A matrix over the type f with m rows and n columns. This just wraps the Matrix constructor and adds size information to the type

Display a matrix as a String using the Show instance of its elements.

O(rows*cols). Similar to force. It copies the matrix content dropping any extra memory.

Useful when using submatrix from a big matrix.

O(rows*cols). Generate a matrix from a generator function. | The elements are 1-indexed, i.e. top-left element is (1,1). Example of usage:

matrix (\(i,j) -> 2*i - j) :: Matrix 2 4 Int
( 1  0 -1 -2 )
( 3  2  1  0 )

O(1). Represent a vector as a one row matrix.

O(1). Represent a vector as a one row matrix.

O(rows*cols). The zero matrix This produces a zero matrix of the size given by the type. Often, the correct dimensions can be inferred by the compiler. If you want a specific size, give a type.

zero :: Matrix 2 2 Int
( 0 0 )
( 0 0 )

O(rows*cols). Identity matrix

identitiy @n =
( 1 0 0 ... 0 0 )
( 0 1 0 ... 0 0 )
(       ...     )
( 0 0 0 ... 1 0 )
( 0 0 0 ... 0 1 )

Similar to diagonalList, but using Vector, which should be more efficient. The size of the vector is not checked and will lead to an exception if it's not of size n.

Similar to diagonalList, but using Vector, which should be more efficient. The size of the vector is not checked and will lead to an exception if it's not of size n.

O(rows*cols). Permutation matrix. The parameters are given as type level Nats. To use this, use -XDataKinds and -XTypeApplications. The first type parameter gives the matrix' size, the two following give the rows (or columns) to permute.

permMatrix @n @i @j =
              i     j       n
  1 ( 1 0 ... 0 ... 0 ... 0 0 )
  2 ( 0 1 ... 0 ... 0 ... 0 0 )
    (     ...   ...   ...     )
  i ( 0 0 ... 0 ... 1 ... 0 0 )
    (     ...   ...   ...     )
  j ( 0 0 ... 1 ... 0 ... 0 0 )
    (     ...   ...   ...     )
    ( 0 0 ... 0 ... 0 ... 1 0 )
  n ( 0 0 ... 0 ... 0 ... 0 1 )

When i == j it reduces to identity n.

O(rows*cols). Permutation matrix. The values of the row and column identifiers are not checked and if they are out of range (not between 1 and n) an exception will be thrown.

permMatrixUnsafe @n i j =
              i     j       n
  1 ( 1 0 ... 0 ... 0 ... 0 0 )
  2 ( 0 1 ... 0 ... 0 ... 0 0 )
    (     ...   ...   ...     )
  i ( 0 0 ... 0 ... 1 ... 0 0 )
    (     ...   ...   ...     )
  j ( 0 0 ... 1 ... 0 ... 0 0 )
    (     ...   ...   ...     )
    ( 0 0 ... 0 ... 0 ... 1 0 )
  n ( 0 0 ... 0 ... 0 ... 0 1 )

When i == j it reduces to identity n.

Create a matrix from a list of elements. The list must have exactly length n*m or this returns Nothing. An example:

fromList [1..9] :: Maybe (Matrix 3 3 Int)
Just ( 1 2 3 )
     ( 4 5 6 )
     ( 7 8 9 )

Create a matrix from a non-empty list given the desired size. The list must have at least rows*cols elements. An example:

fromListUnsafe [1..9] :: Matrix 3 3 Int
( 1 2 3 )
( 4 5 6 )
( 7 8 9 )

Create a matrix from a list of rows. The list must have exactly m lists of length n. Nothing is returned otherwise Example:

fromLists [ [1,2,3]      ( 1 2 3 )
          , [4,5,6]      ( 4 5 6 )
          , [7,8,9] ] =  ( 7 8 9 )

Create a matrix from a list of rows. The list must have exactly m lists of length n. If this does not hold, the resulting Matrix will have different static dimensions that the runtime dimension and will result in hard to debug errors. Use fromLists whenever you're unsure. Example:

fromListsUnsafe [ [1,2,3]      ( 1 2 3 )
                , [4,5,6]      ( 4 5 6 )
                , [7,8,9] ] =  ( 7 8 9 )

Get the elements of a matrix stored in a list.

       ( 1 2 3 )
       ( 4 5 6 )
toList ( 7 8 9 ) = [1..9]

Get the elements of a matrix stored in a list of lists, where each list contains the elements of a single row.

        ( 1 2 3 )   [ [1,2,3]
        ( 4 5 6 )   , [4,5,6]
toLists ( 7 8 9 ) = , [7,8,9] ]

O(1). Get an element of a matrix. Indices range from (1,1) to (m,n). The parameters are given as type level Nats. To use this, use -XDataKinds and -XTypeApplications.

The type parameters are: row, column

Example:

              ( 1 2 )
getElem @2 @1 ( 3 4 ) = 3

Short alias for unsafeGet. Careful: This has no bounds checking This deviates from Data.Matrix, where (!) does check bounds on runtime.

O(1). Unsafe variant of getElem. This will do no bounds checking

Alias for '(!)'. This exists to keep the interface similar to Data.Matrix but serves no other purpose. Use '(!)' (or even better getElem) instead.

Variant of unsafeGet that returns Maybe instead of an error.

Variant of setElem that returns Maybe instead of an error.

O(1). Get a row of a matrix as a vector. The range of the input is not checked and must be between 1 and m

O(1). Get a column of a matrix as a vector. The range of the input is not checked and must be between 1 and n

Varian of getRow that returns a maybe instead of an error Only available when used with matrix >= 0.3.6!

Variant of getCol that returns a maybe instead of an error Only available when used with matrix >= 0.3.6!

O(min rows cols). Diagonal of a not necessarily square matrix.

O(rows*cols). Transform a Matrix to a Vector of size rows*cols. This is equivalent to get all the rows of the matrix using getRow and then append them, but far more efficient.

Type safe matrix multiplication This is called (*) in matrix. Since the dimensions of the input matrices differ, they are not the same type and we cannot use Num's (*)

Type safe scalar multiplication

Replace the value of a cell in a matrix. The position to be replaced is given by TypeLevel Nats. To use this, use -XDataKinds and -XTypeApplications.

Example: setElem 1 2 0 (1 2 3) = (1 0 3)

Unsafe variant of setElem, without bounds checking.

O(rows*cols). The transpose of a matrix. Example:

          ( 1 2 3 )   ( 1 4 7 )
          ( 4 5 6 )   ( 2 5 8 )
transpose ( 7 8 9 ) = ( 3 6 9 )

Set the size of a matrix to given parameters. Use a default element for undefined entries if the matrix has been extended.

Extend a matrix to the expected size adding a default element. If the matrix already has the required size, nothing happens. Example:

                             ( 1 2 3 0 0 )
                 ( 1 2 3 )   ( 4 5 6 0 0 )
                 ( 4 5 6 )   ( 7 8 9 0 0 )
extendTo @4 @5 0 ( 7 8 9 ) = ( 0 0 0 0 0 )

O(rows^4). The inverse of a square matrix Uses naive Gaussian elimination formula.

O(rows*rows*cols*cols). Converts a matrix to reduced row echelon form, thus solving a linear system of equations. This requires that (cols > rows) if cols < rows, then there are fewer variables than equations and the problem cannot be solved consistently. If rows = cols, then it is basically a homogenous system of equations, so it will be reduced to identity or an error depending on whether the marix is invertible (this case is allowed for robustness).

O(rows*cols). Map a function over a row. The row to map is given by a TypeLevel Nat. To use this, use -XDataKinds and -XTypeApplications. Example:

                          ( 1 2 3 )   ( 1 2 3 )
                          ( 4 5 6 )   ( 5 6 7 )
mapRow @2 (\_ x -> x + 1) ( 7 8 9 ) = ( 7 8 9 )

O(rows*cols). Map a function over a row. The bounds of the row parameter is not checked and might throw an error. Example:

                               ( 1 2 3 )   ( 1 2 3 )
                               ( 4 5 6 )   ( 5 6 7 )
mapRowUnsafe (\_ x -> x + 1) 2 ( 7 8 9 ) = ( 7 8 9 )

O(rows*cols). Map a function over a column. The row to map is given by a TypeLevel Nat. To use this, use -XDataKinds and -XTypeApplications. Example:

                          ( 1 2 3 )   ( 1 3 3 )
                          ( 4 5 6 )   ( 4 6 6 )
mapCol @2 (\_ x -> x + 1) ( 7 8 9 ) = ( 7 9 9 )

O(rows*cols). Map a function over a column. The bounds of the row parameter is not checked and might throw an error. Example:

                               ( 1 2 3 )   ( 1 3 3 )
                               ( 4 5 6 )   ( 4 6 6 )
mapColUnsafe (\_ x -> x + 1) 2 ( 7 8 9 ) = ( 7 9 9 )

O(rows*cols). Map a function over elements. Example:

                           ( 1 2 3 )   ( 0 -1 -2 )
                           ( 4 5 6 )   ( 1  0 -1 )
mapPos (\(r,c) _ -> r - c) ( 7 8 9 ) = ( 2  1  0 )

Only available when used with matrix >= 0.3.6!

O(1). Extract a submatrix from the given position. The type parameters expected are the starting and ending indices of row and column elements.

O(1). Extract a submatrix from the given position. The type parameters are the dimension of the returned matrix, the run-time indices are the indiced of the top-left element of the new matrix. Example:

                          ( 1 2 3 )
                          ( 4 5 6 )   ( 2 3 )
submatrixUnsafe @2 @2 1 2 ( 7 8 9 ) = ( 5 6 )

O(rows*cols). Remove a row and a column from a matrix. Example:

                  ( 1 2 3 )
                  ( 4 5 6 )   ( 1 3 )
minorMatrix @2 @2 ( 7 8 9 ) = ( 7 9 )

O(rows*cols). Remove a row and a column from a matrix. Example:

                      ( 1 2 3 )
                      ( 4 5 6 )   ( 1 3 )
minorMatrixUnsafe 2 2 ( 7 8 9 ) = ( 7 9 )

O(1). Make a block-partition of a matrix using a given element as reference. The element will stay in the bottom-right corner of the top-left corner matrix. This means, the ranges of the pivot elements positions are \[ i <- [1..m-1], j <- [1..n-1] \]

                  (             )   ( TR   | TL   )
                  (             )   ( ...  | ...  )
                  (    x        )   (    x |      )
splitBlocks @i @j (             ) = (-------------) , where x = a_{i,j}
                  (             )   ( BL   | BR   )
                  (             )   ( ...  | ...  )
                  (             )   (      |      )

Note that contrary to the matrix version of this function, blocks will never be empty. Also, because of TypeLits not providing proper dependent types, there is no way to have a type safe variant of this functon where the pivot element is given at run-time.

Horizontally join two matrices. Visually:

( A ) <|> ( B ) = ( A | B )

Horizontally join two matrices. Visually:

                  ( A )
( A ) <-> ( B ) = ( - )
                  ( B )

Join blocks of the form detailed in splitBlocks. Precisely:

joinBlocks (tl,tr,bl,br) =
  (tl <|> tr)
      <->
  (bl <|> br)

Perform an operation element-wise. This uses matrix's elementwiseUnsafe since we can guarantee proper dimensions at compile time.

Standard matrix multiplication by definition.

Standard matrix multiplication by definition.

Strassen's matrix multiplication.

Mixed Strassen's matrix multiplication.

Scale a matrix by a given factor. Example:

              ( 1 2 3 )   (  2  4  6 )
              ( 4 5 6 )   (  8 10 12 )
scaleMatrix 2 ( 7 8 9 ) = ( 14 16 18 )

Scale a row by a given factor. The input row is not checked for validity. Example:

              ( 1 2 3 )   (  1  2  3 )
              ( 4 5 6 )   ( 12 15 18 )
scaleRow @2 3 ( 7 8 9 ) = (  7  8  9 )

Scale a row by a given factor. The input row is not checked for validity. Example:

                   ( 1 2 3 )   (  1  2  3 )
                   ( 4 5 6 )   ( 12 15 18 )
scaleRowUnsafe 3 2 ( 7 8 9 ) = (  7  8  9 )

Add to one row a scalar multiple of another row. Example:

                    ( 1 2 3 )   (  1  2  3 )
                    ( 4 5 6 )   (  6  9 12 )
combineRows @2 @1 2 ( 7 8 9 ) = (  7  8  9 )

Add to one row a scalar multiple of another row. Example:

                        ( 1 2 3 )   (  1  2  3 )
                        ( 4 5 6 )   (  6  9 12 )
combineRowsUnsafe 2 2 1 ( 7 8 9 ) = (  7  8  9 )

Switch two rows of a matrix. Example:

                 ( 1 2 3 )   ( 4 5 6 )
                 ( 4 5 6 )   ( 1 2 3 )
switchRows @1 @2 ( 7 8 9 ) = ( 7 8 9 )

Switch two rows of a matrix. The validity of the input row numbers is not checked Example:

                     ( 1 2 3 )   ( 4 5 6 )
                     ( 4 5 6 )   ( 1 2 3 )
switchRowsUnsafe 1 2 ( 7 8 9 ) = ( 7 8 9 )

Switch two coumns of a matrix. Example:

                 ( 1 2 3 )   ( 2 1 3 )
                 ( 4 5 6 )   ( 5 4 6 )
switchCols @1 @2 ( 7 8 9 ) = ( 8 7 9 )

Switch two coumns of a matrix. The validity of the input column numbers is not checked. Example:

                     ( 1 2 3 )   ( 2 1 3 )
                     ( 4 5 6 )   ( 5 4 6 )
switchColsUnsafe 1 2 ( 7 8 9 ) = ( 8 7 9 )

Matrix LU decomposition with partial pivoting. The result for a matrix M is given in the format (U,L,P,d) where:

• U is an upper triangular matrix.
• L is an unit lower triangular matrix.
• P is a permutation matrix.
• d is the determinant of P.
• PM = LU.

These properties are only guaranteed when the input matrix is invertible. An additional property matches thanks to the strategy followed for pivoting:

• L_(i,j) <= 1, for all i,j.

This follows from the maximal property of the selected pivots, which also leads to a better numerical stability of the algorithm.

Example:

         ( 1 2 0 )     ( 2 0  2 )   (   1 0 0 )   ( 0 0 1 )
         ( 0 2 1 )     ( 0 2 -1 )   ( 1/2 1 0 )   ( 1 0 0 )
luDecomp ( 2 0 2 ) = ( ( 0 0  2 ) , (   0 1 1 ) , ( 0 1 0 ) , 1 )

Nothing is returned if no LU decomposition exists.

Unsafe version of luDecomp. It fails when the input matrix is singular.

Matrix LU decomposition with complete pivoting. The result for a matrix M is given in the format (U,L,P,Q,d,e) where:

• U is an upper triangular matrix.
• L is an unit lower triangular matrix.
• P,Q are permutation matrices.
• d,e are the determinants of P and Q respectively.
• PMQ = LU.

These properties are only guaranteed when the input matrix is invertible. An additional property matches thanks to the strategy followed for pivoting:

• L_(i,j) <= 1, for all i,j.

This follows from the maximal property of the selected pivots, which also leads to a better numerical stability of the algorithm.

Example:

          ( 1 0 )    ( 2 1 )  (   1    0 0 )  ( 0 0 1 )
          ( 0 2 )    ( 0 2 )  (   0    1 0 )  ( 0 1 0 )  ( 1 0 )
luDecomp' ( 2 1 ) = (( 0 0 ), ( 1/2 -1/4 1 ), ( 1 0 0 ), ( 0 1 ), -1 , 1 )

Nothing is returned if no LU decomposition exists.

Unsafe version of luDecomp'. It fails when the input matrix is singular.

Simple Cholesky decomposition of a symmetric, positive definite matrix. The result for a matrix M is a lower triangular matrix L such that:

• M = LL^T.

Example:

           (  2 -1  0 )   (  1.41  0     0    )
           ( -1  2 -1 )   ( -0.70  1.22  0    )
cholDecomp (  0 -1  2 ) = (  0.00 -0.81  1.15 )

Sum of the elements in the diagonal. See also getDiag. Example:

      ( 1 2 3 )
      ( 4 5 6 )
trace ( 7 8 9 ) = 15

Product of the elements in the diagonal. See also getDiag. Example:

         ( 1 2 3 )
         ( 4 5 6 )
diagProd ( 7 8 9 ) = 45

Matrix determinant using Laplace expansion. If the elements of the Matrix are instance of Ord and Fractional consider to use detLU in order to obtain better performance. Function detLaplace is extremely slow.

Matrix determinant using LU decomposition. It works even when the input matrix is singular.

Flatten a matrix of matrices.

Apply a map function to the unsafe inner matrix type.

Transform a binary unstatic function to a binary static function.

Forget static information about a matrix. This converts this converts the Matrix type to Data.Matrix.Matrix