Installation
Before installing the Haskell bindings
you need to install the BLAS and LAPACK packages.
Please note, that additionally to the reference implementation in FORTRAN 77
there are alternative optimized implementations
like OpenBLAS, ATLAS, Intel MKL.
Debian, Ubuntu
sudo apt-get install libblas-dev liblapack-dev
MacOS
You may install pkgconfig and LAPACK via MacPorts:
sudo port install pkgconfig lapack
However, the pkg-config files for LAPACK
seem to be installed in a non-standard location.
You must make them visible to pkg-config by
export PKG_CONFIG_PATH=/opt/local/lib/lapack/pkgconfig:$PKG_CONFIG_PATH
You may set the search PATH permanently by adding
the above command line to your $HOME/.profile file.
Alternatively, a solution for all users of your machine
would be to set symbolic links:
sudo ln -s /opt/local/lib/lapack/pkgconfig/blas.pc /opt/local/lib/pkgconfig/blas.pc
sudo ln -s /opt/local/lib/lapack/pkgconfig/lapack.pc /opt/local/lib/pkgconfig/lapack.pc
Introduction
Here is a small example for constructing and formatting matrices:
Prelude> import qualified Numeric.LAPACK.Matrix as Matrix
Prelude Matrix> import Numeric.LAPACK.Format as Fmt ((##))
Prelude Matrix Fmt> let a = Matrix.fromList (Matrix.shapeInt 3) (Matrix.shapeInt 4) [(0::Float)..]
Prelude Matrix Fmt> a ## "%.4f"
0.0000 1.0000 2.0000 3.0000
4.0000 5.0000 6.0000 7.0000
8.0000 9.0000 10.0000 11.0000
Prelude Matrix Fmt> import qualified Numeric.LAPACK.Matrix.Shape as MatrixShape
Prelude Matrix Fmt MatrixShape> import qualified Numeric.LAPACK.Matrix.Triangular as Triangular
Prelude Matrix Fmt MatrixShape Triangular> let u = Triangular.upperFromList MatrixShape.RowMajor (Matrix.shapeInt 4) [(0::Float)..]
Prelude Matrix Fmt MatrixShape Triangular> (u, Triangular.transpose u) ## "%.4f"
0.0000 1.0000 2.0000 3.0000
4.0000 5.0000 6.0000
7.0000 8.0000
9.0000
0.0000
1.0000 4.0000
2.0000 5.0000 7.0000
3.0000 6.0000 8.0000 9.0000
You may find a more complex introductory example at:
http://code.henning-thielemann.de/bob2019/main.pdf
We do not try to do fancy formatting for the Show instance.
The Show instances of matrices emit plain valid Haskell code in one line
where all numbers are printed in full precision.
If matrices are part of larger Haskell data structures
this kind of formatting works best.
For human-friendly formatting in GHCi you need to add something like ## "%.4f"
after a matrix or vector expression.
It means: Print all numbers in fixed point representation
using four digits for the fractional part.
You can use the formatting placeholders provided by printf.
The matrices have Hyper instances for easy usage in
HyperHaskell.
Formatting is based on the Output type class
that currently supports output as
Text boxes
for GHCi and
HTML
for HyperHaskell.
You may tell GHCi to use Format class instead of Show:
Fmt> let lapackPrint x = x##"%.3f"
Fmt> :set -interactive-print lapackPrint
You may permanently configure this one in your local .ghci file.
If you want to display values via Show class,
you can always fall back by:
Fmt> print "Hello"
Matrix vs. Vector
Vectors are Storable.Arrays from the
comfort-array package.
An array can have a fancy shape
like a shape defined by an enumeration type,
the shape of two appended arrays,
a shape that is compatible to a Haskell container type,
a rectangular or triangular shape.
All operations check dynamically
whether corresponding shapes match structurally.
E.g. a|||b === c|||d composes a matrix from four quadrants a, b, c, d.
It is not enough that a|||b and c|||d have the same width,
but the widths of a and c as well as b and d must match.
The type variables for shapes show which dimensions must be compatible.
We recommend to use type variables for the shapes as long as possible
because this increases type safety even
if you eventually use only ShapeInt.
If you use statically sized shapes you get static size checks.
A matrix can have any of these shapes as height and as width.
E.g. it is no problem to define a matrix
that maps a triangular shaped array to a rectangular shaped one.
There are actually practical applications to such matrices.
A matrix can be treated as vector, but there are limitations.
E.g. if you scale a Hermitian matrix by a complex factor
it will in general be no longer Hermitian.
Another problem: Two equally sized rectangular matrices
may differ in the element order (row major vs. column major).
You cannot simply add them by adding the flattened arrays element-wise.
Thus if you want to perform vector operations on a matrix
the package requires you to "unpack" a matrix to a vector
using Matrix.Array.toVector.
This conversion is almost a no-op and preserves most of the shape information.
The reverse operation is Matrix.Array.fromVector.
There are more matrix types that are not based on a single array.
E.g. we provide a symbolic inverse, a scaling matrix, a permutation matrix.
We also support arrays that represent factors of a matrix factorization.
You obtain these by LU and QR decompositions.
You can extract the matrix factors of it,
but you can also multiply the factors to other matrices
or use the decompositions for solving simultaneous linear equations.
Matrix type parameters
LAPACK supports a variety of special matrix types,
e.g. dense, banded, triangular, symmetric, Hermitian matrices
and our Haskell interface supports them, too.
There are two layers:
The low level layer addresses how matrices are stored for LAPACK.
Matrices and vectors are stored in the Array type
from comfort-array:Data.Array.Comfort.Storable
using shape types from Numeric.LAPACK.Matrix.Layout.
The high level layer provides a matrix type
in Numeric.LAPACK.Matrix.Array
with mathematically relevant type parameters.
The matrix type is:
ArrayMatrix pack property lower upper meas vert horiz height width a
The type parameters are from right to left:
-
a - the element type
-
height and width are the vertical and horizontal shapes of the matrix
-
meas vert horiz form a group with following possible assignments:
| meas |
vert |
horiz |
name |
condition |
Shape |
Small |
Small |
Square matrix |
height == width |
Size |
Small |
Small |
Liberal square |
size height == size width |
Size |
Big |
Small |
Tall matrix |
size height >= size width |
Size |
Small |
Big |
Wide matrix |
size height <= size width |
Size |
Big |
Big |
General matrix |
arbitrary height and width |
Think of meas as the measurement
that goes into the relation of dimensions.
You can either compare shapes (meas ~ Shape)
or their sizes (meas ~ Size).
For vert and horiz the possible values mean:
-
Small: The corresponding dimension is equal
to the minimum of height and width.
-
Big: The corresponding dimension has no further restrictions,
but it is of course at least the minimum of height and width.
The names Small and Big fit best to tall and wide matrices.
The remaining combinations Small Small for Square
and Big Big for General appear to be arbitrary, but they help to
e.g. treat square and tall matrices the same way, where sensible.
Turning Shape into Size or Small into Big
relaxes a dimension relation.
-
lower upper count the numbers of non-zero off-diagonals.
Off course, stored off-diagonals can consist entirely of zeros.
Thus more precisely we have to say, that lower and upper tell
that all values outside the numbered bands are zero.
lower and upper can be:
-
Filled - no restriction on the number of off-diagonals.
-
Bands n,
where n is a natural number unarily encoded in types.
Empty is a synonym for Bands U0.
-
property can be
-
Arbitrary -
this type does not make any further promises about the matrix elements
-
Unit - matrix is triangular with a unit diagonal
It can be used for matrices
that always have a unit diagonal by construction.
This property is preserved by some operations
and enables optimizations by LAPACK.
Solving with a unit triangular matrix does not require division
and thus cannot fail due to division by zero.
-
Symmetric - matrix is symmetric
-
Hermitian - matrix is Hermitian (also supported for real elements)
The internal Hermitian property also has three type tags neg zero pos
to restrict the range of values of bilinear-forms.
This way you can denote positive definiteness and semidefiniteness.
-
pack can Packed or Unpacked.
Unpacked means that the full matrix
bounded by height and width is stored.
Packed format is supported
for triangular, symmetric, Hermitian and banded matrices.
For banded matrices you should always prefer the packed format.
For triangular, symmetric and Hermitian matrices
LAPACK does not always support packed format natively
and our Haskell interface converts forth and back silently.
I also think that unpacked triangular formats
enjoy better support by vectorized block algorithms.
Thus, the unpacked triangular format may be better for performance
although it requires twice as much space as the packed format.
The packed triangular format however is still
the default format for conversion to and from lists,
because this prevents the user from declaring non-zero values
in the empty area of a triangular matrix.
Let us examine some examples:
-
Diagonal matrix:
ArrayMatrix Packed Arbitrary Empty Empty Shape Small Small sh sh a
-
Packed upper triangular matrix:
ArrayMatrix Packed Arbitrary Empty Filled Shape Small Small sh sh a
-
Unpacked unit lower triangular matrix:
ArrayMatrix Unpacked Unit Filled Empty Shape Small Small sh sh a
-
Complex-valued symmetric matrix:
ArrayMatrix Packed Symmetric Filled Filled Shape Small Small sh sh (Complex a)
-
Tall banded matrix:
ArrayMatrix Packed Arbitrary (Bands sub) (Bands super) Size Big Small height width a
The type tags have a mathematical meaning
and this pays off for operations on matrices.
E.g. matrix multiplication adds off-diagonals.
Matrix inversion fills non-zero triangular matrix parts.
The five supported relations for matrix dimensions are transitive,
and thus matrix multiplication maintains a dimension relation,
e.g. tall times tall is tall.
Please note, that not all type parameter combinations are supported.
Some restrictions are dictated by mathematics,
e.g. Hermitian matrices must always be square,
matrices with unit diagonal must always be triangular and so on.
Some combinations are simply not supported, because they do not add value.
E.g. a (square) diagonal matrix is always symmetric
but we allow Symmetric only together with Filled.
Forbidden combinations are often not prevented at the type level,
but you will not be able to construct a matrix of a forbidden type.
Infix operators
The package provides fancy infix operators like #*| and \*#.
They symbolize both operands and operations.
E.g. in #*| the hash means Matrix, the star means Multiplication
and the bar means Column Vector.
Possible operations are:
-
a_*_b - a multiplied by b
-
a_/_b - a multiplied by inverse b
-
a_\_b - inverse a multiplied by b
Possible operands are:
For multiplication of equally shaped matrices
we also provide instances of Semigroup.<>.
Precedence of the operators is chosen analogously to plain * and /.
Associativity is chosen such that the same operator can be applied
multiple times without parentheses.
But sometimes this may mean that you have to mix
left and right associative operators,
and thus you may still need parentheses.
Type errors
You might encounter cryptic type errors
that refer to the encoding of particular matrix types
via matrix type parameters.
E.g. the error
Couldn't match type `Numeric.LAPACK.Matrix.Extent.Big`
with `Numeric.LAPACK.Matrix.Extent.Small`
may mean that you passed Square
where General or Tall was expected.
You may solve the problem with a function
like Square.toFull or Square.fromFull.
The error
Couldn't match type `Type.Data.Bool.False`
with `Type.Data.Bool.True`
most likely refers to non-matching definiteness warranties
in a Hermitian matrix.
You may try a function like Hermitian.assureFullRank
or Hermitian.relaxIndefinite to fix the issue.
