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\section{Linear Algebra package} \begin{Introduction}{Linear Algebra package} This section briefly describes what's available in the Linear Algebra package. Note on examples: In the examples throughout this document, the matrix A will be \begin{verbatim} [1 2 3] [4 5 6] [7 8 9]. \end{verbatim} The functions can be divided into four categories: {\bf Basic matrix handling} \nameref{add\_columns}, \nameref{add\_rows}, \nameref{add\_to\_columns}, \nameref{add\_to\_rows}, \nameref{augment\_columns}, \nameref{char\_poly}, \nameref{column\_dim}, \nameref{copy\_into}, \nameref{diagonal}, \nameref{extend}, \nameref{find\_companion}, \nameref{get\_columns}, \nameref{get\_rows}, \nameref{hermitian\_tp}, \nameref{matrix\_augment}, \nameref{matrix\_stack} , \nameref{minor}, \nameref{mult\_columns}, \nameref{mult\_rows}, \nameref{pivot}, \nameref{remove\_columns}, \nameref{remove\_rows}, \nameref{row\_dim}, \nameref{rows\_pivot}, \nameref{stack\_rows}, \nameref{sub\_matrix}, \nameref{swap\_columns}, \nameref{swap\_entries}, \nameref{swap\_rows}. {\bf Constructors -- functions that create matrices} \nameref{band\_matrix}, \nameref{block\_matrix}, \nameref{char\_matrix}, \nameref{coeff\_matrix}, \nameref{companion}, \nameref{hessian}, \nameref{hilbert}, \nameref{jacobian}, \nameref{jordan\_block}, \nameref{make\_identity}, \nameref{random\_matrix}, \nameref{toeplitz}, \nameref{vandermonde}. {\bf High level algorithms} \nameref{char\_poly}, \nameref{cholesky}, \nameref{gram\_schmidt}, \nameref{lu\_decom}, \nameref{pseudo\_inverse}, \nameref{simplex}, \nameref{svd}. {\bf Normal Forms} There is a separate package, NORMFORM, for computing the following matrix normal forms in \REDUCE: \nameref{smithex}, \nameref{smithex\_int}, \nameref{frobenius}, \nameref{ratjordan}, \nameref{jordansymbolic}, \nameref{jordan}. {\bf Predicates} \nameref{matrixp}, \nameref{squarep}, \nameref{symmetricp}. \end{Introduction} \begin{Switch}{fast_la} By turning the \name{fast\_la} switch on, the speed of the following functions will be increased: \nameref{add\_columns}, \nameref{add\_rows}, \nameref{augment\_columns}, \nameref{column\_dim}, \nameref{copy\_into}, \nameref{make\_identity}, \nameref{matrix\_augment}, \nameref{matrix\_stack}, \nameref{minor}, \nameref{mult\_columns}, \nameref{mult\_rows}, \nameref{pivot}, \nameref{remove\_columns}, \nameref{remove\_rows}, \nameref{rows\_pivot}, \nameref{squarep}, \nameref{stack\_rows}, \nameref{sub\_matrix}, \nameref{swap\_columns}, \nameref{swap\_entries}, \nameref{swap\_rows}, \nameref{symmetricp}. The increase in speed will be negligible unless you are making a significant number (i.e. thousands) of calls. When using this switch, error checking is minimized. This means that illegal input may give strange error messages. Beware. \end{Switch} \begin{Operator}{add_columns} Add columns, add rows: \begin{Syntax} \name{add\_columns}\(\meta{matrix},\meta{c1},\meta{c2},\meta{expr}\) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{c1},\meta{c2} :- positive integers. \meta{expr} :- a scalar expression. The Operator \name{add\_columns} replaces column \meta{\meta{c2}} of \meta{matrix} by \meta{expr} * column(\meta{matrix},\meta{c1}) + column(\meta{matrix},\meta{c2}). \name{add\_rows} performs the equivalent task on the rows of \meta{matrix}. \begin{Examples} add_columns(A,1,2,x); & \begin{multilineoutput}{6cm} [1 x + 2 3] [ ] [4 4*x + 5 6] [ ] [7 7*x + 8 9] \end{multilineoutput} \\ add_rows(A,2,3,5); & \begin{multilineoutput}{6cm} [1 2 3 ] [ ] [4 5 6 ] [ ] [27 33 39] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{add\_to\_columns}, \nameref{add\_to\_rows}, \nameref{mult\_columns}, \nameref{mult\_rows}. \end{Operator} \begin{Operator}{add_rows} see: \nameref{add\_columns}. \end{Operator} \begin{Operator}{add_to_columns} Add to columns, add to rows: \begin{Syntax} \name{add\_to\_columns}\(\meta{matrix},\meta{column\_list},\meta{expr}\) \end{Syntax} \meta{matrix} :- a matrix. \meta{column\_list} :- a positive integer or a list of positive integers. \meta{expr} :- a scalar expression. \name{add\_to\_columns} adds \meta{expr} to each column specified in \meta{column\_list} of \meta{matrix}. \name{add\_to\_rows} performs the equivalent task on the rows of \meta{matrix}. \begin{Examples} add_to_columns(A,\{1,2\},10); & \begin{multilineoutput}{6cm} [11 12 3] [ ] [14 15 6] [ ] [17 18 9] \end{multilineoutput}\\ add_to_rows(A,2,-x) & \begin{multilineoutput}{6cm} [ 1 2 3 ] [ ] [ - x + 4 - x + 5 - x + 6] [ ] [ 7 8 9 ] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{add\_columns}, \nameref{add\_rows}, \nameref{mult\_rows}, \nameref{mult\_columns}. \end{Operator} \begin{Operator}{add_to_rows} see: \nameref{add\_to\_columns}. \end{Operator} \begin{Operator}{augment_columns} Augment columns, stack rows: \begin{Syntax} \name{augment\_columns}(\meta{matrix},\meta{column\_list}) \end{Syntax} \meta{matrix} :- a matrix. \meta{column\_list} :- either a positive integer or a list of positive integers. \name{augment\_columns} gets hold of the columns of \meta{matrix} specified in \name{column\_list} and sticks them together. \name{stack\_rows} performs the same task on rows of \meta{matrix}. \begin{Examples} augment_columns(A,\{1,2\}) & \begin{multilineoutput}{6cm} [1 2] [ ] [4 5] [ ] [7 8] \end{multilineoutput} \\ stack_rows(A,\{1,3\}) & \begin{multilineoutput}{6cm} [1 2 3] [ ] [7 8 9] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{get\_columns}, \nameref{get\_rows}, \nameref{sub\_matrix}. \end{Operator} \begin{Operator}{band_matrix} \begin{Syntax} \name{band\_matrix}(\meta{expr\_list},\meta{square\_size}) \end{Syntax} \meta{expr\_list} :- either a single scalar expression or a list of an odd number of scalar expressions. \meta{square\_size} :- a positive integer. \name{band\_matrix} creates a square matrix of dimension \meta{square\_size}. The diagonal consists of the middle expression of the \meta{expr\_list}. The expressions to the left of this fill the required number of sub\_diagonals and the expressions to the right the super\_diagonals. \begin{Examples} band_matrix(\{x,y,z\},6) & \begin{multilineoutput}{6cm} [y z 0 0 0 0] [ ] [x y z 0 0 0] [ ] [0 x y z 0 0] [ ] [0 0 x y z 0] [ ] [0 0 0 x y z] [ ] [0 0 0 0 x y] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{diagonal}. \end{Operator} \begin{Operator}{block_matrix} \begin{Syntax} \name{block\_matrix}(\meta{r},\meta{c},\meta{matrix\_list}) \end{Syntax} \meta{r},\meta{c} :- positive integers. \meta{matrix\_list} :- a list of matrices. \name{block\_matrix} creates a matrix that consists of \meta{r} by \meta{c} matrices filled from the \meta{matrix\_list} row wise. \begin{Examples} B := make_identity(2); & \begin{multilineoutput}{6cm} [1 0] b := [ ] [0 1] \end{multilineoutput} \\ C := mat((5),(5)); & \begin{multilineoutput}{6cm} [5] c := [ ] [5] \end{multilineoutput} \\ D := mat((22,33),(44,55)); & \begin{multilineoutput}{6cm} [22 33] d := [ ] [44 55] \end{multilineoutput} \\ block_matrix(2,3,\{B,C,D,D,C,B\}); & \begin{multilineoutput}{6cm} [1 0 5 22 33] [ ] [0 1 5 44 55] [ ] [22 33 5 1 0 ] [ ] [44 55 5 0 1 ] \end{multilineoutput} \\ \end{Examples} \end{Operator} \begin{Operator}{char_matrix} \begin{Syntax} \name{char\_matrix}(\meta{matrix},\meta{lambda}) \end{Syntax} \meta{matrix} :- a square matrix. \meta{lambda} :- a symbol or algebraic expression. \meta{char\_matrix} creates the characteristic matrix C of \meta{matrix}. This is C = \meta{lambda} * Id - A. Id is the identity matrix. \begin{Examples} char_matrix(A,x); & \begin{multilineoutput}{6cm} [x - 1 -2 -3 ] [ ] [ -4 x - 5 -6 ] [ ] [ -7 -8 x - 9] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{char\_poly}. \end{Operator} \begin{Operator}{char_poly} \begin{Syntax} \name{char\_poly}(\meta{matrix},\meta{lambda}) \end{Syntax} \meta{matrix} :- a square matrix. \meta{lambda} :- a symbol or algebraic expression. \name{char\_poly} finds the characteristic polynomial of \meta{matrix}. This is the determinant of \meta{lambda} * Id - A. Id is the identity matrix. \begin{Examples} char_poly(A,x); & x^3-15*x^2-18*x \end{Examples} Related functions: \nameref{char\_matrix}. \end{Operator} \begin{Operator}{cholesky} \begin{Syntax} \name{cholesky}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a positive definite matrix containing numeric entries. \name{cholesky} computes the cholesky decomposition of \meta{matrix}. It returns \{L,U\} where L is a lower matrix, U is an upper matrix, A = LU, and U = $L^T$. \begin{Examples} F := mat((1,1,0),(1,3,1),(0,1,1)); & \begin{multilineoutput}{6cm} [1 1 0] [ ] f := [1 3 1] [ ] [0 1 1] \end{multilineoutput} \\ on rounded; \\ cholesky(F); & \begin{multilineoutput}{6cm} \{ [1 0 0 ] [ ] [1 1.41421356237 0 ] [ ] [0 0.707106781187 0.707106781187] , [1 1 0 ] [ ] [0 1.41421356237 0.707106781187] [ ] [0 0 0.707106781187] \} \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{lu\_decom}. \end{Operator} \begin{Operator}{coeff_matrix} \begin{Syntax} \name{coeff\_matrix}(\{\meta{lineq\_list}\}) \end{Syntax} (If you are feeling lazy then the braces can be omitted.) \meta{lineq\_list} :- linear equations. Can be of the form equation = number or just equation. \name{coeff\_matrix} creates the coefficient matrix C of the linear equations. It returns \{C,X,B\} such that CX = B. \begin{Examples} coeff_matrix(\{x+y+4*z=10,y+x-z=20,x+y+4\}); & \begin{multilineoutput}{6cm} \{ [4 1 1] [ ] [-1 1 1] [ ] [0 1 1] , [z] [ ] [y] [ ] [x] , [10] [ ] [20] [ ] [-4] \} \end{multilineoutput} \\ \end{Examples} \end{Operator} \begin{Operator}{column_dim} Column dimension, row dimension: \begin{Syntax} \name{column\_dim}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a matrix. \name{column\_dim} finds the column dimension of \meta{matrix}. \name{row\_dim} finds the row dimension of \meta{matrix}. \begin{Examples} column_dim(A); & 3 \\ row_dim(A); & 3 \\ \end{Examples} \end{Operator} \begin{Operator}{companion} \begin{Syntax} \name{companion}(\meta{poly},\meta{x}) \end{Syntax} \meta{poly} :- a monic univariate polynomial in \meta{x}. \meta{x} :- the variable. \name{companion} creates the companion matrix C of \meta{poly}. This is the square matrix of dimension n, where n is the degree of \meta{poly} w.r.t. \meta{x}. The entries of C are: C(i,n) = -coeffn(\meta{poly},\meta{x},i-1) for i = 1 \ldots n, C(i,i-1) = 1 for i = 2 \ldots n and the rest are 0. \begin{Examples} companion(x^4+17*x^3-9*x^2+11,x); & \begin{multilineoutput}{6cm} [0 0 0 -11] [ ] [1 0 0 0 ] [ ] [0 1 0 9 ] [ ] [0 0 1 -17] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{find\_companion}. \end{Operator} \begin{Operator}{copy_into} \begin{Syntax} \name{copy\_into}(\meta{A},\meta{B},\meta{r},\meta{c}) \end{Syntax} \meta{A},\meta{B} :- matrices. \meta{r},\meta{c} :- positive integers. \name{copy\_into} copies matrix \meta{matrix} into \meta{B} with \meta{matrix}(1,1) at \meta{B}(\meta{r},\meta{c}). \begin{Examples} G := mat((0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0)); & \begin{multilineoutput}{6cm} [0 0 0 0 0] [ ] [0 0 0 0 0] [ ] g := [0 0 0 0 0] [ ] [0 0 0 0 0] [ ] [0 0 0 0 0] \end{multilineoutput} \\ copy_into(A,G,1,2); & \begin{multilineoutput}{6cm} [0 1 2 3 0] [ ] [0 4 5 6 0] [ ] [0 7 8 9 0] [ ] [0 0 0 0 0] [ ] [0 0 0 0 0] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{augment\_columns}, \nameref{extend}, \nameref{matrix\_augment}, \nameref{matrix\_stack}, \nameref{stack\_rows}, \nameref{sub\_matrix}. \end{Operator} \begin{Operator}{diagonal} \begin{Syntax} \name{diagonal}(\{\meta{mat\_list}\}) \end{Syntax} (If you are feeling lazy then the braces can be omitted.) \meta{mat\_list} :- each can be either a scalar expression or a square \nameref{matrix}. \name{diagonal} creates a matrix that contains the input on the diagonal. \begin{Examples} H := mat((66,77),(88,99)); & \begin{multilineoutput}{6cm} [66 77] h := [ ] [88 99] \end{multilineoutput} \\ diagonal(\{A,x,H\}); & \begin{multilineoutput}{6cm} [1 2 3 0 0 0 ] [ ] [4 5 6 0 0 0 ] [ ] [7 8 9 0 0 0 ] [ ] [0 0 0 x 0 0 ] [ ] [0 0 0 0 66 77] [ ] [0 0 0 0 88 99] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{jordan\_block}. \end{Operator} \begin{Operator}{extend} \begin{Syntax} \name{extend}(\meta{matrix},\meta{r},\meta{c},\meta{expr}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{r},\meta{c} :- positive integers. \meta{expr} :- algebraic expression or symbol. \name{extend} returns a copy of \meta{matrix} that has been extended by \meta{r} rows and \meta{c} columns. The new entries are made equal to \meta{expr}. \begin{Examples} extend(A,1,2,x); & \begin{multilineoutput}{6cm} [1 2 3 x x] [ ] [4 5 6 x x] [ ] [7 8 9 x x] [ ] [x x x x x] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{copy\_into}, \nameref{matrix\_augment}, \nameref{matrix\_stack}, \nameref{remove\_columns}, \nameref{remove\_rows}. \end{Operator} \begin{Operator}{find_companion} \begin{Syntax} \name{find\_companion}(\meta{matrix},\meta{x}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{x} :- the variable. Given a companion matrix, \name{find\_companion} finds the polynomial from which it was made. \begin{Examples} C := companion(x^4+17*x^3-9*x^2+11,x); & \begin{multilineoutput}{6cm} [0 0 0 -11] [ ] [1 0 0 0 ] c := [ ] [0 1 0 9 ] [ ] [0 0 1 -17] \end{multilineoutput}\\ find_companion(C,x); & x^4+17*x^3-9*x^2+11 \end{Examples} Related functions: \nameref{companion}. \end{Operator} \begin{Operator}{get_columns} Get columns, get rows: \begin{Syntax} \name{get\_columns}(\meta{matrix},\meta{column\_list}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{c} :- either a positive integer or a list of positive integers. \name{get\_columns} removes the columns of \meta{matrix} specified in \meta{column\_list} and returns them as a list of column matrices. \name{get\_rows} performs the same task on the rows of \meta{matrix}. \begin{Examples} get_columns(A,\{1,3\}); & \begin{multilineoutput}{6cm} \{ [1] [ ] [4] [ ] [7] , [3] [ ] [6] [ ] [9] \} \end{multilineoutput} \\ get_rows(A,2); & \begin{multilineoutput}{6cm} \{ [4 5 6] \} \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{augment\_columns}, \nameref{stack\_rows}, \nameref{sub\_matrix}. \end{Operator} \begin{Operator}{get_rows} see: \nameref{get\_columns}. \end{Operator} \begin{Operator}{gram_schmidt} \begin{Syntax} \name{gram\_schmidt}(\{\meta{vec\_list}\}) \end{Syntax} (If you are feeling lazy then the braces can be omitted.) \meta{vec\_list} :- linearly independent vectors. Each vector must be written as a list, eg:\{1,0,0\}. \name{gram\_schmidt} performs the gram\_schmidt orthonormalization on the input vectors. It returns a list of orthogonal normalized vectors. \begin{Examples} gram_schmidt(\{\{1,0,0\},\{1,1,0\},\{1,1,1\}\}); & \{\{1,0,0\},\{0,1,0\},\{0,0,1\}\} \\ gram_schmidt(\{\{1,2\},\{3,4\}\}); & \{\{ \rfrac{1}{{sqrt(5)}} , \rfrac{2}{sqrt(5)} \}, \{ \rfrac{2*sqrt(5)}{5} , \rfrac{-sqrt(5)}{5} \}\} \end{Examples} \end{Operator} \begin{Operator}{hermitian_tp} \begin{Syntax} \name{ hermitian\_tp}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \name{hermitian\_tp} computes the hermitian transpose of \meta{matrix}. This is a \nameref{matrix} in which the (i,j)'th entry is the conjugate of the (j,i)'th entry of \meta{matrix}. \begin{Examples} J := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); & \begin{multilineoutput}{6cm} [i + 1 i + 2 i + 3] [ ] j := [ 4 5 2 ] [ ] [ 1 i 0 ] \end{multilineoutput} \\ hermitian_tp(j); & \begin{multilineoutput}{6cm} [ - i + 1 4 1 ] [ ] [ - i + 2 5 - i] [ ] [ - i + 3 2 0 ] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{tp}. \end{Operator} \begin{Operator}{hessian} \begin{Syntax} \name{hessian}(\meta{expr},\meta{variable\_list}) \end{Syntax} \meta{expr} :- a scalar expression. \meta{variable\_list} :- either a single variable or a list of variables. \name{hessian} computes the hessian matrix of \meta{expr} w.r.t. the variables in \meta{variable\_list}. % Does df exist in the help pages? This is an n by n matrix where n is the number of variables and the (i,j)'th entry is \nameref{df}(\meta{expr},\meta{variable\_list}(i), \meta{variable\_list}(j)). \begin{Examples} hessian(x*y*z+x^2,\{w,x,y,z\}); & \begin{multilineoutput}{6cm} [0 0 0 0] [ ] [0 2 z y] [ ] [0 z 0 x] [ ] [0 y x 0] \end{multilineoutput}{6cm} \end{Examples} Related functions: \nameref{df}. \end{Operator} \begin{Operator}{hilbert} \begin{Syntax} \name{hilbert}(\meta{square\_size},\meta{expr}) \end{Syntax} \meta{square\_size} :- a positive integer. \meta{expr} :- an algebraic expression. \name{hilbert} computes the square hilbert matrix of dimension \meta{square\_size}. This is the symmetric matrix in which the (i,j)'th entry is 1/(i+j-\meta{expr}). \begin{Examples} hilbert(3,y+x); & \begin{multilineoutput}{6cm} [ - 1 - 1 - 1 ] [----------- ----------- -----------] [ x + y - 2 x + y - 3 x + y - 4 ] [ ] [ - 1 - 1 - 1 ] [----------- ----------- -----------] [ x + y - 3 x + y - 4 x + y - 5 ] [ ] [ - 1 - 1 - 1 ] [----------- ----------- -----------] [ x + y - 4 x + y - 5 x + y - 6 ] \end{multilineoutput} \end{Examples} \end{Operator} \begin{Operator}{jacobian} \begin{Syntax} \name{jacobian}(\meta{expr\_list},\meta{variable\_list}) \end{Syntax} \meta{expr\_list} :- either a single algebraic expression or a list of algebraic expressions. \meta{variable\_list} :- either a single variable or a list of variables. \name{jacobian} computes the jacobian matrix of \meta{expr\_list} w.r.t. \meta{variable\_list}. This is a matrix whose (i,j)'th entry is \nameref{df}(\meta{expr\_list} (i),\meta{variable\_list}(j)). The matrix is n by m where n is the number of variables and m the number of expressions. \begin{Examples} jacobian(\{x^4,x*y^2,x*y*z^3\},\{w,x,y,z\}); & \begin{multilineoutput}{6cm} [ 3 ] [0 4*x 0 0 ] [ ] [ 2 ] [0 y 2*x*y 0 ] [ ] [ 3 3 2] [0 y*z x*z 3*x*y*z ] \end{multilineoutput} \end{Examples} Related functions: \nameref{hessian}, \nameref{df}. \end{Operator} \begin{Operator}{jordan_block} \begin{Syntax} \name{jordan\_block}(\meta{expr},\meta{square\_size}) \end{Syntax} \meta{expr} :- an algebraic expression or symbol. \meta{square\_size} :- a positive integer. \name{jordan\_block} computes the square jordan block matrix J of dimension \meta{square\_size}. The entries of J are: J(i,i) = \meta{expr} for i=1 \ldots n, J(i,i+1) = 1 for i=1 \ldots n-1, and all other entries are 0. \begin{Examples} jordan\_block(x,5); & \begin{multilineoutput}{6cm} [x 1 0 0 0] [ ] [0 x 1 0 0] [ ] [0 0 x 1 0] [ ] [0 0 0 x 1] [ ] [0 0 0 0 x] \end{multilineoutput} \end{Examples} Related functions: \nameref{diagonal}, \nameref{companion}. \end{Operator} \begin{Operator}{lu_decom} \begin{Syntax} \name{lu\_decom}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a \nameref{matrix} containing either numeric entries or imaginary entries with numeric coefficients. \name{lu\_decom} performs LU decomposition on \meta{matrix}, ie: it returns \{L,U\} where L is a lower diagonal \nameref{matrix}, U an upper diagonal \nameref{matrix} and A = LU. Caution: The algorithm used can swap the rows of \meta{matrix} during the calculation. This means that LU does not equal \meta{matrix} but a row equivalent of it. Due to this, \name{lu\_decom} returns \{L,U,vec\}. The call \name{convert(\meta{matrix},vec)} will return the matrix that has been decomposed, i.e: LU = convert(\meta{matrix},vec). \begin{Examples} K := mat((1,3,5),(-4,3,7),(8,6,4)); & \begin{multilineoutput}{6cm} [1 3 5] [ ] k := [-4 3 7] [ ] [8 6 4] \end{multilineoutput}\\ on rounded;\\ lu := lu_decom(K); & \begin{multilineoutput}{6cm} lu := \{ [8 0 0 ] [ ] [-4 6.0 0 ] [ ] [1 2.25 1.125] , [1 0.75 0.5] [ ] [0 1 1.5] [ ] [0 0 1 ] , [3 2 3]\} \end{multilineoutput} \\ first lu * second lu; & \begin{multilineoutput}{6cm} [8 6.0 4.0] [ ] [-4 3.0 7.0] [ ] [1 3.0 5.0] \end{multilineoutput}\\ convert(K,third lu); & \begin{multilineoutput} [8 6 4] [ ] [-4 3 7] [ ] [1 3 5] \end{multilineoutput}\\ P := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); & \begin{multilineoutput}{6cm} [i + 1 i + 2 i + 3] [ ] p := [ 4 5 2 ] [ ] [ 1 i 0 ] \end{multilineoutput}\\ lu := lu_decom(P); & \begin{multilineoutput}{6cm} lu := \{ [ 1 0 0 ] [ ] [ 4 - 4*i + 5 0 ] [ ] [i + 1 3 0.414634146341*i + 2.26829268293] , [1 i 0 ] [ ] [0 1 0.19512195122*i + 0.243902439024] [ ] [0 0 1 ] , [3 2 3]\} \end{multilineoutput}\\ first lu * second lu; & \begin{multilineoutput}{6cm} [ 1 i 0 ] [ ] [ 4 5 2.0 ] [ ] [i + 1 i + 2 i + 3.0] \end{multilineoutput}\\ convert(P,third lu); & \begin{multilineoutput}{6cm} [ 1 i 0 ] [ ] [ 4 5 2 ] [ ] [i + 1 i + 2 i + 3] \end{multilineoutput}\\ \end{Examples} Related functions: \nameref{cholesky}. \end{Operator} \begin{Operator}{make_identity} \begin{Syntax} \name{make\_identity}(\meta{square\_size}) \end{Syntax} \meta{square\_size} :- a positive integer. \name{make\_identity} creates the identity matrix of dimension \meta{square\_size}. \begin{Examples} make_identity(4); & \begin{multilineoutput}{6cm} [1 0 0 0] [ ] [0 1 0 0] [ ] [0 0 1 0] [ ] [0 0 0 1] \end{multilineoutput} \end{Examples} Related functions: \nameref{diagonal}. \end{Operator} \begin{Operator}{matrix_augment} Matrix augment, matrix stack: \begin{Syntax} \name{matrix\_augment} \{\meta{matrix\_list}\} \end{Syntax} (If you are feeling lazy then the braces can be omitted.) \meta{matrix\_list} :- matrices. \name{matrix\_augment} sticks the matrices in \meta{matrix\_list} together horizontally. \name{matrix\_stack} sticks the matrices in \meta{matrix\_list} together vertically. \begin{Examples} matrix_augment(\{A,A\}); & \begin{multilineoutput}{6cm} [1 2 3 1 2 3] [ ] [4 5 6 4 5 6] [ ] [7 8 9 7 8 9] \end{multilineoutput}\\ matrix_stack(A,A); & \begin{multilineoutput}{6cm} [1 2 3] [ ] [4 5 6] [ ] [7 8 9] [ ] [1 2 3] [ ] [4 5 6] [ ] [7 8 9] \end{multilineoutput} \\ \end{Examples} Related functions: \nameref{augment\_columns}, \nameref{stack\_rows}, \nameref{sub\_matrix}. \end{Operator} \begin{Operator}{matrixp} \begin{Syntax} \name{matrixp}(\meta{test\_input}) \end{Syntax} \meta{test\_input} :- anything you like. \name{matrixp} is a boolean function that returns t if the input is a matrix and nil otherwise. \begin{Examples} matrixp A; & t \\ matrixp(doodlesackbanana);& nil \end{Examples} Related functions: \nameref{squarep}, \nameref{symmetricp}. \end{Operator} \begin{Operator}{matrix_stack} see: \nameref{matrix\_augment}. \end{Operator} \begin{Operator}{minor} \begin{Syntax} \name{minor}(\meta{matrix},\meta{r},\meta{c}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{r},\meta{c} :- positive integers. \name{minor} computes the (\meta{r},\meta{c})'th minor of \meta{matrix}. This is created by removing the \meta{r}'th row and the \meta{c}'th column from \meta{matrix}. \begin{Examples} minor(A,1,3); & \begin{multilineoutput}{6cm} [4 5] [ ] [7 8] \end{multilineoutput} \end{Examples} Related functions: \nameref{remove\_columns}, \nameref{remove\_rows}. \end{Operator} \begin{Operator}{mult_columns} Mult columns, mult rows: \begin{Syntax} \name{mult\_columns}(\meta{matrix},\meta{column\_list},\meta{expr}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{column\_list} :- a positive integer or a list of positive integers. \meta{expr} :- an algebraic expression. \name{mult\_columns} returns a copy of \meta{matrix} in which the columns specified in \meta{column\_list} have been multiplied by \meta{expr}. \name{mult\_rows} performs the same task on the rows of \meta{matrix}. \begin{Examples} mult_columns(A,\{1,3\},x); & \begin{multilineoutput}{6cm} [ x 2 3*x] [ ] [4*x 5 6*x] [ ] [7*x 8 9*x] \end{multilineoutput}\\ mult_rows(A,2,10); & \begin{multilineoutput}{6cm} [1 2 3 ] [ ] [40 50 60] [ ] [7 8 9 ] \end{multilineoutput} \end{Examples} Related functions: \nameref{add\_to\_columns}, \nameref{add\_to\_rows}. \end{Operator} \begin{Operator}{mult_rows} see: \nameref{mult\_columns}. \end{Operator} \begin{Operator}{pivot} \begin{Syntax} \name{pivot}(\meta{matrix},\meta{r},\meta{c}) \end{Syntax} \meta{matrix} :- a matrix. \meta{r},\meta{c} :- positive integers such that \meta{matrix}(\meta{r}, \meta{c}) neq 0. \name{pivot} pivots \meta{matrix} about it's (\meta{r},\meta{c})'th entry. To do this, multiples of the \meta{r}'th row are added to every other row in the matrix. This means that the \meta{c}'th column will be 0 except for the (\meta{r},\meta{c})'th entry. \begin{Examples} pivot(A,2,3); & \begin{multilineoutput}{6cm} [ - 1 ] [-1 ------ 0] [ 2 ] [ ] [4 5 6] [ ] [ 1 ] [1 --- 0] [ 2 ] \end{multilineoutput} \end{Examples} Related functions: \nameref{rows\_pivot}. \end{Operator} \begin{Operator}{pseudo_inverse} \begin{Syntax} \name{pseudo\_inverse}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \name{pseudo\_inverse}, also known as the Moore-Penrose inverse, computes the pseudo inverse of \meta{matrix}. Given the singular value decomposition of \meta{matrix}, i.e: A = $U*P*V^T$, then the pseudo inverse $A^{-1}$ is defined by $A^{-1} = V^T*P^{-1}*U$. Thus \meta{matrix} * pseudo\_inverse(A) = Id. (Id is the identity matrix). \begin{Examples} R := mat((1,2,3,4),(9,8,7,6)); & \begin{multilineoutput}{6cm} [1 2 3 4] r := [ ] [9 8 7 6] \end{multilineoutput}\\ on rounded; \\ pseudo_inverse(R); & \begin{multilineoutput}{6cm} [ - 0.199999999996 0.100000000013 ] [ ] [ - 0.0499999999988 0.0500000000037 ] [ ] [ 0.0999999999982 - 5.57825497203e-12] [ ] [ 0.249999999995 - 0.0500000000148 ] \end{multilineoutput} \end{Examples} Related functions: \nameref{svd}. \end{Operator} \begin{Operator}{random_matrix} \begin{Syntax} \name{random\_matrix}(\meta{r},\meta{c},\meta{limit}) \end{Syntax} \meta{r},\meta{c},\meta{limit} :- positive integers. \name{random\_matrix} creates an \meta{r} by \meta{c} matrix with random entries in the range -limit < entry < limit. Switches: \name{imaginary} :- if on then matrix entries are x+i*y where -limit < x,y < \meta{limit}. \name{not\_negative} :- if on then 0 < entry < \meta{limit}. In the imaginary case we have 0 < x,y < \meta{limit}. \name{only\_integer} :- if on then each entry is an integer. In the imaginary case x and y are integers. \name{symmetric} :- if on then the matrix is symmetric. \name{upper\_matrix} :- if on then the matrix is upper triangular. \name{lower\_matrix} :- if on then the matrix is lower triangular. \begin{Examples} on rounded; \\ random_matrix(3,3,10); & \begin{multilineoutput}{6cm} [ - 8.11911717343 - 5.71677292768 0.620580830035 ] [ ] [ - 0.032596262422 7.1655452861 5.86742633837 ] [ ] [ - 9.37155438255 - 7.55636708637 - 8.88618627557] \end{multilineoutput}\\ on only_integer, not_negative, upper_matrix, imaginary; \\ random_matrix(4,4,10); & \begin{multilineoutput}{6cm} [70*i + 15 28*i + 8 2*i + 79 27*i + 44] [ ] [ 0 46*i + 95 9*i + 63 95*i + 50] [ ] [ 0 0 31*i + 75 14*i + 65] [ ] [ 0 0 0 5*i + 52 ] \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{remove_columns} Remove columns, remove rows: \begin{Syntax} \name{remove\_columns}(\meta{matrix},\meta{column\_list}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{column\_list} :- either a positive integer or a list of positive integers. \name{remove\_columns} removes the columns specified in \meta{column\_list} from \meta{matrix}. \name{remove\_rows} performs the same task on the rows of \meta{matrix}. \begin{Examples} remove_columns(A,2); & \begin{multilineoutput}{6cm} [1 3] [ ] [4 6] [ ] [7 9] \end{multilineoutput}\\ remove_rows(A,\{1,3\}); & \begin{multilineoutput}{6cm} [4 5 6] \end{multilineoutput}\\ \end{Examples} Related functions: \nameref{minor}. \end{Operator} \begin{Operator}{remove_rows} see: \nameref{remove\_columns}. \end{Operator} \begin{Operator}{row_dim} see: \nameref{column\_dim}. \end{Operator} \begin{Operator}{rows_pivot} \begin{Syntax} \name{rows\_pivot}(\meta{matrix},\meta{r},\meta{c},\{\meta{row\_list}\}) \end{Syntax} \meta{matrix} :- a nameref{matrix}. \meta{r},\meta{c} :- positive integers such that \meta{matrix}(\meta{r}, \meta{c}) neq 0. \meta{row\_list} :- positive integer or a list of positive integers. \name{rows\_pivot} performs the same task as \name{pivot} but applies the pivot only to the rows specified in \meta{row\_list}. \begin{Examples} N := mat((1,2,3),(4,5,6),(7,8,9),(1,2,3),(4,5,6)); & \begin{multilineoutput}{6cm} [1 2 3] [ ] [4 5 6] [ ] n := [7 8 9] [ ] [1 2 3] [ ] [4 5 6] \end{multilineoutput}\\ rows_pivot(N,2,3,{4,5}); & \begin{multilineoutput}{6cm} [1 2 3] [ ] [4 5 6] [ ] [7 8 9] [ ] [ - 1 ] [-1 ------ 0] [ 2 ] [ ] [0 0 0] \end{multilineoutput}\\ \end{Examples} Related functions: \nameref{pivot}. \end{Operator} \begin{Operator}{simplex} \begin{Syntax} \name{simplex}(\meta{max/min},\meta{objective function}, \{\meta{linear inequalities}\}) \end{Syntax} \meta{max/min} :- either max or min (signifying maximize and minimize). \meta{objective function} :- the function you are maximizing or minimizing. \meta{linear inequalities} :- the constraint inequalities. Each one must be of the form {\it sum of variables ( <=,=,>=) number}. \name{simplex} applies the revised simplex algorithm to find the optimal(either maximum or minimum) value of the \meta{objective function} under the linear inequality constraints. It returns \{optimal value,\{ values of variables at this optimal\}\}. The algorithm implies that all the variables are non-negative. \begin{Examples} simplex(max,x+y,\{x>=10,y>=20,x+y<=25\}); & ***** Error in simplex: Problem has no feasible solution\\ simplex(max,10x+5y+5.5z,\{5x+3z<=200,x+0.1y+0.5z<=12, 0.1x+0.2y+0.3z<=9, 30x+10y+50z<=1500\}); & \{525.0,\{x=40.0,y=25.0,z=0\}\}\\ \end{Examples} \end{Operator} \begin{Operator}{squarep} \begin{Syntax} \name{squarep}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \name{squarep} is a predicate that returns t if the \meta{matrix} is square and nil otherwise. \begin{Examples} squarep(mat((1,3,5))); & nil \\ squarep(A); t\\ \end{Examples} Related functions: \nameref{matrixp}, \nameref{symmetricp}. \end{Operator} \begin{Operator}{stack_rows} see: \nameref{augment\_columns}. \end{Operator} \begin{Operator}{sub_matrix} \begin{Syntax} \name{sub\_matrix}(\meta{matrix},\meta{row\_list},\meta{column\_list}) \end{Syntax} \meta{matrix} :- a matrix. \meta{row\_list}, \meta{column\_list} :- either a positive integer or a list of positive integers. name{sub\_matrix} produces the matrix consisting of the intersection of the rows specified in \meta{row\_list} and the columns specified in \meta{column\_list}. \begin{Examples} sub_matrix(A,\{1,3\},\{2,3\}); & \begin{multilineoutput}{6cm} [2 3] [ ] [8 9] \end{multilineoutput} \end{Examples} Related functions: \nameref{augment\_columns}, \nameref{stack\_rows}. \end{Operator} \begin{Operator}{svd} \index{singular value decomposition} Singular value decomposition: \begin{Syntax} \name{svd}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a \nameref{matrix} containing only numeric entries. \name{svd} computes the singular value decomposition of \meta{matrix}. It returns \{U,P,V\} where A = $U*P*V^T$ and P = diag(sigma(1) ... sigma(n)). sigma(i) for i= 1 ... n are the singular values of \meta{matrix}. n is the column dimension of \meta{matrix}. The singular values of \meta{matrix} are the non-negative square roots of the eigenvalues of $A^T*A$. U and V are such that $U*U^T = V*V^T = V^T*V$ = Id. Id is the identity matrix. \begin{Examples} Q := mat((1,3),(-4,3)); & \begin{multilineoutput}{6cm} [1 3] q := [ ] [-4 3] \end{multilineoutput}\\ on rounded; \\ svd(Q); & \begin{multilineoutput}{6cm} \{ [ 0.289784137735 0.957092029805] [ ] [ - 0.957092029805 0.289784137735] , [5.1491628629 0 ] [ ] [ 0 2.9130948854] , [ - 0.687215403194 0.726453707825 ] [ ] [ - 0.726453707825 - 0.687215403194] \} \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{swap_columns} Swap columns, swap rows: \begin{Syntax} \name{swap\_columns} (\meta{matrix},\meta{c1},\meta{c2}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{c1},\meta{c1} :- positive integers. \name{swap\_columns} swaps column \meta{c1} of \meta{matrix} with column \meta{c2}. \name{swap\_rows} performs the same task on two rows of \meta{matrix}. \begin{Examples} swap_columns(A,2,3); & \begin{multilineoutput}{6cm} [1 3 2] [ ] [4 6 5] [ ] [7 9 8] \end{multilineoutput}\\ swap_rows(A,1,3); & \begin{multilineoutput}{6cm} [7 8 9] [ ] [4 5 6] [ ] [1 2 3] \end{multilineoutput} \end{Examples} Related functions: \nameref{swap\_entries}. \end{Operator} \begin{Operator}{swap_entries} \begin{Syntax} \name{swap\_entries}(\meta{matrix},\{\meta{r1},\meta{c1}\},\{\meta{r2}, \meta{c2}\}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \meta{r1},\meta{c1},\meta{r2},\meta{c2} :- positive integers. \name{swap\_entries} swaps \meta{matrix}(\meta{r1},\meta{c1}) with \meta{matrix}(\meta{r2},\meta{c2}). \begin{Examples} swap_entries(A,\{1,1\},\{3,3\}); & \begin{multilineoutput}{6cm} [9 2 3] [ ] [4 5 6] [ ] [7 8 1] \end{multilineoutput} \end{Examples} Related functions: \nameref{swap\_columns}, \nameref{swap\_rows}. \end{Operator} \begin{Operator}{swap_rows} see: \nameref{swap\_columns}. \end{Operator} \begin{Operator}{symmetricp} \begin{Syntax} \name{symmetricp}(\meta{matrix}) \end{Syntax} \meta{matrix} :- a \nameref{matrix}. \name{symmetricp} is a predicate that returns t if the matrix is symmetric and nil otherwise. \begin{Examples} symmetricp(make_identity(11)); & t \\ symmetricp(A); & nil\\ \end{Examples} Related functions: \nameref{matrixp}, \nameref{squarep}. \end{Operator} \begin{Operator}{toeplitz} \begin{Syntax} \name{toeplitz}(\meta{expr\_list}) \end{Syntax} (If you are feeling lazy then the braces can be omitted.) \meta{expr\_list} :- list of algebraic expressions. \name{toeplitz} creates the toeplitz matrix from the \meta{expr\_list}. This is a square symmetric matrix in which the first expression is placed on the diagonal and the i'th expression is placed on the (i-1)'th sub and super diagonals. It has dimension n where n is the number of expressions. \begin{Examples} toeplitz({w,x,y,z}); & \begin{multilineoutput}{6cm} [w x y z] [ ] [x w x y] [ ] [y x w x] [ ] [z y x w] \end{multilineoutput} \end{Examples} \end{Operator} \begin{Operator}{vandermonde} \begin{Syntax} \name{vandermonde}(\{\meta{expr\_list}\}) \end{Syntax} (If you are feeling lazy then the braces can be omitted.) \meta{expr\_list} :- list of algebraic expressions. \name{vandermonde} creates the vandermonde matrix from the \meta{expr\_list}. This is the square matrix in which the (i,j)'th entry is \meta{expr\_list}$(i)^(j-1)$. It has dimension n where n is the number of expressions. \begin{Examples} vandermonde({x,2*y,3*z}); & \begin{multilineoutput}{6cm} [ 2 ] [1 x x ] [ ] [ 2] [1 2*y 4*y ] [ ] [ 2] [1 3*z 9*z ] \end{multilineoutput} \end{Examples} \end{Operator}