D^      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~(c) Alberto Ruiz 2007-15BSD3 Alberto Ruiz provisionalNone 34<specialized fromIntegral specialized fromIntegralNumber of elementscreates a Vector from a list: +> fromList [2,3,5,7] 4 |> [2.0,3.0,5.0,7.0] Create a vector from a list of elements and explicit dimension. The input list is truncated if it is too long, so it may safely be used, for instance, with infinite lists. 5 |> [1..]fromList [1.0,2.0,3.0,4.0,5.0] ECreate a vector of indexes, useful for matrix extraction using '(??)'4takes a number of consecutive elements from a Vector subVector 2 3 (fromList [1..10])fromList [3.0,4.0,5.0]Reads a vector position:fromList [0..9] @> 77.00access to Vector elements without range checkingconcatenate a list of vectors*vjoin [fromList [1..5::Double], konst 1 3]*fromList [1.0,2.0,3.0,4.0,5.0,1.0,1.0,1.0]2Extract consecutive subvectors of the given sizes.)takesV [3,4] (linspace 10 (1,10::Double))3[fromList [1.0,2.0,3.0],fromList [4.0,5.0,6.0,7.0]]Ytransforms a complex vector into a real vector with alternating real and imaginary parts Xtransforms a real vector into a complex vector with alternating real and imaginary partsmap on VectorszipWith for VectorsunzipWith for Vectors'monadic map over Vectors the monad m must be strictmonadic map over Vectorsbmonadic map over Vectors with the zero-indexed index passed to the mapping function the monad m must be strictSmonadic map over Vectors with the zero-indexed index passed to the mapping functionycreates a Vector of the specified length using the supplied function to to map the index to the value at that index. 3> buildVector 4 fromIntegral 4 |> [0.0,1.0,2.0,3.0]zip for Vectors unzip for Vectors+   index of the starting elementnumber of elements to extractsourceresult *    +      (c) Alberto Ruiz 2007-15BSD3 Alberto Ruiz provisionalNone >KL clear the fpu&postfix function application (flip ($))@error codes for the auxiliary functions required by the wrappers'check the error code(postfix error code check%Error capture and conversion to Maybe!"#$%&'(!"#$%&'(!"#$%&'($%&((c) Alberto Ruiz 2007-15BSD3 Alberto Ruiz provisionalNone >KL$*uniform distribution in [0,1)+=normal distribution with mean zero and standard deviation onesum of elementssum of elementssum of elementssum of elementsproduct of elementsproduct of elementsproduct of elementsproduct of elementsUobtains different functions of a vector: norm1, norm2, max, min, posmax, posmin, etc.Uobtains different functions of a vector: norm1, norm2, max, min, posmax, posmin, etc.:obtains different functions of a vector: only norm1, norm2:obtains different functions of a vector: only norm1, norm2Uobtains different functions of a vector: norm1, norm2, max, min, posmax, posmin, etc.Uobtains different functions of a vector: norm1, norm2, max, min, posmax, posmin, etc.'map of real vectors with given function*map of complex vectors with given function'map of real vectors with given function'map of real vectors with given function'map of real vectors with given function'map of real vectors with given function'map of real vectors with given function*map of complex vectors with given function'map of real vectors with given function*map of complex vectors with given function'map of real vectors with given function'map of real vectors with given function%elementwise operation on real vectors(elementwise operation on complex vectors %elementwise operation on real vectors (elementwise operation on complex vectors %elementwise operation on CInt vectors %elementwise operation on CInt vectors-NObtains a vector of pseudorandom elements (use randomIO to get a random seed)./range 5fromList [0,1,2,3,4] )*+, !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    - distribution vector size./ )*+, !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    -./ )*+, !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    -./x (c) Alberto Ruiz 2007-15BSD3 Alberto Ruiz provisionalNone   345>KL0Supported matrix elements.1<Matrix representation suitable for BLAS/LAPACK computations.Matrix transpose.;mCreates a vector by concatenation of rows. If the matrix is ColumnMajor, this operation requires a transpose.flatten (ident 3).fromList [1.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,1.0]<the inverse of  common value with "adaptable" 1=Create a matrix from a list of vectors. All vectors must have the same dimension, or dimension 1, which is are automatically expanded.>2extracts the rows of a matrix as a list of vectors?3Creates a matrix from a list of vectors, as columns@6Creates a list of vectors from the columns of a matrixReads a matrix position.DCreates a matrix from a vector by grouping the elements in rows with the desired number of columns. (GNU-Octave groups by columns. To do it you can define reshapeF r = tr' . reshape r( where r is the desired number of rows.)reshape 4 (fromList [1..12])(3><4) [ 1.0, 2.0, 3.0, 4.0 , 5.0, 6.0, 7.0, 8.0 , 9.0, 10.0, 11.0, 12.0 ]EAapplication of a vector function on the flattened matrix elementsFCapplication of a vector function on the flattened matrices elementsG;reference to a rectangular slice of a matrix (no data copy)H!save a matrix as a 2D ASCII table0123456789:;<=>?@ABCDEFG(r0,c0) starting position(rt,ct) dimensions of submatrix input matrixresult     H)"printf" format (e.g. "%.2f", "%g", etc.) !"#0123456789:;<=>?@ABCDEFG     H0 123456789:;<=>?@ABCDEFG     H !"# (c) Alberto Ruiz 2010BSD3 Alberto Ruiz provisionalNone  34579>LI+Structures that may contain complex numbersJSupported real types$,Supported single-double precision type pairs%Ccreates a complex vector from vectors with real and imaginary parts&the inverse of  toComplexI'()*+J$,-%&./0123456789:;<=>I'()*+J I'()*+J$,-%&./01234 None <?If we use unsafePerformIO, it may not get inlined, so in a function that returns IO (which are all safe uses of app* in this module), there would be unecessary calls to unsafePerformIO or its internals.KGOnly useful since it is left associated with a precedence of 1, unlike @#, which is right associative. e.g. msomeFunction `appMatrixLen` m `appVectorLen` v `app` other `app` arguments `app` go here One could also write: c(someFunction `appMatrixLen` m `appVectorLen` v) other arguments (go here) RThis will disregard the order of the matrix, and simply return it as-is. If the order of the matrix is RowMajor, this function is identical to ;. ?KLMNOPQRS KLMNOPQRS ?KLMNOPQRSKLNP(c) Alberto Ruiz 2008BSD3 Alberto Ruiz provisionalNone  INTr0 c0 height width=TUVWXYZ[\]^_`abcdAeBCDfghijklmnEopqrFGstuvwxyz{HI|}~JKL3MNTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~/TUVWXYZ[\]^_`abcdAeBCDfghijklmnEopqrFGstuvwxyz{HI|}~JKL(c) Alberto Ruiz 2007-10BSD3 Alberto Ruiz provisionalNone  3457>L*Specification of indexes for the operator .General matrix slicing.m(4><5) [ 0, 1, 2, 3, 4 , 5, 6, 7, 8, 9 , 10, 11, 12, 13, 14 , 15, 16, 17, 18, 19 ]m ?? (Take 3, DropLast 2)(3><3) [ 0, 1, 2 , 5, 6, 7 , 10, 11, 12 ]m ?? (Pos (idxs[2,1]), All)(2><5) [ 10, 11, 12, 13, 14 , 5, 6, 7, 8, 9 ]+m ?? (PosCyc (idxs[-7,80]), Range 4 (-2) 0)(2><3) [ 9, 7, 5 , 4, 2, 0 ]O0obtains the common value of a property of a listP1creates a matrix from a vertical list of matricesQ3creates a matrix from a horizontal list of matricesACreate a matrix from blocks given as a list of lists of matrices.iSingle row-column components are automatically expanded to match the corresponding common row and column: disp = putStr . dispf 2 Bdisp $ fromBlocks [[ident 5, 7, row[10,20]], [3, diagl[1,2,3], 0]] 8x101 0 0 0 0 7 7 7 10 200 1 0 0 0 7 7 7 10 200 0 1 0 0 7 7 7 10 200 0 0 1 0 7 7 7 10 200 0 0 0 1 7 7 7 10 203 3 3 3 3 1 0 0 0 03 3 3 3 3 0 2 0 0 03 3 3 3 3 0 0 3 0 0create a block diagonal matrix<disp 2 $ diagBlock [konst 1 (2,2), konst 2 (3,5), col [5,7]]7x81 1 0 0 0 0 0 01 1 0 0 0 0 0 00 0 2 2 2 2 2 00 0 2 2 2 2 2 00 0 2 2 2 2 2 00 0 0 0 0 0 0 50 0 0 0 0 0 0 75diagBlock [(0><4)[], konst 2 (2,3)] :: Matrix Double(2><7)$ [ 0.0, 0.0, 0.0, 0.0, 2.0, 2.0, 2.0& , 0.0, 0.0, 0.0, 0.0, 2.0, 2.0, 2.0 ] Reverse rowsReverse columns&creates a rectangular diagonal matrix:5diagRect 7 (fromList [10,20,30]) 4 5 :: Matrix Double(4><5) [ 10.0, 7.0, 7.0, 7.0, 7.0 , 7.0, 20.0, 7.0, 7.0, 7.0 , 7.0, 7.0, 30.0, 7.0, 7.0 , 7.0, 7.0, 7.0, 7.0, 7.0 ]/extracts the diagonal from a rectangular matrix'Create a matrix from a list of elements"(2><3) [2, 4, 7+2*iC, -3, 11, 0](2><3), [ 2.0 :+ 0.0, 4.0 :+ 0.0, 7.0 :+ 2.0. , (-3.0) :+ (-0.0), 11.0 :+ 0.0, 0.0 :+ 0.0 ]yThe input list is explicitly truncated, so that it can safely be used with lists that are too long (like infinite lists). (2><3)[1..](2><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 ]bThis is the format produced by the instances of Show (Matrix a), which can also be used for input.R7Creates a matrix with the last n rows of another matrixS2Creates a copy of a matrix without the last n rowsT:Creates a matrix with the last n columns of another matrixU5Creates a copy of a matrix without the last n columns Creates a 1+ from a list of lists (considered as rows).fromLists [[1,2],[3,4],[5,6]](3><2) [ 1.0, 2.0 , 3.0, 4.0 , 5.0, 6.0 ]$creates a 1-row matrix from a vectorasRow (fromList [1..5]) (1><5) [ 1.0, 2.0, 3.0, 4.0, 5.0 ]'creates a 1-column matrix from a vectorasColumn (fromList [1..5])(5><1) [ 1.0 , 2.0 , 3.0 , 4.0 , 5.0 ]Vcreates a Matrix of the specified size using the supplied function to to map the row/column position to the value at that row/column position. > buildMatrix 3 4 (\(r,c) -> fromIntegral r * fromIntegral c) (3><4) [ 0.0, 0.0, 0.0, 0.0, 0.0 , 0.0, 1.0, 2.0, 3.0, 4.0 , 0.0, 2.0, 4.0, 6.0, 8.0]Hilbert matrix of order N: Ihilb n = buildMatrix n n (\(i,j)->1/(fromIntegral i + fromIntegral j +1))WSrearranges the rows of a matrix according to the order given in a list of integers.XSrearranges the rows of a matrix according to the order given in a list of integers.Kcreates matrix by repetition of a matrix a given number of rows and columnsrepmat (ident 2) 2 3(4><6) [ 1.0, 0.0, 1.0, 0.0, 1.0, 0.0 , 0.0, 1.0, 0.0, 1.0, 0.0, 1.0 , 1.0, 0.0, 1.0, 0.0, 1.0, 0.0! , 0.0, 1.0, 0.0, 1.0, 0.0, 1.0 ] A version of Fl which automatically adapt matrices with a single row or column to match the dimensions of the other matrix.yPartition a matrix into blocks with the given numbers of rows and columns. The remaining rows and columns are discarded.Fully partition a matrix into blocks of the same size. If the dimensions are not a multiple of the given size the last blocks will be smaller.cmapMatrixWithIndexM_ (\(i,j) v -> printf "m[%d,%d] = %.f\n" i j v :: IO()) ((2><3)[1 :: Double ..]) m[0,0] = 1 m[0,1] = 2 m[0,2] = 3 m[1,0] = 4 m[1,1] = 5 m[1,2] = 6mmapMatrixWithIndexM (\(i,j) v -> Just $ 100*v + 10*fromIntegral i + fromIntegral j) (ident 3:: Matrix Double) Just (3><3) [ 100.0, 1.0, 2.0 , 10.0, 111.0, 12.0 , 20.0, 21.0, 122.0 ]emapMatrixWithIndex (\(i,j) v -> 100*v + 10*fromIntegral i + fromIntegral j) (ident 3:: Matrix Double)(3><3) [ 100.0, 1.0, 2.0 , 10.0, 111.0, 12.0 , 20.0, 21.0, 122.0 ]=YZ[\]^_`OPQabRSTUVWXcdefghijk:YZ[\]^_`OPQabRSTUVWXcdefgh5YZ[\]^_`OPQabRSTUVWXcdefghijk (c) Alberto Ruiz 2010BSD3 Alberto Ruiz provisionalNone lFormatting toolCreates a string from a matrix given a separator and a function to show each entry. Using this function the user can easily define any desired display function: import Text.Printf(printf) +disp = putStr . format " " (printf "%.2f")FShow a matrix with "autoscaling" and a given number of decimal places.%putStr . disps 2 $ 120 * (3><4) [1..]3x4 E3 0.12 0.24 0.36 0.48 0.60 0.72 0.84 0.96 1.08 1.20 1.32 1.444Show a matrix with a given number of decimal places.dispf 2 (1/3 + ident 3)="3x3\n1.33 0.33 0.33\n0.33 1.33 0.33\n0.33 0.33 1.33\n""putStr . dispf 2 $ (3><4)[1,1.5..]3x41.00 1.50 2.00 2.503.00 3.50 4.00 4.505.00 5.50 6.00 6.50EputStr . unlines . tail . lines . dispf 2 . asRow $ linspace 10 (0,1):0.00 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00m4Show a vector using a function for showing matrices..putStr . vecdisp (dispf 2) $ linspace 10 (0,1)@10 |> 0.00 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00+Tool to display matrices with latex syntax.)latexFormat "bmatrix" (dispf 2 $ ident 2):"\\begin{bmatrix}\n1 & 0\n\\\\\n0 & 1\n\\end{bmatrix}"n<Pretty print a complex number with at most n decimal digits.<Pretty print a complex matrix with at most n decimal digits.9load a matrix from an ASCII file formatted as a 2D table.lopqrm4type of braces: "matrix", "bmatrix", "pmatrix", etc.@Formatted matrix, with elements separated by spaces and newlinesnstuvw Hmlopqrmnstuvw(c) Alberto Ruiz 2006-14BSD3 Alberto Ruiz provisionalNone KL:xMatrix product based on BLAS's dgemm.yMatrix product based on BLAS's zgemm.zMatrix product based on BLAS's sgemm.{Matrix product based on BLAS's cgemm.|)Full SVD of a real matrix using LAPACK's dgesvd.})Full SVD of a real matrix using LAPACK's dgesdd.~,Full SVD of a complex matrix using LAPACK's zgesvd.,Full SVD of a complex matrix using LAPACK's zgesdd.*Thin SVD of a real matrix, using LAPACK's dgesvd with jobu == jobvt == 'S'.-Thin SVD of a complex matrix, using LAPACK's zgesvd with jobu == jobvt == 'S'.*Thin SVD of a real matrix, using LAPACK's dgesdd with jobz == 'S'.-Thin SVD of a complex matrix, using LAPACK's zgesdd with jobz == 'S'.1Singular values of a real matrix, using LAPACK's dgesvd with jobu == jobvt == 'N'.4Singular values of a complex matrix, using LAPACK's zgesvd with jobu == jobvt == 'N'.1Singular values of a real matrix, using LAPACK's dgesdd with jobz == 'N'.4Singular values of a complex matrix, using LAPACK's zgesdd with jobz == 'N'.PSingular values and all right singular vectors of a real matrix, using LAPACK's dgesvd# with jobu == 'N' and jobvt == 'A'.SSingular values and all right singular vectors of a complex matrix, using LAPACK's zgesvd# with jobu == 'N' and jobvt == 'A'.OSingular values and all left singular vectors of a real matrix, using LAPACK's dgesvd$ with jobu == 'A' and jobvt == 'N'.RSingular values and all left singular vectors of a complex matrix, using LAPACK's zgesvd# with jobu == 'A' and jobvt == 'N'.OEigenvalues and right eigenvectors of a general complex matrix, using LAPACK's zgeevI. The eigenvectors are the columns of v. The eigenvalues are not sorted.8Eigenvalues of a general complex matrix, using LAPACK's zgeev3 with jobz == 'N'. The eigenvalues are not sorted.LEigenvalues and right eigenvectors of a general real matrix, using LAPACK's dgeevI. The eigenvectors are the columns of v. The eigenvalues are not sorted.5Eigenvalues of a general real matrix, using LAPACK's dgeev3 with jobz == 'N'. The eigenvalues are not sorted.NEigenvalues and right eigenvectors of a symmetric real matrix, using LAPACK's dsyev_. The eigenvectors are the columns of v. The eigenvalues are sorted in descending order (use  for ascending order). in ascending orderQEigenvalues and right eigenvectors of a hermitian complex matrix, using LAPACK's zheev_. The eigenvectors are the columns of v. The eigenvalues are sorted in descending order (use  for ascending order). in ascending order7Eigenvalues of a symmetric real matrix, using LAPACK's dsyevC with jobz == 'N'. The eigenvalues are sorted in descending order.:Eigenvalues of a hermitian complex matrix, using LAPACK's zheevC with jobz == 'N'. The eigenvalues are sorted in descending order.Solve a real linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition, based on LAPACK's dgesv6. For underconstrained or overconstrained systems use  or  . See also .Solve a complex linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition, based on LAPACK's zgesv6. For underconstrained or overconstrained systems use  or  . See also .wSolves a symmetric positive definite system of linear equations using a precomputed Cholesky factorization obtained by .wSolves a Hermitian positive definite system of linear equations using a precomputed Cholesky factorization obtained by .Least squared error solution of an overconstrained real linear system, or the minimum norm solution of an underconstrained system, using LAPACK's dgels!. For rank-deficient systems use .Least squared error solution of an overconstrained complex linear system, or the minimum norm solution of an underconstrained system, using LAPACK's zgels!. For rank-deficient systems use .kMinimum norm solution of a general real linear least squares problem Ax=B using the SVD, based on LAPACK's dgelss6. Admits rank-deficient systems but it is slower than . The effective rank of A is determined by treating as zero those singular valures which are less than rcond times the largest singular value. If rcond == Nothing machine precision is used.nMinimum norm solution of a general complex linear least squares problem Ax=B using the SVD, based on LAPACK's zgelss6. Admits rank-deficient systems but it is slower than . The effective rank of A is determined by treating as zero those singular valures which are less than rcond times the largest singular value. If rcond == Nothing machine precision is used.WCholesky factorization of a complex Hermitian positive definite matrix, using LAPACK's zpotrf.TCholesky factorization of a real symmetric positive definite matrix, using LAPACK's dpotrf.WCholesky factorization of a complex Hermitian positive definite matrix, using LAPACK's zpotrf ( version).TCholesky factorization of a real symmetric positive definite matrix, using LAPACK's dpotrf ( version).2QR factorization of a real matrix, using LAPACK's dgeqr2.5QR factorization of a complex matrix, using LAPACK's zgeqr2.build rotation from reflectorsbuild rotation from reflectorsAHessenberg factorization of a square real matrix, using LAPACK's dgehrd.DHessenberg factorization of a square complex matrix, using LAPACK's zgehrd.<Schur factorization of a square real matrix, using LAPACK's dgees.?Schur factorization of a square complex matrix, using LAPACK's zgees.:LU factorization of a general real matrix, using LAPACK's dgetrf.=LU factorization of a general complex matrix, using LAPACK's zgetrf.@Solve a real linear system from a precomputed LU decomposition (), using LAPACK's dgetrs.CSolve a complex linear system from a precomputed LU decomposition (), using LAPACK's zgetrs.=LDL factorization of a symmetric real matrix, using LAPACK's dsytrf.@LDL factorization of a hermitian complex matrix, using LAPACK's zhetrf.ASolve a real linear system from a precomputed LDL decomposition (), using LAPACK's dsytrs.DSolve a complex linear system from a precomputed LDL decomposition (), using LAPACK's zsytrs.xyz{|}~rcondcoefficient matrixright hand sides (as columns)solution vectors (as columns)rcondcoefficient matrixright hand sides (as columns)solution vectors (as columns)xyz{|}~xyz{|}~(c) Alberto Ruiz 2010-14BSD3 Alberto Ruiz provisionalNone  34579>L#conjugate transpose transpose$Matrix product and related functionsmatrix product@sum of absolute value of elements (differs in complex case from norm1)!sum of absolute value of elementseuclidean normelement of maximum magnitudekonst 7 3 :: Vector FloatfromList [7.0,7.0,7.0]konst i (3::Int,4::Int)(3><4)1 [ 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.01 , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.03 , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 ]9Basic element-by-element functions for numeric containers!element by element multiplication6scale the element by element reciprocal of the object: @scaleRecip 2 (fromList [5,i]) == 2 |> [0.4 :+ 0.0,0.0 :+ (-2.0)]element by element division(create a structure with a single elementlet v = fromList [1..3::Double]v / scalar (norm2 v)CfromList [0.2672612419124244,0.5345224838248488,0.8017837257372732]complex conjugate for integer arrayscmod 3 (range 5)fromList [0,1,2,0,1]2fromInt ((2><2) [0..3]) :: Matrix (Complex Double)(2><2)[ 0.0 :+ 0.0, 1.0 :+ 0.0, 2.0 :+ 0.0, 3.0 :+ 0.0 ]like H (cannot implement instance Functor because of Element class constraint)generic indexing functionvector [1,2,3] `atIndex` 12.0matrix 3 [0..8] `atIndex` (2,0)6.0index of minimum elementindex of maximum elementvalue of minimum elementvalue of maximum elementthe sum of elementsthe product of elements#A more efficient implementation of !cmap (\x -> if x>0 then 1 else 0) step $ linspace 5 (-1,1::Double)5 |> [0.0,0.0,0.0,1.0,1.0]Element by element version of /case compare a b of {LT -> l; EQ -> e; GT -> g}.<Arguments with any dimension = 1 are automatically expanded:Econd ((1><4)[1..]) ((3><1)[1..]) 0 100 ((3><4)[1..]) :: Matrix Double(3><4)[ 100.0, 2.0, 3.0, 4.0, 0.0, 100.0, 7.0, 8.0, 0.0, 0.0, 100.0, 12.0 ]$let chop x = cond (abs x) 1E-6 0 0 x0Find index of elements which satisfy a predicate$find (>0) (ident 3 :: Matrix Double)[(0,0),(1,1),(2,2)]+Create a structure from an association list(assoc 5 0 [(3,7),(1,4)] :: Vector DoublefromList [0.0,4.0,0.0,7.0,0.0]Bassoc (2,3) 0 [((0,2),7),((1,0),2*i-3)] :: Matrix (Complex Double)(2><3)( [ 0.0 :+ 0.0, 0.0 :+ 0.0, 7.0 :+ 0.0* , (-3.0) :+ 2.0, 0.0 :+ 0.0, 0.0 :+ 0.0 ]+Modify a structure using an update function:accum (ident 5) (+) [((1,1),5),((0,3),3)] :: Matrix Double(5><5) [ 1.0, 0.0, 0.0, 3.0, 0.0 , 0.0, 6.0, 0.0, 0.0, 0.0 , 0.0, 0.0, 1.0, 0.0, 0.0 , 0.0, 0.0, 0.0, 1.0, 0.0 , 0.0, 0.0, 0.0, 0.0, 1.0 ]computation of histogram:Iaccum (konst 0 7) (+) (map (flip (,) 1) [4,5,4,1,5,2,5]) :: Vector Double&fromList [0.0,1.0,1.0,0.0,2.0,3.0,0.0]unconjugated dot productOuter product of two vectors.)fromList [1,2,3] `outer` fromList [5,2,3](3><3) [ 5.0, 2.0, 3.0 , 10.0, 4.0, 6.0 , 15.0, 6.0, 9.0 ]"Kronecker product of two matrices. m1=(2><3) [ 1.0, 2.0, 0.0 , 0.0, -1.0, 3.0 ] m2=(4><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 , 7.0, 8.0, 9.0 , 10.0, 11.0, 12.0 ]kronecker m1 m2 (8><9): [ 1.0, 2.0, 3.0, 2.0, 4.0, 6.0, 0.0, 0.0, 0.0: , 4.0, 5.0, 6.0, 8.0, 10.0, 12.0, 0.0, 0.0, 0.0: , 7.0, 8.0, 9.0, 14.0, 16.0, 18.0, 0.0, 0.0, 0.0: , 10.0, 11.0, 12.0, 20.0, 22.0, 24.0, 0.0, 0.0, 0.0: , 0.0, 0.0, 0.0, -1.0, -2.0, -3.0, 3.0, 6.0, 9.0: , 0.0, 0.0, 0.0, -4.0, -5.0, -6.0, 12.0, 15.0, 18.0: , 0.0, 0.0, 0.0, -7.0, -8.0, -9.0, 21.0, 24.0, 27.0< , 0.0, 0.0, 0.0, -10.0, -11.0, -12.0, 30.0, 33.0, 36.0 ].Creates a square matrix with a given diagonal..creates the identity matrix of given dimension      !"size default valueassociation listresult#initial structureupdate functionassociation listresult$%&'()*+,ablegresultsize default valueassociation listresultinitial structureupdate functionassociation listresult-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwr      !"#$%&'()*+,-./0123456789:;<=x   !   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwNone 357yGeneral matrix with specialized internal representations for dense, sparse, diagonal, banded, and constant elements./let m = mkSparse [((0,999),1.0),((1,1999),2.0)]m3SparseR {gmCSR = CSR {csrVals = fromList [1.0,2.0],5 csrCols = fromList [1000,2000],1 csrRows = fromList [1,2,3],# csrNRows = 2,' csrNCols = 2000}, nRows = 2,# nCols = 2000}let m = mkDense (mat 2 [1..4])mDense {gmDense = (2><2) [ 1.0, 2.0$ , 3.0, 4.0 ], nRows = 2, nCols = 2}general matrix - vector product/let m = mkSparse [((0,999),1.0),((1,1999),2.0)]m !#> vector [1..2000]fromList [1000.0,4000.0]+xyz{|}~xyz{|}~(c) Alberto Ruiz 2014BSD3 Alberto Ruiz provisionalNone {  !"#$%&'(234789:ABCEFKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~KLMNOPQRS C!"#$%23479:B'&(A erfmhopklds{u|}xy~VWXYZ[\]^_`abcTUqijgnvwtz EF8(c) Alberto Ruiz 2006-14BSD3 Alberto Ruiz provisionalNone  3457>LAVA matrix that, by construction, it is known to be complex Hermitian or real symmetric.It can be created using 5, 6, or 7(, and the matrix can be extracted using 4._QR decomposition of a matrix in compact form. (The orthogonal matrix is not explicitly formed.)VLDL decomposition of a complex Hermitian or real symmetric matrix in a compact format.1LU decomposition of a matrix in a compact format.PGeneric linear algebra functions for double precision real and complex matrices..(Single precision data can be converted using  and )."Full singular value decomposition. a = (5><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 , 7.0, 8.0, 9.0 , 10.0, 11.0, 12.0 , 13.0, 14.0, 15.0 ] :: Matrix Double let (u,s,v) = svd adisp 3 u5x5&-0.101 0.768 0.614 0.028 -0.149&-0.249 0.488 -0.503 0.172 0.646&-0.396 0.208 -0.405 -0.660 -0.449&-0.543 -0.072 -0.140 0.693 -0.447&-0.690 -0.352 0.433 -0.233 0.398sEfromList [35.18264833189422,1.4769076999800903,1.089145439970417e-15]disp 3 v3x3-0.519 -0.751 0.408-0.576 -0.046 -0.816-0.632 0.659 0.408let d = diagRect 0 s 5 3disp 3 d5x335.183 0.000 0.000 0.000 1.477 0.000 0.000 0.000 0.000 0.000 0.000 0.000disp 3 $ u <> d <> tr v5x3 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.00010.000 11.000 12.00013.000 14.000 15.000 A version of  which returns only the min (rows m) (cols m) singular vectors of m.If (u,s,v) = thinSVD m then m == u <> diag s <> tr v. a = (5><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 , 7.0, 8.0, 9.0 , 10.0, 11.0, 12.0 , 13.0, 14.0, 15.0 ] :: Matrix Double let (u,s,v) = thinSVD adisp 3 u5x3-0.101 0.768 0.614-0.249 0.488 -0.503-0.396 0.208 -0.405-0.543 -0.072 -0.140-0.690 -0.352 0.433sEfromList [35.18264833189422,1.4769076999800903,1.089145439970417e-15]disp 3 v3x3-0.519 -0.751 0.408-0.576 -0.046 -0.816-0.632 0.659 0.408disp 3 $ u <> diag s <> tr v5x3 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.00010.000 11.000 12.00013.000 14.000 15.000Singular values only. A version of G which returns an appropriate diagonal matrix with the singular values.If (u,d,v) = fullSVD m then m == u <> d <> tr v. Similar to T, returning only the nonzero singular values and the corresponding singular vectors. a = (5><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 , 7.0, 8.0, 9.0 , 10.0, 11.0, 12.0 , 13.0, 14.0, 15.0 ] :: Matrix Double let (u,s,v) = compactSVD adisp 3 u5x2-0.101 0.768-0.249 0.488-0.396 0.208-0.543 -0.072-0.690 -0.352s/fromList [35.18264833189422,1.4769076999800903]disp 3 u5x2-0.101 0.768-0.249 0.488-0.396 0.208-0.543 -0.072-0.690 -0.352disp 3 $ u <> diag s <> tr v5x3 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.00010.000 11.000 12.00013.000 14.000 15.000<Singular values and all right singular vectors (as columns).;Singular values and all left singular vectors (as columns). RObtains the LU decomposition of a matrix in a compact data structure suitable for  . mSolution of a linear system (for several right hand sides) from the precomputed LU factorization obtained by  .Solve a linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition. For underconstrained or overconstrained systems use   or  . It is similar to   .  , but  linearSolve0 raises an error if called on a singular system.Solve a linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition, returning Nothing for a singular system. For underconstrained or overconstrained systems use   or  . vSolve a symmetric or Hermitian positive definite linear system using a precomputed Cholesky decomposition obtained by . Minimum norm solution of a general linear least squares problem Ax=B using the SVD. Admits rank-deficient systems but it is slower than  g. The effective rank of A is determined by treating as zero those singular valures which are less than " times the largest singular value. Least squared error solution of an overconstrained linear system, or the minimum norm solution of an underconstrained system. For rank-deficient systems use  . Similar to l, without checking that the input matrix is hermitian or symmetric. It works with the lower triangular part.SObtains the LDL decomposition of a matrix in a compact data structure suitable for .lSolution of a linear system (for several right hand sides) from a precomputed LDL factorization obtained by .ONote: this can be slower than the general solver based on the LU decomposition.SEigenvalues (not ordered) and eigenvectors (as columns) of a general square matrix.If  (s,v) = eig m then m <> v == v <> diag s Ba = (3><3) [ 3, 0, -2 , 4, 5, -1 , 3, 1, 0 ] :: Matrix Double let (l, v) = eig aputStr . dispcf 3 . asRow $ l1x3!1.925+1.523i 1.925-1.523i 4.151putStr . dispcf 3 $ v3x3$-0.455+0.365i -0.455-0.365i 0.181$ 0.603 0.603 -0.978$ 0.033+0.543i 0.033-0.543i -0.104"putStr . dispcf 3 $ complex a <> v3x3$-1.432+0.010i -1.432-0.010i 0.753$ 1.160+0.918i 1.160-0.918i -4.059$-0.763+1.096i -0.763-1.096i -0.433putStr . dispcf 3 $ v <> diag l3x3$-1.432+0.010i -1.432-0.010i 0.753$ 1.160+0.918i 1.160-0.918i -4.059$-0.763+1.096i -0.763-1.096i -0.4335Eigenvalues (not ordered) of a general square matrix. Similar to k without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part. Similar to k without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.oEigenvalues and eigenvectors (as columns) of a complex hermitian or real symmetric matrix, in descending order.If (s,v) = eigSH m then m == v <> diag s <> tr v @a = (3><3) [ 1.0, 2.0, 3.0 , 2.0, 4.0, 5.0 , 3.0, 5.0, 6.0 ] let (l, v) = eigSH alEfromList [11.344814282762075,0.17091518882717918,-0.5157294715892575]disp 3 $ v <> diag l <> tr v3x31.000 2.000 3.0002.000 4.000 5.0003.000 5.000 6.000REigenvalues (in descending order) of a complex hermitian or real symmetric matrix.QR factorization.If  (q,r) = qr m then  m == q <> r/, where q is unitary and r is upper triangular.9Compute the QR decomposition of a matrix in compact form.Zgenerate a matrix with k orthogonal columns from the compact QR decomposition obtained by .RQ factorization.If  (r,q) = rq m then  m == r <> q/, where q is unitary and r is upper triangular.Hessenberg factorization.If (p,h) = hess m then m == p <> h <> tr pj, where p is unitary and h is in upper Hessenberg form (it has zero entries below the first subdiagonal).Schur factorization.If (u,s) = schur m then m == u <> s <> tr u, where u is unitary and s is a Shur matrix. A complex Schur matrix is upper triangular. A real Schur matrix is upper triangular in 2x2 blocks.c"Anything that the Jordan decomposition can do, the Schur decomposition can do better!" (Van Loan) Similar to V, but instead of an error (e.g., caused by a matrix not positive definite) it returns . Similar to l, without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.LCholesky factorization of a positive definite hermitian or symmetric matrix.If  c = chol m then c is upper triangular and m == tr c <> c. Similar to V, but instead of an error (e.g., caused by a matrix not positive definite) it returns . MJoint computation of inverse and logarithm of determinant of a square matrix.!LDeterminant of a square matrix. To avoid possible overflow or underflow use  .".Explicit LU factorization of a general matrix.If (l,u,p,s) = lu m then m == p <> l <> u{, where l is lower triangular, u is upper triangular, p is a permutation matrix and s is the signature of the permutation.#%Inverse of a square matrix. See also  .$:Pseudoinverse of a general matrix with default tolerance (% 1, similar to GNU-Octave).% pinvTol r7 computes the pseudoinverse of a matrix with tolerance tol=r*g*eps*(max rows cols)), where g is the greatest singular value. [m = (3><3) [ 1, 0, 0 , 0, 1, 0 , 0, 0, 1e-10] :: Matrix Double pinv m1. 0. 0.0. 1. 0.0. 0. 10000000000. pinvTol 1E8 m1. 0. 0.0. 1. 0.0. 0. 1.4Numeric rank of a matrix from the SVD decomposition.&2Numeric rank of a matrix from its singular values.#The machine precision of a Double: eps = 2.22044604925031e-16 (the value used by GNU-Octave).'%1 + 0.5*peps == 1, 1 + 0.6*peps /= 1(AThe nullspace of a matrix from its precomputed SVD decomposition.$The nullspace of a matrix. See also (.QThe nullspace of a matrix, assumed to be one-dimensional, with machine precision.)@The range space a matrix from its precomputed SVD decomposition.:Return an orthonormal basis of the range space of a matrix+YReciprocal of the 2-norm condition number of a matrix, computed from the singular values.,9Number of linearly independent rows or columns. See also &-CGeneric matrix functions for diagonalizable matrices. For instance: logm = matFunc log.Matrix exponential. It uses a direct translation of Algorithm 11.3.1 in Golub & Van Loan, based on a scaled Pade approximation./Matrix square root. Currently it uses a simple iterative algorithm described in Wikipedia. It only works with invertible matrices that have a real solution. 1m = (2><2) [4,9 ,0,4] :: Matrix Doublesqrtm m(2><2) [ 2.0, 2.25 , 0.0, 2.0 ](For diagonalizable matrices you can try - sqrt: matFunc sqrt ((2><2) [1,0,0,-1])(2><2) [ 1.0 :+ 0.0, 0.0 :+ 0.0 , 0.0 :+ 0.0, 0.0 :+ 1.0 ]0GCompute the explicit LU decomposition from the compact one obtained by  .;Approximate number of common digits in the maximum element.2jGeneralized symmetric positive definite eigensystem Av = lBv, for A and B symmetric, B positive definite.4"Extract the general matrix from a ; structure, forgetting its symmetric or Hermitian property.5ICompute the complex Hermitian or real symmetric part of a square matrix ( (x + tr x)/2).6Compute the contraction  tr x <> x of a general matrix.7_At your own risk, declare that a matrix is complex Hermitian or real symmetric for usage in , 9, etc. Only a triangular part of the matrix will be used.   0Cholesky decomposition of the coefficient matrixright hand sidessolution   .(inverse, (log abs det, sign or phase of det))!"#$%numeric zero (e.g. 1*)input matrix msv of m rank of m&numeric zero (e.g. 1*)maximum dimension of the matrixsingular values rank of m'(Left "numeric" zero (eg. 1*)), or Right "theoretical" matrix rank.input matrix m of m nullspacerelative tolerance in  units (e.g., use 3 to get 3*) input matrix.list of unitary vectors spanning the nullspace)Left "numeric" zero (eg. 1*)), or Right "theoretical" matrix rank.input matrix m of morth*+,-./012AB3AB4567w      !"#$%&'()*+,-./01234567h      !"#$%&'()*+,-./01234567(c) Alberto Ruiz 2010-14BSD3 Alberto Ruiz provisionalNone 34579 9build 5 (**2) :: Vector DoublefromList [0.0,1.0,4.0,9.0,16.0]Hilbert matrix of order N:=let hilb n = build (n,n) (\i j -> 1/(i+j+1)) :: Matrix DoubleputStr . dispf 2 $ hilb 33x31.00 0.50 0.330.50 0.33 0.250.33 0.25 0.209Matrix-matrix, matrix-vector, and vector-matrix products.;3Creates a real vector containing a range of values:linspace 5 (-3,7::Double)!fromList [-3.0,-0.5,2.0,4.5,7.0]@-linspace 5 (8,2+i) :: Vector (Complex Double)CfromList [8.0 :+ 0.0,6.5 :+ 0.25,5.0 :+ 0.5,3.5 :+ 0.75,2.0 :+ 1.0].Logarithmic spacing can be defined as follows: )logspace n (a,b) = 10 ** linspace n (a,b)<An infix synonym for @&vector [1,2,3,4] <.> vector [-2,0,1,1]5.0let V = 0:+1 :: C%fromList [1+V,1] <.> fromList [1,1+V] 2.0 :+ 0.0=dense matrix-vector productlet m = (2><3) [1..]m(2><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 ]let v = vector [10,20,30]m #> vfromList [140.0,320.0]dense matrix-vector product>dense vector-matrix product?oLeast squares solution of a linear system, similar to the \ operator of Matlab/Octave (based on linearSolveSVD) 4a = (3><2) [ 1.0, 2.0 , 2.0, 4.0 , 2.0, -1.0 ]  v = vector [13.0,27.0,1.0] let x = a <\> vx/fromList [3.0799999999999996,5.159999999999999]a #> x4fromList [13.399999999999999,26.799999999999997,1.0]GIt also admits multiple right-hand sides stored as columns in a matrix.BBCompute mean vector and covariance matrix of the rows of a matrix.>meanCov $ gaussianSample 666 1000 (fromList[4,5]) (diagl[2,3])1(fromList [4.010341078059521,5.0197204699640405],(2><2)1 [ 1.9862461923890056, -1.0127225830525157e-24 , -1.0127225830525157e-2, 3.0373954915729318 ])Dm <- randn 4 10disp 2 m4x10D-0.31 0.41 0.43 -0.19 -0.17 -0.23 -0.17 -1.04 -0.07 -1.24D 0.26 0.19 0.14 0.83 -1.54 -0.09 0.37 -0.63 0.71 -0.50D-0.11 -0.10 -1.29 -1.40 -1.04 -0.89 -0.68 0.35 -1.46 1.86D 1.04 -0.29 0.19 -0.75 -2.20 -0.01 1.06 0.11 -2.09 -1.58*disp 2 $ m ?? (All, Pos $ sortIndex (m!1))4x10D-0.17 -1.04 -1.24 -0.23 0.43 0.41 -0.31 -0.17 -0.07 -0.19D-1.54 -0.63 -0.50 -0.09 0.14 0.19 0.26 0.37 0.71 0.83D-1.04 0.35 1.86 -0.89 -1.29 -0.10 -0.11 -0.68 -1.46 -1.40D-2.20 0.11 -1.58 -0.01 0.19 -0.29 1.04 1.06 -2.09 -0.75EFExtract elements from positions given in matrices of rows and columns.r(3><3) [ 1, 1, 1 , 1, 2, 2 , 1, 2, 3 ]c(3><3) [ 0, 1, 5 , 2, 2, 1 , 4, 4, 1 ]m(4><6) [ 0, 1, 2, 3, 4, 5 , 6, 7, 8, 9, 10, 11 , 12, 13, 14, 15, 16, 17 , 18, 19, 20, 21, 22, 23 ] remap r c m(3><3) [ 6, 7, 11 , 8, 14, 13 , 10, 16, 19 ] The indexes are autoconformable.c'(3><1) [ 1 , 2 , 4 ] remap r c' m(3><3) [ 7, 7, 7 , 8, 14, 14 , 10, 16, 22 ]89:;<=>?@ABCDE89:;<=>?@ABCDE89:;<=>?@ABCDE<=>?(c) Alberto Ruiz 2009-14BSD3 Alberto Ruiz provisionalNone F`Obtains a matrix whose rows are pseudorandom samples from a multivariate Gaussian distribution.G_Obtains a matrix whose rows are pseudorandom samples from a multivariate uniform distribution.9pseudorandom matrix with uniform elements between 0 and 1H9pseudorandom matrix with uniform elements between 0 and 1I(pseudorandom matrix with normal elementsdisp 3 =<< randn 3 53x5%0.386 -1.141 0.491 -0.510 1.512%0.069 -0.919 1.022 -0.181 0.745%0.313 -0.670 -0.097 -1.575 -0.583Fnumber of rows mean vectorcovariance matrixresultGnumber of rowsranges for each columnresultrowscolumnsHI )*+,-FGHIFGHI(c) Alberto Ruiz 2012BSD3 Alberto Ruiz provisionalNone 4J correlation)corr (fromList[1,2,3]) (fromList [1..10])2fromList [14.0,20.0,26.0,32.0,38.0,44.0,50.0,56.0]K convolution (JI with reversed kernel and padded input, equivalent to polynomial product)&conv (fromList[1,1]) (fromList [-1,1])fromList [-1.0,0.0,1.0]L similar to J, using  instead of (*)M 2D correlation (without padding):disp 5 $ corr2 (konst 1 (3,3)) (ident 10 :: Matrix Double) 8x83 2 1 0 0 0 0 02 3 2 1 0 0 0 01 2 3 2 1 0 0 00 1 2 3 2 1 0 00 0 1 2 3 2 1 00 0 0 1 2 3 2 10 0 0 0 1 2 3 20 0 0 0 0 1 2 3N2D convolution:disp 5 $ conv2 (konst 1 (3,3)) (ident 10 :: Matrix Double) 12x12"1 1 1 0 0 0 0 0 0 0 0 0"1 2 2 1 0 0 0 0 0 0 0 0"1 2 3 2 1 0 0 0 0 0 0 0"0 1 2 3 2 1 0 0 0 0 0 0"0 0 1 2 3 2 1 0 0 0 0 0"0 0 0 1 2 3 2 1 0 0 0 0"0 0 0 0 1 2 3 2 1 0 0 0"0 0 0 0 0 1 2 3 2 1 0 0"0 0 0 0 0 0 1 2 3 2 1 0"0 0 0 0 0 0 0 1 2 3 2 1"0 0 0 0 0 0 0 0 1 2 2 1"0 0 0 0 0 0 0 0 0 1 1 1ORmatrix computation implemented as separated vector operations by rows and columns.JkernelsourceKLMNkernelOJKLMNOJKLMNO(c) Vivian McPhail 2010BSD36Vivian McPhail <haskell.vivian.mcphail <at> gmail.com> provisionalportableNone 4lProvide optimal association order for a chain of matrix multiplications and apply the multiplications.xThe algorithm is the well-known O(n^3) dynamic programming algorithm that builds a pyramid of optimal associations. mm1, m2, m3, m4 :: Matrix Double m1 = (10><15) [1..] m2 = (15><20) [1..] m3 = (20><5) [1..] m4 = (5><10) [1..] >>> optimiseMult [m1,m2,m3,m4] will perform ((m1  (m2  m3))  m4)2The naive left-to-right multiplication would take 4500? scalar multiplications whereas the optimised version performs 2750 scalar multiplications. The complexity in this case is 32 (= 4^3/2) * (2 comparisons, 3 scalar multiplications, 3 scalar additions, 5 lookups, 2 updates) + a constant (= three table allocations)(c) Alberto Ruiz 2011BSD3 Alberto Ruiz provisionalNone  3457>L  None 345Rconjugate gradientSresidualTsquared norm of residualUcurrent solutionVnormalized size of correctionWYSolve a sparse linear system using the conjugate gradient method with default parameters.XYSolve a sparse linear system using the conjugate gradient method with default parameters.PQRSTUVW is symmetriccoefficient matrixright-hand sidesolutionX symmetric/relative tolerance for the residual (e.g. 1E-4)%relative tolerance for x (e.g. 1E-3)maximum number of iterationscoefficient matrixinitial solutionright-hand sidesolution PQRSTUVWX PQRSTUVWX(c) Alberto Ruiz 2014BSD3 Alberto Ruiz provisionalNone  3457>L          (c) Alberto Ruiz 2013BSD3 Alberto Ruiz provisionalNone 34579YAlternative indexing function.vector [1..10] ! 34.0-On a matrix it gets the k-th row as a vector:matrix 5 [1..15] ! 1fromList [6.0,7.0,8.0,9.0,10.0]matrix 5 [1..15] ! 1 ! 39.0bimaginary unitcCreate a real vector. vector [1..5]fromList [1.0,2.0,3.0,4.0,5.0]dCreate a real matrix.matrix 5 [1..15](3><5) [ 1.0, 2.0, 3.0, 4.0, 5.0 , 6.0, 7.0, 8.0, 9.0, 10.0! , 11.0, 12.0, 13.0, 14.0, 15.0 ]eGprint a real matrix with given number of digits after the decimal pointdisp 5 $ ident 2 / 32x20.33333 0.000000.00000 0.33333f)create a real diagonal matrix from a list diagl [1,2,3](3><3) [ 1.0, 0.0, 0.0 , 0.0, 2.0, 0.0 , 0.0, 0.0, 3.0 ]a real matrix of zerosa real matrix of onesconcatenation of real vectorsghorizontal concatenationident 3 ||| konst 7 (3,4)(3><7)$ [ 1.0, 0.0, 0.0, 7.0, 7.0, 7.0, 7.0$ , 0.0, 1.0, 0.0, 7.0, 7.0, 7.0, 7.0& , 0.0, 0.0, 1.0, 7.0, 7.0, 7.0, 7.0 ]ha synonym for (g) (unicode 0x00a6, broken bar)ivertical concatenationja synonym for (i) (unicode 0x2014, em dash)k+create a single row real matrix from a list row [2,3,1,8](1><4) [ 2.0, 3.0, 1.0, 8.0 ]l.create a single column real matrix from a list col [7,-2,4](3><1) [ 7.0 , -2.0 , 4.0 ]m extract rows(20><4) [1..] ? [2,1,1](3><4) [ 9.0, 10.0, 11.0, 12.0 , 5.0, 6.0, 7.0, 8.0 , 5.0, 6.0, 7.0, 8.0 ]nextract columns2(unicode 0x00bf, inverted question mark, Alt-Gr ?)(3><4) [1..] [3,0](3><2) [ 4.0, 1.0 , 8.0, 5.0 , 12.0, 9.0 ]o)cross product (for three-element vectors)2-norm of real vectorrRCheck if the absolute value or complex magnitude is greater than a given thresholdmagnit 1E-6 (1E-12 :: R)Falsemagnit 1E-6 (3+iC :: C)Truemagnit 0 (3 :: I ./. 5)Trues4Obtains a vector in the same direction with 2-norm=1 trans . invtsize $ vector [1..10]10size $ (2><5)[1..10::Double](2,5)ugMatrix of pairwise squared distances of row vectors (using the matrix product trick in blog.smola.org)vouter products of rowsa(3><2) [ 1.0, 2.0 , 10.0, 20.0 , 100.0, 200.0 ]b(3><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 , 7.0, 8.0, 9.0 ]rowOuters a (b ||| 1)(3><8)< [ 1.0, 2.0, 3.0, 1.0, 2.0, 4.0, 6.0, 2.0< , 40.0, 50.0, 60.0, 10.0, 80.0, 100.0, 120.0, 20.0> , 700.0, 800.0, 900.0, 100.0, 1400.0, 1600.0, 1800.0, 200.0 ]w5solution of overconstrained homogeneous linear systemx?solution of overconstrained homogeneous symmetric linear system8generic reference implementation of gaussian elimination a <> gaussElim a b = b|Experimental implementation of  1 for any Fractional element type, including  n  and  n .3let m = ident 5 + (5><5) [0..] :: Matrix (Z ./. 17)(5><5) [ 1, 1, 2, 3, 4 , 5, 7, 7, 8, 9 , 10, 11, 13, 13, 14 , 15, 16, 0, 2, 2 , 3, 4, 5, 6, 8 ]$let (l,u,p,s) = luFact $ luPacked' ml(5><5) [ 1, 0, 0, 0, 0 , 6, 1, 0, 0, 0 , 12, 7, 1, 0, 0 , 7, 10, 7, 1, 0 , 8, 2, 6, 11, 1 ]u(5><5) [ 15, 16, 0, 2, 2 , 0, 13, 7, 13, 14 , 0, 0, 15, 0, 11 , 0, 0, 0, 15, 15 , 0, 0, 0, 0, 1 ]}Experimental implementation of  , for any Fractional element type, including  n  and  n .-let a = (2><2) [1,2,3,5] :: Matrix (Z ./. 13)(2><2) [ 1, 2 , 3, 5 ]b(2><3) [ 5, 1, 3 , 8, 6, 3 ]luSolve' (luPacked' a) b(2><3) [ 4, 7, 4 , 7, 10, 6 ]aYZ[\]^_`abcdnumber of columnselements in row orderefrowscolumnsrowscolumnsghijklmnopqrstuvwxyz{ !"#$%&'()*+,|-./01234567}89:;<=>?@ABCDEFGHIJKEHIJKLMNOYZ[\]^_`abcdefghijklmnopqrstuvwxyz{ !"#$'*,|05}:ZYZ[\]^_`abcdefghijklmnopqrstuvwxyz{ !"#$%&'()*+,|-./01234567}89:;<=>?@ABCDEFGHIJK Z ghijm n (c) Alberto Ruiz 2015BSD3 experimentalNone !"(34579>CIKLNIWrapper with a phantom integer for statically checked modular arithmetic.L'this instance is only valid for prime m+~MNOPQRSTUVWXYZ[\]^_`abcdefghijklmnLopqrst~)~MNOPQRSTUVWXYZ[\]^_`abcdefghijklmnLopqrst~(c) Alberto Ruiz 2015BSD3 Alberto Ruiz provisionalNone KL56789:;<=> ./156;<=>?@DGH89;CDEOYZcdefgiklmntyz{~~ c/ dt56<kl;D=>?@YZ89;fmnGEgiCDeHyz{.O1(c) Alberto Ruiz 2006-15BSD3 Alberto Ruiz provisionalNone 4dense matrix productlet a = (3><5) [1..]a(3><5) [ 1.0, 2.0, 3.0, 4.0, 5.0 , 6.0, 7.0, 8.0, 9.0, 10.0! , 11.0, 12.0, 13.0, 14.0, 15.0 ])let b = (5><2) [1,3, 0,2, -1,5, 7,7, 6,0]b(5><2) [ 1.0, 3.0 , 0.0, 2.0 , -1.0, 5.0 , 7.0, 7.0 , 6.0, 0.0 ]a <> b(3><2) [ 56.0, 50.0 , 121.0, 135.0 , 186.0, 220.0 ]Solve a linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition, returning Nothing for a singular system. For underconstrained or overconstrained systems use   or  . %a = (2><2) [ 1.0, 2.0 , 3.0, 5.0 ]  3b = (2><3) [ 6.0, 1.0, 10.0 , 15.0, 3.0, 26.0 ] linearSolve a b Just (2><3)E [ -1.4802973661668753e-15, 0.9999999999999997, 1.999999999999997G , 3.000000000000001, 1.6653345369377348e-16, 4.000000000000002 ]let Just x = itdisp 5 x2x3-0.00000 1.00000 2.00000 3.00000 0.00000 4.00000a <> x(2><3) [ 6.0, 1.0, 10.0 , 15.0, 3.0, 26.0 ]Dreturn an orthonormal basis of the null space of a matrix. See also (.Ereturn an orthonormal basis of the range space of a matrix. See also ). 56789:;<=> )*+,-./0156;<=>?@DGHIJ      !"#$%&'()*+,-./012456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_bcdefgiklmnopqrstuvwxyz{|}~@<=>o?    }| WX#$%+,! [\]^_pqwx2"0./-JKLMN,)*+-HIFGBvus'1r*A()&b56740:IJPQRSTUV(c) Alberto Ruiz 2006-14BSD3 Alberto Ruiz provisionalNone 56789:;<=> )*+,-./0156;<=>?@DGHIJ      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ hja`3(c) Alberto Ruiz 2006-14BSD3 provisionalNone  (34579>CIKLNNuvwxyz{|}~4uvwxyz{|}~Buvwxyz{|}~(c) Alberto Ruiz 2014BSD3 experimentalNone  !"(34579=>CIKLNYLLO ! " # $%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOP Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s tuuvwxyz{z|}~~        !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghiijklmnopqrstuvwxyz{|}~l}$%&ylY*{|TPVUXNOM7/_aR       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                    R                                                                                                                                        !"#!"$!"%!"&!"'!"(!")!"*!"+!", -!./0123456789!:;!:<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~!.!!.      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghhijklmnopqrstuvwx!.yz{|}s~l Zfp      !"#$%&'()*+,-./0123456789:$%;<=>?@ABCDEFGHIJKLMNf%&OOxPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~hmatrix-0.17.0.1Numeric.LinearAlgebra.DataNumeric.LinearAlgebra.DevelNumeric.LinearAlgebraNumeric.LinearAlgebra.HMatrixNumeric.LinearAlgebra.StaticInternal.VectorInternal.DevelInternal.VectorizedInternal.MatrixData.Packed.Matrix fromListsInternal.ConversionInternal.Foreign Internal.STInternal.Element Internal.IOInternal.LAPACKInternal.NumericInternal.SparseInternal.AlgorithmsInternal.ContainerInternal.RandomInternal.ConvolutionInternal.ChainNumeric.Vector Internal.CGNumeric.Matrix Internal.UtilInternal.ModularInternal.Staticvecto_E0SPW1C3C7W31vFROmsMtHData.Vector.StorableVectorunsafeToForeignPtrunsafeFromForeignPtrfromListCRZIfiti createVectortoList|>idxs subVectorat'vjointakesV zipVectorWithunzipVectorWith foldVectorfoldVectorWithIndexfoldLoop foldVectorG mapVectorM mapVectorM_mapVectorWithIndexMmapVectorWithIndexM_mapVectorWithIndex toByteStringfromByteString zipVector unzipVector TransArrayTransTransRawapplyapplyRaw//check#|RandDistUniformGaussianSeed randomVector roundVectorrangeElementMatrix MatrixOrderRowMajor ColumnMajorrowscolsorderOf showInternalcmatfmatflattentoListsfromRowstoRows fromColumns toColumnsatM'matrixFromVector createMatrixreshape liftMatrix liftMatrix2 subMatrix saveMatrix Complexable RealElementapp appVector appVectorLen appMatrix appMatrixLen appMatrixRawappMatrixRawLenunsafeMatrixToVectorunsafeMatrixToForeignPtrSliceRowOperAXPYSCALSWAPRowRangeAllRowsRowFromRowColRangeAllColsColFromColSTMatrixSTVector thawVectorunsafeThawVector runSTVectorunsafeReadVectorunsafeWriteVector modifyVector liftSTVector freezeVectorunsafeFreezeVector readVector writeVectornewUndefinedVector newVector thawMatrixunsafeThawMatrix runSTMatrixunsafeReadMatrixunsafeWriteMatrix modifyMatrix liftSTMatrixunsafeFreezeMatrix freezeMatrix readMatrix writeMatrix setMatrixnewUndefinedMatrix newMatrixrowOper extractMatrixgemmmmutable ExtractorAllRangePosPosCycTakeTakeLastDropDropLast?? fromBlocks diagBlockflipudfliprldiagRecttakeDiag><takeRowsdropRows takeColumns dropColumnsasRowasColumn fromArray2DrepmatliftMatrix2AutotoBlocks toBlocksEverymapMatrixWithIndexM_mapMatrixWithIndexMmapMatrixWithIndexformatdispsdispf latexFormatdispcf loadMatrix loadMatrix'TestablecheckTioCheckTLinearscaleAdditiveadd Transposabletrtr'DoubleOfSingleOf ComplexOfRealOfConvertrealcomplexsingledouble toComplex fromComplexProductNumericKonstkonst ContainerIndexOfscalarconjarctan2cmodfromInttoIntfromZtoZcmapatIndexminIndexmaxIndex minElement maxElement sumElements prodElementsstepcondfindassocaccumudotouter kroneckerdiagidentGMatrixSparseRgmCSRnRowsnColsSparseCgmCSCDiagdiagValsDensegmDenseCSRcsrValscsrColscsrRowscsrNRowscsrNCols AssocMatrixmkCSRmkDensemkSparsefromCSRmkDiagR!#>toDenseHermQRLDLLUFieldsvdthinSVDsingularValues compactSVDrightSVleftSVluPackedluSolve cholSolvelinearSolveSVD linearSolveLS ldlPackedldlSolveeig eigenvalueseigSH'eigenvaluesSH'eigSH eigenvaluesSHqrqrRawqrgrrqhessschurmbCholSHcholSHcholmbCholinvlndetdetluinvpinvpinvTolranksvpeps nullspaceSVDorthSVD haussholderrcondrankmatFuncexpmsqrtmluFact relativeErrorgeigSHgeigSH'unSymsymmTmtrustSymBuildbuildLSDivlinspace<.>#><#<\>dot optimiseMultmeanCov sortVector sortIndexremapgaussianSample uniformSamplerandrandncorrconvcorrMincorr2conv2 separableCGStatecgpcgrcgr2cgxcgdxcgSolvecgSolve' Indexable!Normednorm_0norm_1norm_2norm_InfℂℝiCvectormatrixdispdiagl|||¦===——rowcol?¿cross norm_Frob norm_nuclearmagnitunitarysize pairwiseD2 rowOutersnull1null1symdispDots dispBlanks dispShort luPacked'luSolve'./.Mod<> linearSolve nullspaceorth<·>mulDispSizedunwrapextractcreateMLDomaindiagRHerSymEigen eigensystemSq&#vec2vec3vec4dimeyeblockAtunrowuncollinSolvesvdTallsvdFlat𝑖her withNullspacewithCompactSVDsplitheadTail splitRows splitCols withVector withMatrixmeansafeRead@>asReal asComplex mapVector buildVectoravecinlinePerformIOchunkchunks putVector getVector$fBinaryVector unsafeWithfinit errorCodembCatch..>::>:>OkCIdxsOMCVCM$fTransArrayVectorsumFsumRsumQsumCprodFprodRprodQprodC toScalarR toScalarF toScalarC toScalarQ toScalarI toScalarL vectorMapR vectorMapC vectorMapF vectorMapQ vectorMapI vectorMapL vectorMapValR vectorMapValC vectorMapValF vectorMapValQ vectorMapValI vectorMapValL vectorZipR vectorZipC vectorZipF vectorZipQ vectorZipI vectorZipLTConstTVVVTVVFunCodeSNorm2AbsSumMaxIdxMaxMinIdxMin FunCodeVVAddSubMulDivPowATan2 FunCodeSVScaleRecip AddConstantNegatePowSVPowVSModSVModVSFunCodeVSinCosTanAbsASinACosATanSinhCoshTanhASinhACoshATanhExpLogSignSqrt cconstantL cconstantI cconstantC cconstantQ cconstantR cconstantF c_conjugateC c_conjugateQc_stepLc_stepIc_stepDc_stepF c_long2int c_int2long c_float2int c_int2float c_double2long c_long2double c_double2int c_int2doublec_double2floatc_float2doublec_range_vectorc_round_vectorc_random_vector c_vectorScan c_vectorZipL c_vectorZipI c_vectorZipQ c_vectorZipF c_vectorZipC 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