Heptadiagonal matrix

In linear algebra, a heptadiagonal matrix is a matrix that is nearly diagonal; to be exact, it is a matrix in which the only nonzero entries are on the main diagonal, and the first three diagonals above and below it. So it is of the form

${\displaystyle {\begin{bmatrix}B_{11}&B_{12}&B_{13}&B_{14}&0&\cdots &0&0\\B_{21}&B_{22}&B_{23}&B_{24}&B_{25}&0&\ddots &0\\B_{31}&B_{32}&B_{33}&B_{34}&B_{35}&B_{36}&\ddots &\vdots \\B_{41}&B_{42}&B_{43}&B_{44}&B_{45}&B_{46}&B_{47}&0\\0&B_{52}&B_{53}&B_{54}&B_{55}&B_{56}&B_{57}&B_{58}\\\vdots &\ddots &B_{63}&B_{64}&B_{65}&B_{66}&B_{67}&B_{68}\\0&\cdots &0&B_{74}&B_{75}&B_{76}&B_{77}&B_{78}\\0&0&\cdots &0&B_{85}&B_{86}&B_{87}&B_{88}\end{bmatrix}}}$

It follows that a heptadiagonal matrix has at most ${\displaystyle 7n-12}$ nonzero entries, where n is the size of the matrix. Hence, heptadiagonal matrices are sparse. This makes them useful in numerical analysis.