlet boundary conditions. Chen's author notes define this nodal basis and illustrate it in one and two dimensions.^[ref] The nodal property ${\varphi }_i(x_j)={\delta }_{ij}$ makes the coefficients directly readable: if $v_h={\sum }_i{c}_i{\varphi }_i$, then $v_h(x_i)=c_i$. Unlike the global Lagrange cardinal polynomials, each of these functions is confined to two neighboring elements.

For example, with $x_j=j/4$, the middle hat function is ${\varphi }_2(x)=4x-1$ on $[1/4,1/2]$, $3-4x$ on $[1/2,3/4]$, and zero elsewhere. Hence ${\varphi }_2(3/8)=1/2$, while ${\varphi }_1(3/8)=1/2$ and ${\varphi }_3(3/8)=0$. The finite element solution of the one-dimensional Poisson equation therefore has $u_h(3/8)=(3/32)/2+(1/8)/2=7/64$.

Local assembly makes this locality computational. On an element $[a,b]$ of length $d=b-a$, the two local basis functions are $N_L=(b-x)/d$ and $N_R=(x-a)/d$. For the Poisson equation with unit forcing, their local stiffness matrix and load vector areAdd these four matrix entries and two loads to the rows and columns belonging to the element's endpoints, then impose the prescribed boundary values. An interior node receives contributions from its two neighboring elements. Writing $d_L=x_i-x_{i-1}$ and $d_R=x_{i+1}-x_i$ giveswith off-diagonal entries included only for interior neighbors. On a uniform mesh these reduce to $2/h$, $-1/h$ and $h$. Nonadjacent hat functions have no derivative overlap of positive length, so all distant entries vanish. This produces a sparse matrix with only three diagonals: the global coupling is built entirely from local two-node interactions.

Power iteration finds a dominant eigenvalue and its eigenvector by repeatedly applying a matrix and rescaling the resulting vector. The essential idea is selective amplification: components along different eigenvectors grow at different rates, so one direction can eventually dominate. It is the computational form of the power method and is useful when applying a large sparse matrix to a vector is cheap but computing its whole spectral decomposition is expensive.

For a square matrix over the real numbers or complex numbers $A$, choose a unit vector $x_0$ and repeatHere $\Vert \cdot {\Vert }_2$ is the Euclidean norm. If $y_{k+1}=0$, stop before division: the current vector is an eigenvector for the zero eigenvalue, and this iteration cannot continue. Otherwise, in exact arithmetic,Repeated normalization keeps the stored vectors at manageable size without changing their directions; it does not guarantee that an individual multiplication cannot exceed the range of floating-point arithmetic. The dominant eigenvalue means largest modulus, which need not mean largest signed real value.^[ref]

Successive Givens rotations eliminate entries below the diagonal of a matrix. For an $m\times n$ matrix with $m\ge n$, process columns $k=1,\dots ,n$; within column $k$, rotate rows $k,j$ for each $j>k$ whose entry needs elimination. Apply each Givens row-pair update to columns $k,\dots ,n$. Both rows are zero in earlier columns, so these operations preserve earlier eliminations. No assumption of full column rank is needed to perform the eliminations.

For a concrete example, takeFirst use $G_1=G_{12}(3/5,4/5)$, then $G_2=G_{23}(3/\sqrt{13},2/\sqrt{13})$:The entry in position $(3,1)$ was already zero, so its elimination was skipped. The second Givens rotation combines the remaining pair $(3,2{)}^T$ in column two. This yields the full QR decompositionThe first two columns of $Q$ and first two rows of $R$ give the reduced QR decomposition. Direct multiplication verifies both $QR=A$ and $Q^{T}Q=I$. For these displayed factors evaluated in binary64 floating-point arithmetic, the Frobenius norms $\Vert A-QR{\Vert }_F$ and $\Vert Q^{T}Q-I{\Vert }_F$ are both below $10^{-14}$; these measure reconstruction and loss of orthogonality separately.

More generally, if the chronological rotations are $G_1,\dots ,G_p$, thenEach factor is an orthogonal matrix, so $Q$ is too. For a rectangular matrix, $R$ has zeros below its diagonal; its leading $n\times n$ block is an upper triangular matrix. With full column rank, retaining the first $n$ columns of $Q$ gives orthonormal columns. To impose a positive diagonal when necessary, flip matching columns of $Q$ and rows of $R$, preserving their product.

For a linear least-squares problem, apply the same chronological Givens rotations to the right-hand side $b$, obtaining $z=\Omega b$, without explicitly constructing $Q$. Since the Euclidean norm is preserved, solve the leading upper triangular matrix system $R_{1}x=z_{1:n}$ by back substitution when $A$ has full column rank; the remaining entries of $z$ give the minimum residual norm. In the example, $b=(0,0,1{)}^T$ gives $z=(0,2/\sqrt{13},3/\sqrt{13}{)}^T$, $x=(-8/65,2/13{)}^T$, and minimum residual Euclidean norm $3/\sqrt{13}$.

The locality is useful for a sparse matrix or an existing QR decomposition needing an extra row: only two rows change at each step, and the new row can be eliminated against the stored upper triangular matrix. For an $n\times n$ factor, this extra-row procedure takes $n$ rotations and $O(n^2)$ arithmetic operations. Locality does not guarantee preservation of sparsity: mixing rows can create nonzeros wherever either row had one. For a fully populated $m\times n$ input, the straightforward elimination uses ${\sum }_{k=1}^n(m-k)$ rotations, each touching $n-k+1$ columns, giving $O(m{n}^2)$ work. Storage layout, zero patterns and implementation affect speed; this count alone establishes no universal advantage over other QR decomposition algorithms.

The Euclidean norm preservation explains the appeal for numerical stability: exact rotations do not amplify a perturbation's length. Rounded coefficients and Givens row-pair updates still introduce roundoff error. Small reconstruction and orthogonality residuals check the computed factors, while the condition number of the original problem still controls how accurately a linear least-squares problem determines its solution.

Sparse Gaussian elimination solves a linear system by Gaussian elimination while exploiting its sparse matrix structure to avoid storing zeros and performing unnecessary arithmetic operations. The catch is that zeros in the input need not remain zero: eliminating a shared unknown can couple two equations that previously had no direct coupling. These new positions are fill-in, and their number can depend strongly on the order of elimination.

For a real symmetric positive-definite matrix, the usual direct method is a Cholesky decomposition. Choose an ordering, use symbolic factorization to allocate the predicted factor entries, compute their values, then solve two triangular systems. Compressed sparse storage makes the reduced storage useful. General LU decomposition may also need numerical partial pivoting, so an ordering chosen for sparsity can change during factorization.^[ref]

For a fixed ordering, let $P$ be the permutation matrix that puts the unknowns in that order. Permute both rows and columns:Here $C$ is the lower triangular matrix in the Cholesky decomposition of $B$; $z=Px$. Forward substitution and back substitution give the original solution. The simultaneous permutation preserves the symmetric positive-definite matrix property because $v^{T}Bv=(P^{T}v{)}^{T}A(P^{T}v)>0$ for every nonzero vector $v$. A row permutation alone generally loses the symmetric matrix structure.

The local mechanism is a Schur complement. If the active pivot element is $a_{kk}$, then for remaining indices $i,j$,Even when $a_{ij}=0$, two nonzero couplings through $k$ make the second term nonzero. Thus the elimination graph removes $k$ and joins its remaining neighbors into a clique. The fill-path criterion for symmetric elimination extends this local rule across several steps. The elimination tree organizes the resulting dependencies; the minimum degree algorithm and nested dissection choose orderings to control them.

This connection between graphs and linear systems was already explicit in Donald J. Rose, R. Endre Tarjan and George S. Lueker's 1976 study of fill-in and perfect elimination orderings.^[ref]

Symbolic factorization computes possible nonzero positions and update dependencies from a sparse matrix pattern and ordering before calculating numerical factor values. The elimination graph predicts fill and the elimination tree organizes dependencies. Numerical pivoting or exceptional cancellation can make the computed factor pattern differ from this structural prediction.