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854 lines
34 KiB
OpenSCAD
854 lines
34 KiB
OpenSCAD
//////////////////////////////////////////////////////////////////////
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// LibFile: linalg.scad
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// This file provides linear algebra, with support for matrix construction,
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// solutions to linear systems of equations, QR and Cholesky factorizations, and
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// matrix inverse.
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// Includes:
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// include <BOSL2/std.scad>
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// FileGroup: Math
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// FileSummary: Linear Algebra: solve linear systems, construct and modify matrices.
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// FileFootnotes: STD=Included in std.scad
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//////////////////////////////////////////////////////////////////////
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// Section: Matrices
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// The matrix, a rectangular array of numbers which represents a linear transformation,
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// is the fundamental object in linear algebra. In OpenSCAD a matrix is a list of lists of numbers
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// with a rectangular structure. Because OpenSCAD treats all data the same, most of the functions that
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// index matrices or construct them will work on matrices (lists of lists) whose elements are not numbers but may be
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// arbitrary data: strings, booleans, or even other lists. It may even be acceptable in some cases if the structure is non-rectangular.
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// Of course, linear algebra computations and solutions require true matrices with rectangular structure, where all the entries are
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// finite numbers.
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// .
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// Matrices in OpenSCAD are lists of row vectors. However, a potential source of confusion is that OpenSCAD
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// treats vectors as either column vectors or row vectors as demanded by
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// context. Thus both `v*M` and `M*v` are valid if `M` is square and `v` has the right length. If you want to multiply
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// `M` on the left by `v` and `w` you can do this with `[v,w]*M` but if you want to multiply on the right side with `v` and `w` as
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// column vectors, you now need to use {{transpose()}} because OpenSCAD doesn't adjust matrices
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// contextually: `A=M*transpose([v,w])`. The solutions are now columns of A and you must extract
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// them with {{column()}} or take the transpose of `A`.
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// Section: Matrix testing and display
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// Function: is_matrix()
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// Synopsis: Check if input is a numeric matrix, optionally of specified size
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// Topics: Matrices
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// See Also: is_matrix_symmetric(), is_rotation()
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// Usage:
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// test = is_matrix(A, [m], [n], [square])
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// Description:
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// Returns true if A is a numeric matrix of height m and width n with finite entries. If m or n
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// are omitted or set to undef then true is returned for any positive dimension.
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// Arguments:
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// A = The matrix to test.
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// m = If given, requires the matrix to have this height.
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// n = Is given, requires the matrix to have this width.
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// square = If true, matrix must have height equal to width. Default: false
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function is_matrix(A,m,n,square=false) =
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is_list(A)
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&& (( is_undef(m) && len(A) ) || len(A)==m)
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&& (!square || len(A) == len(A[0]))
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&& is_vector(A[0],n)
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&& is_consistent(A);
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// Function: is_matrix_symmetric()
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// Synopsis: Checks if matrix is symmetric
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// Topics: Matrices
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// See Also: is_matrix(), is_rotation()
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// Usage:
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// b = is_matrix_symmetric(A, [eps])
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// Description:
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// Returns true if the input matrix is symmetric, meaning it approximately equals its transpose.
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// The matrix can have arbitrary entries.
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// Arguments:
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// A = matrix to test
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// eps = epsilon for comparing equality. Default: 1e-12
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function is_matrix_symmetric(A,eps=1e-12) =
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approx(A,transpose(A), eps);
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// Function: is_rotation()
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// Synopsis: Check if a transformation matrix represents a rotation.
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// Topics: Affine, Matrices, Transforms
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// See Also: is_matrix(), is_matrix_symmetric(), is_rotation()
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// Usage:
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// b = is_rotation(A, [dim], [centered])
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// Description:
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// Returns true if the input matrix is a square affine matrix that is a rotation around any point,
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// or around the origin if `centered` is true.
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// The matrix must be 3x3 (representing a 2d transformation) or 4x4 (representing a 3d transformation).
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// You can set `dim` to 2 to require a 2d transform (3x3 matrix) or to 3 to require a 3d transform (4x4 matrix).
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// Arguments:
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// A = matrix to test
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// dim = if set, specify dimension in which the transform operates (2 or 3)
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// centered = if true then require rotation to be around the origin. Default: false
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function is_rotation(A,dim,centered=false) =
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let(n=len(A))
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is_matrix(A,square=true)
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&& ( n==3 || n==4 && (is_undef(dim) || dim==n-1))
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&&
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(
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let(
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rotpart = [for(i=[0:n-2]) [for(j=[0:n-2]) A[j][i]]]
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)
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approx(determinant(rotpart),1)
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)
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&&
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(!centered || [for(row=[0:n-2]) if (!approx(A[row][n-1],0)) row]==[]);
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// Function&Module: echo_matrix()
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// Synopsis: Print a matrix neatly to the console.
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// Topics: Matrices
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// See Also: is_matrix(), is_matrix_symmetric(), is_rotation()
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// Usage:
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// echo_matrix(M, [description], [sig], [sep], [eps]);
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// dummy = echo_matrix(M, [description], [sig], [sep], [eps]),
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// Description:
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// Display a numerical matrix in a readable columnar format with `sig` significant
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// digits. Values smaller than eps display as zero. If you give a description
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// it is displayed at the top. You can change the space between columns by
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// setting `sep` to a number of spaces, which will use wide figure spaces the same
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// width as digits, or you can set it to any string to separate the columns.
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// Values that are NaN or INF will display as "nan" and "inf". Values which are
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// otherwise non-numerica display as two dashes. Note that this includes lists, so
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// a 3D array will display as a list of dashes.
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// Arguments:
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// M = matrix to display, which should be numerical
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// description = optional text to print before the matrix
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// sig = number of digits to display. Default: 4
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// sep = number of spaces between columns or a text string to separate columns. Default: 1
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// eps = numbers smaller than this display as zero. Default: 1e-9
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function echo_matrix(M,description,sig=4,sep=1,eps=1e-9) =
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let(
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horiz_line = chr(8213),
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matstr = _format_matrix(M,sig=sig,sep=sep,eps=eps),
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separator = str_join(repeat(horiz_line,10)),
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dummy=echo(str(separator,is_def(description) ? str(" ",description) : ""))
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[for(row=matstr) echo(row)]
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)
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echo(separator);
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module echo_matrix(M,description,sig=4,sep=1,eps=1e-9)
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{
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dummy = echo_matrix(M,description,sig,sep,eps);
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}
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// Section: Matrix indexing
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// Function: column()
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// Synopsis: Extract a column from a matrix.
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// Topics: Matrices, List Handling, Arrays
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// See Also: select(), slice()
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// Usage:
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// list = column(M, i);
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// Description:
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// Extracts entry `i` from each list in M, or equivalently column i from the matrix M, and returns it as a vector.
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// This function will return `undef` at all entry positions indexed by i not found in M.
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// Arguments:
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// M = The given list of lists.
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// i = The index to fetch
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// Example:
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// M = [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]];
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// a = column(M,2); // Returns [3, 7, 11, 15]
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// b = column(M,0); // Returns [1, 5, 9, 13]
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// N = [ [1,2], [3], [4,5], [6,7,8] ];
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// c = column(N,1); // Returns [1,undef,5,7]
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// data = [[1,[3,4]], [3, [9,3]], [4, [3,1]]]; // Matrix with non-numeric entries
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// d = column(data,0); // Returns [1,3,4]
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// e = column(data,1); // Returns [[3,4],[9,3],[3,1]]
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function column(M, i) =
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assert( is_list(M), "The input is not a list." )
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assert( is_int(i) && i>=0, "Invalid index")
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[for(row=M) row[i]];
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// Function: submatrix()
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// Synopsis: Extract a submatrix from a matrix
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// Topics: Matrices, Arrays
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// See Also: column(), block_matrix(), submatrix_set()
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// Usage:
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// mat = submatrix(M, idx1, idx2);
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// Description:
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// The input must be a list of lists (a matrix or 2d array). Returns a submatrix by selecting the rows listed in idx1 and columns listed in idx2.
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// Arguments:
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// M = Given list of lists
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// idx1 = rows index list or range
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// idx2 = column index list or range
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// Example:
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// M = [[ 1, 2, 3, 4, 5],
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// [ 6, 7, 8, 9,10],
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// [11,12,13,14,15],
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// [16,17,18,19,20],
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// [21,22,23,24,25]];
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// submatrix(M,[1:2],[3:4]); // Returns [[9, 10], [14, 15]]
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// submatrix(M,[1], [3,4])); // Returns [[9,10]]
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// submatrix(M,1, [3,4])); // Returns [[9,10]]
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// submatrix(M,1,3)); // Returns [[9]]
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// submatrix(M, [3,4],1); // Returns [[17],[22]]);
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// submatrix(M, [1,3],[2,4]); // Returns [[8,10],[18,20]]);
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// A = [[true, 17, "test"],
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// [[4,2], 91, false],
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// [6, [3,4], undef]];
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// submatrix(A,[0,2],[1,2]); // Returns [[17, "test"], [[3, 4], undef]]
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function submatrix(M,idx1,idx2) =
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[for(i=idx1) [for(j=idx2) M[i][j] ] ];
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// Section: Matrix construction and modification
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// Function: ident()
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// Synopsis: Return identity matrix.
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// Topics: Affine, Matrices, Transforms
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// See Also: IDENT, submatrix(), column()
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// Usage:
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// mat = ident(n);
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// Description:
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// Create an `n` by `n` square identity matrix.
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// Arguments:
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// n = The size of the identity matrix square, `n` by `n`.
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// Example:
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// mat = ident(3);
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// // Returns:
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// // [
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// // [1, 0, 0],
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// // [0, 1, 0],
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// // [0, 0, 1]
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// // ]
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// Example:
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// mat = ident(4);
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// // Returns:
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// // [
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// // [1, 0, 0, 0],
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// // [0, 1, 0, 0],
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// // [0, 0, 1, 0],
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// // [0, 0, 0, 1]
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// // ]
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function ident(n) = [
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for (i = [0:1:n-1]) [
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for (j = [0:1:n-1]) (i==j)? 1 : 0
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]
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];
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// Function: diagonal_matrix()
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// Synopsis: Make a diagonal matrix.
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// Topics: Affine, Matrices
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// See Also: column(), submatrix()
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// Usage:
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// mat = diagonal_matrix(diag, [offdiag]);
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// Description:
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// Creates a square matrix with the items in the list `diag` on
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// its diagonal. The off diagonal entries are set to offdiag,
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// which is zero by default.
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// Arguments:
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// diag = A list of items to put in the diagnal cells of the matrix.
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// offdiag = Value to put in non-diagonal matrix cells.
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function diagonal_matrix(diag, offdiag=0) =
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assert(is_list(diag) && len(diag)>0)
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[for(i=[0:1:len(diag)-1]) [for(j=[0:len(diag)-1]) i==j?diag[i] : offdiag]];
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// Function: transpose()
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// Synopsis: Transpose a matrix
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// Topics: Linear Algebra, Matrices
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// See Also: submatrix(), block_matrix(), hstack(), flatten()
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// Usage:
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// M = transpose(M, [reverse]);
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// Description:
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// Returns the transpose of the given input matrix. The input can be a matrix with arbitrary entries or
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// a numerical vector. If you give a vector then transpose returns it unchanged.
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// When reverse=true, the transpose is done across to the secondary diagonal. (See example below.)
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// By default, reverse=false.
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// Example:
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// M = [
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// [1, 2, 3],
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// [4, 5, 6],
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// [7, 8, 9]
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// ];
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// t = transpose(M);
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// // Returns:
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// // [
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// // [1, 4, 7],
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// // [2, 5, 8],
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// // [3, 6, 9]
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// // ]
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// Example:
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// M = [
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// [1, 2, 3],
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// [4, 5, 6]
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// ];
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// t = transpose(M);
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// // Returns:
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// // [
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// // [1, 4],
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// // [2, 5],
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// // [3, 6],
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// // ]
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// Example:
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// M = [
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// [1, 2, 3],
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// [4, 5, 6],
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// [7, 8, 9]
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// ];
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// t = transpose(M, reverse=true);
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// // Returns:
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// // [
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// // [9, 6, 3],
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// // [8, 5, 2],
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// // [7, 4, 1]
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// // ]
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// Example: Transpose on a list of numbers returns the list unchanged
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// transpose([3,4,5]); // Returns: [3,4,5]
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// Example: Transpose on non-numeric input
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// arr = [
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// [ "a", "b", "c"],
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// [ "d", "e", "f"],
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// [[1,2],[3,4],[5,6]]
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// ];
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// t = transpose(arr);
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// // Returns:
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// // [
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// // ["a", "d", [1,2]],
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// // ["b", "e", [3,4]],
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// // ["c", "f", [5,6]],
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// // ]
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function transpose(M, reverse=false) =
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assert( is_list(M) && len(M)>0, "Input to transpose must be a nonempty list.")
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is_list(M[0])
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? let( len0 = len(M[0]) )
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assert([for(a=M) if(!is_list(a) || len(a)!=len0) 1 ]==[], "Input to transpose has inconsistent row lengths." )
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reverse
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? [for (i=[0:1:len0-1])
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[ for (j=[0:1:len(M)-1]) M[len(M)-1-j][len0-1-i] ] ]
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: [for (i=[0:1:len0-1])
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[ for (j=[0:1:len(M)-1]) M[j][i] ] ]
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: assert( is_vector(M), "Input to transpose must be a vector or list of lists.")
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M;
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// Function: outer_product()
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// Synopsis: Compute the outer product of two vectors.
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// Topics: Linear Algebra, Matrices
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// See Also: submatrix(), determinant()
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// Usage:
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// x = outer_product(u,v);
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// Description:
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// Compute the outer product of two vectors, which is a matrix.
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// Usage:
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// M = outer_product(u,v);
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function outer_product(u,v) =
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assert(is_vector(u) && is_vector(v), "The inputs must be vectors.")
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[for(ui=u) ui*v];
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// Function: submatrix_set()
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// Synopsis: Takes a matrix as input and change values in a submatrix.
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// Topics: Matrices, Arrays
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// See Also: column(), submatrix()
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// Usage:
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// mat = submatrix_set(M, A, [m], [n]);
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// Description:
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// Sets a submatrix of M equal to the matrix A. By default the top left corner of M is set to A, but
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// you can specify offset coordinates m and n. If A (as adjusted by m and n) extends beyond the bounds
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// of M then the extra entries are ignored. You can pass in `A=[[]]`, a null matrix, and M will be
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// returned unchanged. This function works on arbitrary lists of lists and the input M need not be rectangular in shape.
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// Arguments:
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// M = Original matrix.
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// A = Submatrix of new values to write into M
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// m = Row number of upper-left corner to place A at. Default: 0
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// n = Column number of upper-left corner to place A at. Default: 0
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function submatrix_set(M,A,m=0,n=0) =
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assert(is_list(M))
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assert(is_list(A))
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assert(is_int(m))
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assert(is_int(n))
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let( badrows = [for(i=idx(A)) if (!is_list(A[i])) i])
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assert(badrows==[], str("Input submatrix malformed rows: ",badrows))
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[for(i=[0:1:len(M)-1])
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assert(is_list(M[i]), str("Row ",i," of input matrix is not a list"))
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[for(j=[0:1:len(M[i])-1])
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i>=m && i <len(A)+m && j>=n && j<len(A[0])+n ? A[i-m][j-n] : M[i][j]]];
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// Function: hstack()
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// Synopsis: Make a new matrix by stacking matrices horizontally.
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// Topics: Matrices, Arrays
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// See Also: column(), submatrix(), block_matrix()
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// Usage:
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// A = hstack(M1, M2)
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// A = hstack(M1, M2, M3)
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// A = hstack([M1, M2, M3, ...])
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// Description:
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// Constructs a matrix by horizontally "stacking" together compatible matrices or vectors. Vectors are treated as columsn in the stack.
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// This command is the inverse of `column`. Note: strings given in vectors are broken apart into lists of characters. Strings given
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// in matrices are preserved as strings. If you need to combine vectors of strings use {{list_to_matrix()}} as shown below to convert the
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// vector into a column matrix. Also note that vertical stacking can be done directly with concat.
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// Arguments:
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// M1 = If given with other arguments, the first matrix (or vector) to stack. If given alone, a list of matrices/vectors to stack.
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// M2 = Second matrix/vector to stack
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// M3 = Third matrix/vector to stack.
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// Example:
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// M = ident(3);
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// v1 = [2,3,4];
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// v2 = [5,6,7];
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// v3 = [8,9,10];
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// a = hstack(v1,v2); // Returns [[2, 5], [3, 6], [4, 7]]
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// b = hstack(v1,v2,v3); // Returns [[2, 5, 8],
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// // [3, 6, 9],
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// // [4, 7, 10]]
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// c = hstack([M,v1,M]); // Returns [[1, 0, 0, 2, 1, 0, 0],
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// // [0, 1, 0, 3, 0, 1, 0],
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// // [0, 0, 1, 4, 0, 0, 1]]
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// d = hstack(column(M,0), submatrix(M,idx(M),[1 2])); // Returns M
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// strvec = ["one","two"];
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// strmat = [["three","four"], ["five","six"]];
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// e = hstack(strvec,strvec); // Returns [["o", "n", "e", "o", "n", "e"],
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// // ["t", "w", "o", "t", "w", "o"]]
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// f = hstack(list_to_matrix(strvec,1), list_to_matrix(strvec,1));
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// // Returns [["one", "one"],
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// // ["two", "two"]]
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// g = hstack(strmat,strmat); // Returns: [["three", "four", "three", "four"],
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// // [ "five", "six", "five", "six"]]
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function hstack(M1, M2, M3) =
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(M3!=undef)? hstack([M1,M2,M3]) :
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(M2!=undef)? hstack([M1,M2]) :
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assert(all([for(v=M1) is_list(v)]), "One of the inputs to hstack is not a list")
|
|
let(
|
|
minlen = min_length(M1),
|
|
maxlen = max_length(M1)
|
|
)
|
|
assert(minlen==maxlen, "Input vectors to hstack must have the same length")
|
|
[for(row=[0:1:minlen-1])
|
|
[for(matrix=M1)
|
|
each matrix[row]
|
|
]
|
|
];
|
|
|
|
|
|
// Function: block_matrix()
|
|
// Synopsis: Make a new matrix from a block of matrices.
|
|
// Topics: Matrices, Arrays
|
|
// See Also: column(), submatrix()
|
|
// Usage:
|
|
// bmat = block_matrix([[M11, M12,...],[M21, M22,...], ... ]);
|
|
// Description:
|
|
// Create a block matrix by supplying a matrix of matrices, which will
|
|
// be combined into one unified matrix. Every matrix in one row
|
|
// must have the same height, and the combined width of the matrices
|
|
// in each row must be equal. Strings will stay strings.
|
|
// Example:
|
|
// A = [[1,2],
|
|
// [3,4]];
|
|
// B = ident(2);
|
|
// C = block_matrix([[A,B],[B,A],[A,B]]);
|
|
// // Returns:
|
|
// // [[1, 2, 1, 0],
|
|
// // [3, 4, 0, 1],
|
|
// // [1, 0, 1, 2],
|
|
// // [0, 1, 3, 4],
|
|
// // [1, 2, 1, 0],
|
|
// // [3, 4, 0, 1]]);
|
|
// D = block_matrix([[A,B], ident(4)]);
|
|
// // Returns:
|
|
// // [[1, 2, 1, 0],
|
|
// // [3, 4, 0, 1],
|
|
// // [1, 0, 0, 0],
|
|
// // [0, 1, 0, 0],
|
|
// // [0, 0, 1, 0],
|
|
// // [0, 0, 0, 1]]);
|
|
// E = [["one", "two"], [3,4]];
|
|
// F = block_matrix([[E,E]]);
|
|
// // Returns:
|
|
// // [["one", "two", "one", "two"],
|
|
// // [ 3, 4, 3, 4]]
|
|
function block_matrix(M) =
|
|
let(
|
|
bigM = [for(bigrow = M) each hstack(bigrow)],
|
|
len0 = len(bigM[0]),
|
|
badrows = [for(row=bigM) if (len(row)!=len0) 1]
|
|
)
|
|
assert(badrows==[], "Inconsistent or invalid input")
|
|
bigM;
|
|
|
|
|
|
// Section: Solving Linear Equations and Matrix Factorizations
|
|
|
|
// Function: linear_solve()
|
|
// Synopsis: Solve Ax=b or, for overdetermined case, solve the least square problem.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// solv = linear_solve(A,b,[pivot])
|
|
// Description:
|
|
// Solves the linear system Ax=b. If `A` is square and non-singular the unique solution is returned. If `A` is overdetermined
|
|
// the least squares solution is returned. If `A` is underdetermined, the minimal norm solution is returned.
|
|
// If `A` is rank deficient or singular then linear_solve returns `[]`. If `b` is a matrix that is compatible with `A`
|
|
// then the problem is solved for the matrix valued right hand side and a matrix is returned. Note that if you
|
|
// want to solve Ax=b1 and Ax=b2 that you need to form the matrix `transpose([b1,b2])` for the right hand side and then
|
|
// transpose the returned value. The solution is computed using QR factorization. If `pivot` is set to true (the default) then
|
|
// pivoting is used in the QR factorization, which is slower but expected to be more accurate.
|
|
// Arguments:
|
|
// A = Matrix describing the linear system, which need not be square
|
|
// b = right hand side for linear system, which can be a matrix to solve several cases simultaneously. Must be consistent with A.
|
|
// pivot = if true use pivoting when computing the QR factorization. Default: true
|
|
function linear_solve(A,b,pivot=true) =
|
|
assert(is_matrix(A), "Input should be a matrix.")
|
|
let(
|
|
m = len(A),
|
|
n = len(A[0])
|
|
)
|
|
assert(is_vector(b,m) || is_matrix(b,m),"Invalid right hand side or incompatible with the matrix")
|
|
let (
|
|
qr = m<n? qr_factor(transpose(A),pivot) : qr_factor(A,pivot),
|
|
maxdim = max(n,m),
|
|
mindim = min(n,m),
|
|
Q = submatrix(qr[0],[0:maxdim-1], [0:mindim-1]),
|
|
R = submatrix(qr[1],[0:mindim-1], [0:mindim-1]),
|
|
P = qr[2],
|
|
zeros = [for(i=[0:mindim-1]) if (approx(R[i][i],0)) i]
|
|
)
|
|
zeros != [] ? [] :
|
|
m<n ? Q*back_substitute(R,transpose(P)*b,transpose=true) // Too messy to avoid input checks here
|
|
: P*_back_substitute(R, transpose(Q)*b); // Calling internal version skips input checks
|
|
|
|
|
|
// Function: linear_solve3()
|
|
// Synopsis: Fast solution to Ax=b where A is 3x3.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// x = linear_solve3(A,b)
|
|
// Description:
|
|
// Fast solution to a 3x3 linear system using Cramer's rule (which appears to be the fastest
|
|
// method in OpenSCAD). The input `A` must be a 3x3 matrix. Returns undef if `A` is singular.
|
|
// The input `b` must be a 3-vector. Note that Cramer's rule is not a stable algorithm, so for
|
|
// the highest accuracy on ill-conditioned problems you may want to use the general solver, which is about ten times slower.
|
|
// Arguments:
|
|
// A = 3x3 matrix for linear system
|
|
// b = length 3 vector, right hand side of linear system
|
|
function linear_solve3(A,b) =
|
|
// Arg sanity checking adds 7% overhead
|
|
assert(b*0==[0,0,0], "Input b must be a 3-vector")
|
|
assert(A*0==[[0,0,0],[0,0,0],[0,0,0]],"Input A must be a 3x3 matrix")
|
|
let(
|
|
Az = [for(i=[0:2])[A[i][0], A[i][1], b[i]]],
|
|
Ay = [for(i=[0:2])[A[i][0], b[i], A[i][2]]],
|
|
Ax = [for(i=[0:2])[b[i], A[i][1], A[i][2]]],
|
|
detA = det3(A)
|
|
)
|
|
detA==0 ? undef : [det3(Ax), det3(Ay), det3(Az)] / detA;
|
|
|
|
|
|
// Function: matrix_inverse()
|
|
// Synopsis: General matrix inverse.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve(), linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// mat = matrix_inverse(A)
|
|
// Description:
|
|
// Compute the matrix inverse of the square matrix `A`. If `A` is singular, returns `undef`.
|
|
// Note that if you just want to solve a linear system of equations you should NOT use this function.
|
|
// Instead use {{linear_solve()}}, or use {{qr_factor()}}. The computation
|
|
// will be faster and more accurate.
|
|
function matrix_inverse(A) =
|
|
assert(is_matrix(A) && len(A)==len(A[0]),"Input to matrix_inverse() must be a square matrix")
|
|
linear_solve(A,ident(len(A)));
|
|
|
|
|
|
// Function: rot_inverse()
|
|
// Synopsis: Invert 2d or 3d rotation transformations.
|
|
// Topics: Matrices, Linear Algebra, Affine
|
|
// See Also: linear_solve(), linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// B = rot_inverse(A)
|
|
// Description:
|
|
// Inverts a 2d (3x3) or 3d (4x4) rotation matrix. The matrix can be a rotation around any center,
|
|
// so it may include a translation. This is faster and likely to be more accurate than using `matrix_inverse()`.
|
|
function rot_inverse(T) =
|
|
assert(is_matrix(T,square=true),"Matrix must be square")
|
|
let( n = len(T))
|
|
assert(n==3 || n==4, "Matrix must be 3x3 or 4x4")
|
|
let(
|
|
rotpart = [for(i=[0:n-2]) [for(j=[0:n-2]) T[j][i]]],
|
|
transpart = [for(row=[0:n-2]) T[row][n-1]]
|
|
)
|
|
assert(approx(determinant(T),1),"Matrix is not a rotation")
|
|
concat(hstack(rotpart, -rotpart*transpart),[[for(i=[2:n]) 0, 1]]);
|
|
|
|
|
|
|
|
|
|
// Function: null_space()
|
|
// Synopsis: Return basis for the null space of A.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve(), linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// x = null_space(A)
|
|
// Description:
|
|
// Returns an orthonormal basis for the null space of `A`, namely the vectors {x} such that Ax=0.
|
|
// If the null space is just the origin then returns an empty list.
|
|
function null_space(A,eps=1e-12) =
|
|
assert(is_matrix(A))
|
|
let(
|
|
Q_R = qr_factor(transpose(A),pivot=true),
|
|
R = Q_R[1],
|
|
zrows = [for(i=idx(R)) if (all_zero(R[i],eps)) i]
|
|
)
|
|
len(zrows)==0 ? [] :
|
|
select(transpose(Q_R[0]), zrows);
|
|
|
|
// Function: qr_factor()
|
|
// Synopsis: Compute QR factorization of a matrix.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve(), linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// qr = qr_factor(A,[pivot]);
|
|
// Description:
|
|
// Calculates the QR factorization of the input matrix A and returns it as the list [Q,R,P]. This factorization can be
|
|
// used to solve linear systems of equations. The factorization is `A = Q*R*transpose(P)`. If pivot is false (the default)
|
|
// then P is the identity matrix and A = Q*R. If pivot is true then column pivoting results in an R matrix where the diagonal
|
|
// is non-decreasing. The use of pivoting is supposed to increase accuracy for poorly conditioned problems, and is necessary
|
|
// for rank estimation or computation of the null space, but it may be slower.
|
|
function qr_factor(A, pivot=false) =
|
|
assert(is_matrix(A), "Input must be a matrix." )
|
|
let(
|
|
m = len(A),
|
|
n = len(A[0])
|
|
)
|
|
let(
|
|
qr = _qr_factor(A, Q=ident(m),P=ident(n), pivot=pivot, col=0, m = m, n = n),
|
|
Rzero = let( R = qr[1]) [
|
|
for(i=[0:m-1]) [
|
|
let( ri = R[i] )
|
|
for(j=[0:n-1]) i>j ? 0 : ri[j]
|
|
]
|
|
]
|
|
) [qr[0], Rzero, qr[2]];
|
|
|
|
function _qr_factor(A,Q,P, pivot, col, m, n) =
|
|
col >= min(m-1,n) ? [Q,A,P] :
|
|
let(
|
|
swap = !pivot ? 1
|
|
: _swap_matrix(n,col,col+max_index([for(i=[col:n-1]) sqr([for(j=[col:m-1]) A[j][i]])])),
|
|
A = pivot ? A*swap : A,
|
|
x = [for(i=[col:1:m-1]) A[i][col]],
|
|
alpha = (x[0]<=0 ? 1 : -1) * norm(x),
|
|
u = x - concat([alpha],repeat(0,m-1)),
|
|
v = alpha==0 ? u : u / norm(u),
|
|
Qc = ident(len(x)) - 2*outer_product(v,v),
|
|
Qf = [for(i=[0:m-1]) [for(j=[0:m-1]) i<col || j<col ? (i==j ? 1 : 0) : Qc[i-col][j-col]]]
|
|
)
|
|
_qr_factor(Qf*A, Q*Qf, P*swap, pivot, col+1, m, n);
|
|
|
|
// Produces an n x n matrix that swaps column i and j (when multiplied on the right)
|
|
function _swap_matrix(n,i,j) =
|
|
assert(i<n && j<n && i>=0 && j>=0, "Swap indices out of bounds")
|
|
[for(y=[0:n-1]) [for (x=[0:n-1])
|
|
x==i ? (y==j ? 1 : 0)
|
|
: x==j ? (y==i ? 1 : 0)
|
|
: x==y ? 1 : 0]];
|
|
|
|
|
|
|
|
// Function: back_substitute()
|
|
// Synopsis: Solve an upper triangular system, Rx=b.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve(), linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// x = back_substitute(R, b, [transpose]);
|
|
// Description:
|
|
// Solves the problem Rx=b where R is an upper triangular square matrix. The lower triangular entries of R are
|
|
// ignored. If transpose==true then instead solve transpose(R)*x=b.
|
|
// You can supply a compatible matrix b and it will produce the solution for every column of b. Note that if you want to
|
|
// solve Rx=b1 and Rx=b2 you must set b to transpose([b1,b2]) and then take the transpose of the result. If the matrix
|
|
// is singular (e.g. has a zero on the diagonal) then it returns [].
|
|
function back_substitute(R, b, transpose = false) =
|
|
assert(is_matrix(R, square=true))
|
|
let(n=len(R))
|
|
assert(is_vector(b,n) || is_matrix(b,n),str("R and b are not compatible in back_substitute ",n, len(b)))
|
|
transpose
|
|
? reverse(_back_substitute(transpose(R, reverse=true), reverse(b)))
|
|
: _back_substitute(R,b);
|
|
|
|
function _back_substitute(R, b, x=[]) =
|
|
let(n=len(R))
|
|
len(x) == n ? x
|
|
: let(ind = n - len(x) - 1)
|
|
R[ind][ind] == 0 ? []
|
|
: let(
|
|
newvalue = len(x)==0
|
|
? b[ind]/R[ind][ind]
|
|
: (b[ind]-list_tail(R[ind],ind+1) * x)/R[ind][ind]
|
|
)
|
|
_back_substitute(R, b, concat([newvalue],x));
|
|
|
|
|
|
|
|
// Function: cholesky()
|
|
// Synopsis: Compute the Cholesky factorization of a matrix.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: linear_solve(), linear_solve3(), matrix_inverse(), rot_inverse(), back_substitute(), cholesky()
|
|
// Usage:
|
|
// L = cholesky(A);
|
|
// Description:
|
|
// Compute the cholesky factor, L, of the symmetric positive definite matrix A.
|
|
// The matrix L is lower triangular and `L * transpose(L) = A`. If the A is
|
|
// not symmetric then an error is displayed. If the matrix is symmetric but
|
|
// not positive definite then undef is returned.
|
|
function cholesky(A) =
|
|
assert(is_matrix(A,square=true),"A must be a square matrix")
|
|
assert(is_matrix_symmetric(A),"Cholesky factorization requires a symmetric matrix")
|
|
_cholesky(A,ident(len(A)), len(A));
|
|
|
|
function _cholesky(A,L,n) =
|
|
A[0][0]<0 ? undef : // Matrix not positive definite
|
|
len(A) == 1 ? submatrix_set(L,[[sqrt(A[0][0])]], n-1,n-1):
|
|
let(
|
|
i = n+1-len(A)
|
|
)
|
|
let(
|
|
sqrtAii = sqrt(A[0][0]),
|
|
Lnext = [for(j=[0:n-1])
|
|
[for(k=[0:n-1])
|
|
j<i-1 || k<i-1 ? (j==k ? 1 : 0)
|
|
: j==i-1 && k==i-1 ? sqrtAii
|
|
: j==i-1 ? 0
|
|
: k==i-1 ? A[j-(i-1)][0]/sqrtAii
|
|
: j==k ? 1 : 0]],
|
|
Anext = submatrix(A,[1:n-1], [1:n-1]) - outer_product(list_tail(A[0]), list_tail(A[0]))/A[0][0]
|
|
)
|
|
_cholesky(Anext,L*Lnext,n);
|
|
|
|
|
|
// Section: Matrix Properties: Determinants, Norm, Trace
|
|
|
|
// Function: det2()
|
|
// Synopsis: Compute determinant of 2x2 matrix.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: det2(), det3(), det4(), determinant(), norm_fro(), matrix_trace()
|
|
// Usage:
|
|
// d = det2(M);
|
|
// Description:
|
|
// Rturns the determinant for the given 2x2 matrix.
|
|
// Arguments:
|
|
// M = The 2x2 matrix to get the determinant of.
|
|
// Example:
|
|
// M = [ [6,-2], [1,8] ];
|
|
// det = det2(M); // Returns: 50
|
|
function det2(M) =
|
|
assert(is_def(M) && M*0==[[0,0],[0,0]], "Expected square matrix (2x2)")
|
|
cross(M[0],M[1]);
|
|
|
|
|
|
// Function: det3()
|
|
// Synopsis: Compute determinant of 3x3 matrix.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: det2(), det3(), det4(), determinant(), norm_fro(), matrix_trace()
|
|
// Usage:
|
|
// d = det3(M);
|
|
// Description:
|
|
// Returns the determinant for the given 3x3 matrix.
|
|
// Arguments:
|
|
// M = The 3x3 square matrix to get the determinant of.
|
|
// Example:
|
|
// M = [ [6,4,-2], [1,-2,8], [1,5,7] ];
|
|
// det = det3(M); // Returns: -334
|
|
function det3(M) =
|
|
assert(is_def(M) && M*0==[[0,0,0],[0,0,0],[0,0,0]], "Expected square matrix (3x3).")
|
|
M[0][0] * (M[1][1]*M[2][2]-M[2][1]*M[1][2]) -
|
|
M[1][0] * (M[0][1]*M[2][2]-M[2][1]*M[0][2]) +
|
|
M[2][0] * (M[0][1]*M[1][2]-M[1][1]*M[0][2]);
|
|
|
|
// Function: det4()
|
|
// Synopsis: Compute determinant of 4x4 matrix.
|
|
// Topics: Matrices, Linear Algebra
|
|
// See Also: det2(), det3(), det4(), determinant(), norm_fro(), matrix_trace()
|
|
// Usage:
|
|
// d = det4(M);
|
|
// Description:
|
|
// Returns the determinant for the given 4x4 matrix.
|
|
// Arguments:
|
|
// M = The 4x4 square matrix to get the determinant of.
|
|
// Example:
|
|
// M = [ [6,4,-2,1], [1,-2,8,-3], [1,5,7,4], [2,3,4,7] ];
|
|
// det = det4(M); // Returns: -1773
|
|
function det4(M) =
|
|
assert(is_def(M) && M*0==[[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]], "Expected square matrix (4x4).")
|
|
M[0][0]*M[1][1]*M[2][2]*M[3][3] + M[0][0]*M[1][2]*M[2][3]*M[3][1] + M[0][0]*M[1][3]*M[2][1]*M[3][2]
|
|
+ M[0][1]*M[1][0]*M[2][3]*M[3][2] + M[0][1]*M[1][2]*M[2][0]*M[3][3] + M[0][1]*M[1][3]*M[2][2]*M[3][0]
|
|
+ M[0][2]*M[1][0]*M[2][1]*M[3][3] + M[0][2]*M[1][1]*M[2][3]*M[3][0] + M[0][2]*M[1][3]*M[2][0]*M[3][1]
|
|
+ M[0][3]*M[1][0]*M[2][2]*M[3][1] + M[0][3]*M[1][1]*M[2][0]*M[3][2] + M[0][3]*M[1][2]*M[2][1]*M[3][0]
|
|
- M[0][0]*M[1][1]*M[2][3]*M[3][2] - M[0][0]*M[1][2]*M[2][1]*M[3][3] - M[0][0]*M[1][3]*M[2][2]*M[3][1]
|
|
- M[0][1]*M[1][0]*M[2][2]*M[3][3] - M[0][1]*M[1][2]*M[2][3]*M[3][0] - M[0][1]*M[1][3]*M[2][0]*M[3][2]
|
|
- M[0][2]*M[1][0]*M[2][3]*M[3][1] - M[0][2]*M[1][1]*M[2][0]*M[3][3] - M[0][2]*M[1][3]*M[2][1]*M[3][0]
|
|
- M[0][3]*M[1][0]*M[2][1]*M[3][2] - M[0][3]*M[1][1]*M[2][2]*M[3][0] - M[0][3]*M[1][2]*M[2][0]*M[3][1];
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// Function: determinant()
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// Synopsis: compute determinant of an arbitrary square matrix.
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// Topics: Matrices, Linear Algebra
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// See Also: det2(), det3(), det4(), determinant(), norm_fro(), matrix_trace()
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// Usage:
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// d = determinant(M);
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// Description:
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// Returns the determinant for the given square matrix.
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// Arguments:
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// M = The NxN square matrix to get the determinant of.
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// Example:
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// M = [ [6,4,-2,9], [1,-2,8,3], [1,5,7,6], [4,2,5,1] ];
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// det = determinant(M); // Returns: 2267
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function determinant(M) =
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assert(is_list(M), "Input must be a square matrix." )
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len(M)==1? M[0][0] :
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len(M)==2? det2(M) :
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len(M)==3? det3(M) :
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len(M)==4? det4(M) :
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assert(is_matrix(M, square=true), "Input must be a square matrix." )
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sum(
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[for (col=[0:1:len(M)-1])
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((col%2==0)? 1 : -1) *
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M[col][0] *
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determinant(
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[for (r=[1:1:len(M)-1])
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[for (c=[0:1:len(M)-1])
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if (c!=col) M[c][r]
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]
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]
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)
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]
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);
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// Function: norm_fro()
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// Synopsis: Compute Frobenius norm of a matrix
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// Topics: Matrices, Linear Algebra
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// See Also: det2(), det3(), det4(), determinant(), norm_fro(), matrix_trace()
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// Usage:
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// norm_fro(A)
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// Description:
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// Computes frobenius norm of input matrix. The frobenius norm is the square root of the sum of the
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// squares of all of the entries of the matrix. On vectors it is the same as the usual 2-norm.
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// This is an easily computed norm that is convenient for comparing two matrices.
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function norm_fro(A) =
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assert(is_matrix(A) || is_vector(A))
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norm(flatten(A));
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// Function: matrix_trace()
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// Synopsis: Compute the trace of a square matrix.
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// Topics: Matrices, Linear Algebra
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// See Also: det2(), det3(), det4(), determinant(), norm_fro(), matrix_trace()
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// Usage:
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// matrix_trace(M)
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// Description:
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// Computes the trace of a square matrix, the sum of the entries on the diagonal.
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function matrix_trace(M) =
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assert(is_matrix(M,square=true), "Input to trace must be a square matrix")
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[for(i=[0:1:len(M)-1])1] * [for(i=[0:1:len(M)-1]) M[i][i]];
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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