Numerical Linear Algebra using LAPACK
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.
sudo apt-get install libblas-dev liblapack-dev
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
You may set the search PATH permanently by adding the above command line to your
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
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.
You may tell GHCi to use
Format class instead of
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
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
d. It is not enough that
c|||d have the same width, but the widths of
c as well as
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
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
widthare the vertical and horizontal shapes of the matrix
meas vert horizform a group with following possible assignments:
meas vert horiz name condition
Square matrix height == width
Liberal square size height == size width
Tall matrix size height >= size width
Wide matrix size height <= size width
General matrix arbitrary height and width
measas the measurement that goes into the relation of dimensions. You can either compare shapes (
meas ~ Shape) or their sizes (
meas ~ Size). For
horizthe possible values mean:
Small: The corresponding dimension is equal to the minimum of
Big: The corresponding dimension has no further restrictions, but it is of course at least the minimum of
Bigfit best to tall and wide matrices. The remaining combinations
Generalappear to be arbitrary, but they help to e.g. treat square and tall matrices the same way, where sensible. Turning
Bigrelaxes a dimension relation.
lower uppercount 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
uppertell that all values outside the numbered bands are zero.
Filled- no restriction on the number of off-diagonals.
Bands n, where
nis a natural number unarily encoded in types.
Emptyis a synonym for
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)
Hermitianproperty also has three type tags
neg zero posto restrict the range of values of bilinear-forms. This way you can denote positive definiteness and semidefiniteness.
Unpacked means that the full matrix bounded by
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:
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.
The package provides fancy infix operators like
\*#. 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:
inverse amultiplied by
Possible operands are:
#- a matrix that is generalized through a type class
##- a full matrix
\- a diagonal matrix represented by a
-- a row vector
|- a column vector
.- a scalar
For multiplication of equally shaped matrices we also provide instances of
Precedence of the operators is chosen analogously to plain
/. 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.
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
Tall was expected. You may solve the problem with a function like
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.relaxIndefinite to fix the issue.