BandedMats.jl

不依赖与任何包的的带状矩阵算法库, 为了后期将代码移植到GEE。
A band matrix algrithm library, which doesn't rely on any other packages, for later porting to GEE to speed up matrix solving.

相比于稀疏矩阵，带状矩阵可使Whittaker提速8倍左右。

Band Matrix

• $p$: 下行带宽 (lower binwidth)
• $q$: 上行带宽 (upper binwidth)
• $A$: 原始带状矩阵 (Matrix, or Band Matrix)
• $B$: 压缩后的带状矩阵 (Banded Matrix)

Band Matrix Storage

• lapack A[i, j] => B[j, i - j + q + 1]

• kong A[i, j] => B[i, j - i + p + 1]

Julia有一种更简洁的代码设计方式，通过修改修改getindex和setindex方式。 但GEE不支持函数重载，因此暂时没有实现这种方式。

Band Matrix Functions

• LU_band: $A = LU$
• L_solve, U_solve: $L U x = b$, $x = U^{-1} (L^{-1} b)$
• LDL_band: $A = LDL'$
• LDL_solve: $L D L' x = b$, $x = (D L')^{-1} (L^{-1} b)$
• inv_diag: fast $(L D L')^{-1}$

See also

1. BandedMatrices.jl, https://github.com/JuliaLinearAlgebra/BandedMatrices.jl

2. Band Matrix Storage, https://www.netlib.org/lapack/lug/node124.html

3. The Algorithm of Doolittle's Method for LU Decompositions, http://mathonline.wikidot.com/the-algorithm-for-doolittle-s-method-for-lu-decompositions

4. https://en.wikipedia.org/wiki/Band_matrix

5. 矩阵计算（二）：带状矩阵结构、效率及存储，https://zhuanlan.zhihu.com/p/400460201