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Memory layout of matrices
Different from ndarray format which consists of shape and strides, matrix format in LAPACK consists of row or column size and leading dimension.
ndarray format and stride
Let us consider 3-dimensional array for explaining ndarray structure.
The address of (x,y,z)-element in ndarray satisfies following relation:
shape = [Nx, Ny, Nz]
where Nx > 0, Ny > 0, Nz > 0
stride = [Sx, Sy, Sz]
&data[(x, y, z)] = &data[(0, 0, 0)] + Sx*x + Sy*y + Sz*z
for x < Nx, y < Ny, z < NzThe array is called
- C-continuous if
[Sx, Sy, Sz] = [Nz*Ny, Nz, 1] - F(Fortran)-continuous if
[Sx, Sy, Sz] = [1, Nx, Nx*Ny]
Strides of ndarray [Sx, Sy, Sz] take arbitrary value,
e.g. it can be non-ordered Sy > Sx > Sz, or can be negative Sx < 0.
If the minimum of [Sx, Sy, Sz] equals to 1,
the value of elements fills data memory region and called “continuous”.
Non-continuous ndarray is useful to get sub-array without copying data.
Matrix layout for LAPACK
LAPACK interface focuses on the linear algebra operations for F-continuous 2-dimensional array.
Under this restriction, stride becomes far simpler; we only have to consider the case [1, S]
This S for a matrix A is called “leading dimension of the array A” in LAPACK document, and denoted by lda.