# HSL

These are the beginnings of a set of interfaces to HSL packages for sparse linear algebra.

Certain HSL packages are freely available to all, others are freely available to academics only. Please refer to the website above for licensing information. In all cases, users are responsible for obtaining HSL packages.

## Installing

```
julia> ]
pkg> add HSL
```

At this point, make sure that there isn't a stray METIS library on your library
path. You especially want to make sure that METIS 5 is not accessible because
the HSL library currently interfaced only supports METIS 4. If you have such
library accessible, it is important to remove it from the library path, at
least temporarily. For example, if you are on OSX and are using Homebrew, you
can hide METIS 5 with `brew unlink metis`

. After the install procedure is
complete, it is fine to link `metis`

again with `brew link metis`

.

Set the environment variables `HSL_MA57_PATH`

and `HSL_MA97_PATH`

to specify where the source archives `tar.gz`

are stored.
The `HSL`

Julia module will take care of compilation.
Once the source archives have been placed in the locations indicated by the environment variables, run

```
julia> ]
pkg> build HSL
pkg> test HSL
```

Note that a C and Fortran compilers are required.

## Supported Packages

### HSL_MA97

HSL_MA97 version 2.6.0: an OpenMP-based direct solver for symmetric linear systems. Example:

```
using MatrixMarket
using HSL
K = MatrixMarket.mmread("K.mtx") # only the lower triangle
rhs = readdlm("rhs.rhs")
LBL = Ma97(K)
ma97_factorize!(LBL)
x = ma97_solve(LBL, rhs) # or x = LBL \ rhs
```

#### Rectangular Systems

There is a convenience interface to solve rectangular systems that complements the sparse QR factorization in Julia.

When *A* is *m*-by-*n* with *m* < *n* and has full row rank,

`(x, y) = ma97_solve(A, b)`

solves for the minimum-norm solution, i.e., *x* such that *Ax = b* and *x +
Aᵀ y = 0*. The call

`(x, y) = ma97_min_norm(A, b)`

is also defined, and is equivalent to the above.

When *m* > *n* and has full column rank,

`(r, x) = ma97_solve(A, b)`

solves for the least-squares solution, i.e., *x* such that *r = b - Ax*
satisfies *Aᵀ r = 0*. The call

`(r, x) = ma97_least_squares(A, b)`

is also defined, and is equivalent to the above.

### HSL_MA57

HSL_MA57 version 5.2.0: a sparse, multifrontal solver for symmetric linear systems. Example:

```
using MatrixMarket
using HSL
K = MatrixMarket.mmread("examples/K_0.mtx") # only the lower triangle
rhs = readdlm("examples/rhs_0.rhs")
rhss = hcat(rhs, rhs)
## factorize and solve
LDL = Ma57(K)
ma57_factorize(LDL)
LDL.info.rank
x = ma57_solve(LDL, rhs) # or x = LBL \ rhs
norm(K*x - rhs)
xx = ma57_solve(LDL, rhss) # or x = LBL \ rhss
```

#### Rectangular Systems

There is a convenience interface to solve rectangular systems that complements the sparse QR factorization in Julia.

When *A* is *m*-by-*n* with *m* < *n* and has full row rank,

`(x, y) = ma57_solve(A, b)`

solves for the minimum-norm solution, i.e., *x* such that *Ax = b* and *x +
Aᵀ y = 0*. The call

`(x, y) = ma57_min_norm(A, b)`

is also defined, and is equivalent to the above.

When *m* > *n* and has full column rank,

`(r, x) = ma57_solve(A, b)`

solves for the least-squares solution, i.e., *x* such that *r = b - Ax*
satisfies *Aᵀ r = 0*. The call

`(r, x) = ma57_least_squares(A, b)`

is also defined, and is equivalent to the above. Example:

```
using MatrixMarket
using HSL
K = MatrixMarket.mmread("examples/K_0.mtx") # only the lower triangle
rhs = readdlm("examples/rhs_0.rhs")
## solve min norm
K_mn = K[1:200,:]
x_mn, y_mn = ma57_min_norm(K_mn, rhs[1:200]) # == ma57_solve(K_mn, rhs[1:200])
## solve least squares
K_ls = K[:,1:200]
r_ls, x_ls = ma57_least_squares(K_ls, rhs) # == ma57_solve(K_ls, rhs)
```