The following is an explanation of the items presented as *Matrix Statistics*
on individual matrix home pages.

This gives the number of rows (M) and columns (N) in the matrix, expressed as N x M. The number of entries in the matrix files is also given The number of entries is equal to the number of nonzeros except when zeros are explicity represented in the matrix files. Such structural zeros are counted as entries, but not as nonzeros (see Nonzeros).

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This gives the field from which its elements are drawn, and its symmetry properties.

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This provides the number of nonzero elements on the diagonal, below the
diagonal and above the diagonal, as well as their total, n(A).
Note that the value of n(A) is the number of nonzero elements in the full
matrix, and is independent of storage mode. Also provided is the number of
nonzeros in the matrix minus its own transpose, n(A-A^{T}).
This number, provides a measure of symmetry: a symmetric matrix will have
n(A-A^{T})=0, while a matrix with no symmetric elements will
have n(A-A^{T})=n(A). ** Note:** In the notation on this page, A' denotes
the transpose of A, while A^{*} denotes the conjugate transpose
of A.

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This provides the average and standard deviation of the number of nonzeros per column, without regard to storage format. Also provided is the index and number of nonzeros in the longest and shortest columns, i.e. the columns with the largest and smallest number of nonzeros.

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This provides the average and standard deviation of the number of nonzeros per row, without regard to storage format. Also provided is the index and number of nonzeros in the longest and shortest rows, i.e., the rows with the largest and smallest number of nonzeros.

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This provides the lower and upper bandwidths, as well as the average and
standard deviation of the distance of nonzero matrix elements from the main
diagonal. The lower bandwidth, l, of a matrix A is the largest |
i-j | such that i<j and A_{ij} is nonzero. The
upper bandwidth, u, of a matrix A is the largest | i-j | such
that i>j and A_{ij} is nonzero.

The number of diagonal bands required to represent the lower triangle of A in band storage mode is l, while the number of diagonal bands required to represent the upper triangle in band storage mode is u. Thus, the amount of storage necessary to represent an n x n nonsymmetric matrix in band storage mode is approximately (l+u+1)n. For a symmetric matrix, i=u and the requirement drops to (l+1)n.

The average and standard deviation of | i-j | for nonzero
A_{ij} provides a measure of the clustering of nonzeros about
the main diagonal.

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This data presents information related to the use of profile or skyline storage
mode. In this mode, all matrix elements from the first nonzero in each row to
the last nonzero in the row are explicitly stored. The quantities relevant to
this storage mode are based on the rowwise upper and lower bandwidths.
We define l_{i} to be the lower bandwidth of row i,

We provide the following additional summary statistics.

- minimum lower bandwidth : min
_{i}[ l_{i}] - minimum upper bandwidth : min
_{i}[ u_{i}] - maximum lower bandwidth : max
_{i}[ l_{i}] - maximum upper bandwidth : max
_{i}[ u_{i}] - average lower bandwidth : average l
_{i} - average upper bandwidth : average u
_{i} - standard deviation of lower bandwidths l
_{i} - standard deviation of upper bandwidths u
_{i}

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This section provides an indication of which matrix diagonals contain the most nonzeros. These are the so-call "heaviest" diagonals. We list the 10 heaviest diagonals. The diagonal is indicated by its offset from the main diagonal; thus 0 is the main diagonal, -1 is the first lower diagonal and 2 is the second upper diagonal. For each of these diagonals we list the number of nonzeros, and the accumulated percent of matrix nonzeros found on this diagonal and all heavier ones. The number of nonvoid diagonals is also given. (A nonvoid diagonal is a diagonal with at least one nonzero entried.)

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In this section we provide numerical indicators relating to the values of matrix elements rather than nonzero structure. In particular, we give

- Frobenius norm : ||A||
_{F}= Sqrt ( sum_{ij}| A_{ij}|^{2}) - 2-norm : ||A||
_{2}= sup [ ||Ax||_{2}] over all vectors ||x||_{2}=1 - condition number : ||A
^{-1}|| ||A|| - diagonal dominance : 2|A
_{ii}| > sum_{j}|A_{ij}|.

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Last change in this page : *Auguest 20, 2001*.
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