BOSL2/linalg.scad
2021-12-28 17:05:37 -05:00

715 lines
27 KiB
OpenSCAD

//////////////////////////////////////////////////////////////////////
// LibFile: linalg.scad
// This file provides linear algebra, with support for matrix construction,
// solutions to linear systems of equations, QR and Cholesky factorizations, and
// matrix inverse.
// Includes:
// include <BOSL2/std.scad>
// FileGroup: Math
// FileSummary: Linear Algebra: solve linear systems, construct and modify matrices.
// FileFootnotes: STD=Included in std.scad
//////////////////////////////////////////////////////////////////////
// Section: Matrices
// The matrix, a rectangular array of numbers which represents a linear transformation,
// is the fundamental object in linear algebra. In OpenSCAD a matrix is a list of lists of numbers
// with a rectangular structure. Because OpenSCAD treats all data the same, most of the functions that
// index matrices or construct them will work on matrices (lists of lists) whose elements are not numbers but may be
// arbitrary data: strings, booleans, or even other lists. It may even be acceptable in some cases if the structure is non-rectangular.
// Of course, linear algebra computations and solutions require true matrices with rectangular structure, where all the entries are
// finite numbers.
// .
// Matrices in OpenSCAD are lists of row vectors. However, a potential source of confusion is that OpenSCAD
// treats vectors as either column vectors or row vectors as demanded by
// 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
// `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
// column vectors, you now need to use {{transpose()}} because OpenSCAD doesn't adjust matrices
// contextually: `A=M*transpose([v,w])`. The solutions are now columns of A and you must extract
// them with {{column()}} or take the transpose of `A`.
// Section: Matrix testing and display
// Function: is_matrix()
// Usage:
// test = is_matrix(A, [m], [n], [square])
// Description:
// Returns true if A is a numeric matrix of height m and width n with finite entries. If m or n
// are omitted or set to undef then true is returned for any positive dimension.
// Arguments:
// A = The matrix to test.
// m = If given, requires the matrix to have this height.
// n = Is given, requires the matrix to have this width.
// square = If true, matrix must have height equal to width. Default: false
function is_matrix(A,m,n,square=false) =
is_list(A)
&& (( is_undef(m) && len(A) ) || len(A)==m)
&& (!square || len(A) == len(A[0]))
&& is_vector(A[0],n)
&& is_consistent(A);
// Function: is_matrix_symmetric()
// Usage:
// b = is_matrix_symmetric(A, [eps])
// Description:
// Returns true if the input matrix is symmetric, meaning it approximately equals its transpose.
// The matrix can have arbitrary entries.
// Arguments:
// A = matrix to test
// eps = epsilon for comparing equality. Default: 1e-12
function is_matrix_symmetric(A,eps=1e-12) =
approx(A,transpose(A), eps);
// Function&Module: echo_matrix()
// Usage:
// echo_matrix(M, [description=], [sig=], [eps=]);
// dummy = echo_matrix(M, [description=], [sig=], [eps=]),
// Description:
// Display a numerical matrix in a readable columnar format with `sig` significant
// digits. Values smaller than eps display as zero. If you give a description
// it is displayed at the top.
function echo_matrix(M,description,sig=4,eps=1e-9) =
let(
horiz_line = chr(8213),
matstr = matrix_strings(M,sig=sig,eps=eps),
separator = str_join(repeat(horiz_line,10)),
dummy=echo(str(separator," ",is_def(description) ? description : ""))
[for(row=matstr) echo(row)]
)
echo(separator);
module echo_matrix(M,description,sig=4,eps=1e-9)
{
dummy = echo_matrix(M,description,sig,eps);
}
// Section: Matrix indexing
// Function: column()
// Usage:
// list = column(M, i);
// Topics: Matrices, List Handling
// See Also: select(), slice()
// Description:
// Extracts entry `i` from each list in M, or equivalently column i from the matrix M, and returns it as a vector.
// This function will return `undef` at all entry positions indexed by i not found in M.
// Arguments:
// M = The given list of lists.
// i = The index to fetch
// Example:
// M = [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]];
// a = column(M,2); // Returns [3, 7, 11, 15]
// b = column(M,0); // Returns [1, 5, 9, 13]
// N = [ [1,2], [3], [4,5], [6,7,8] ];
// c = column(N,1); // Returns [1,undef,5,7]
// data = [[1,[3,4]], [3, [9,3]], [4, [3,1]]]; // Matrix with non-numeric entries
// d = column(data,0); // Returns [1,3,4]
// e = column(data,1); // Returns [[3,4],[9,3],[3,1]]
function column(M, i) =
assert( is_list(M), "The input is not a list." )
assert( is_int(i) && i>=0, "Invalid index")
[for(row=M) row[i]];
// Function: submatrix()
// Usage:
// mat = submatrix(M, idx1, idx2);
// Topics: Matrices
// See Also: column(), block_matrix(), submatrix_set()
// Description:
// 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.
// Arguments:
// M = Given list of lists
// idx1 = rows index list or range
// idx2 = column index list or range
// Example:
// M = [[ 1, 2, 3, 4, 5],
// [ 6, 7, 8, 9,10],
// [11,12,13,14,15],
// [16,17,18,19,20],
// [21,22,23,24,25]];
// submatrix(M,[1:2],[3:4]); // Returns [[9, 10], [14, 15]]
// submatrix(M,[1], [3,4])); // Returns [[9,10]]
// submatrix(M,1, [3,4])); // Returns [[9,10]]
// submatrix(M,1,3)); // Returns [[9]]
// submatrix(M, [3,4],1); // Returns [[17],[22]]);
// submatrix(M, [1,3],[2,4]); // Returns [[8,10],[18,20]]);
// A = [[true, 17, "test"],
// [[4,2], 91, false],
// [6, [3,4], undef]];
// submatrix(A,[0,2],[1,2]); // Returns [[17, "test"], [[3, 4], undef]]
function submatrix(M,idx1,idx2) =
[for(i=idx1) [for(j=idx2) M[i][j] ] ];
// Section: Matrix construction and modification
// Function: ident()
// Usage:
// mat = ident(n);
// Topics: Affine, Matrices
// Description:
// Create an `n` by `n` square identity matrix.
// Arguments:
// n = The size of the identity matrix square, `n` by `n`.
// Example:
// mat = ident(3);
// // Returns:
// // [
// // [1, 0, 0],
// // [0, 1, 0],
// // [0, 0, 1]
// // ]
// Example:
// mat = ident(4);
// // Returns:
// // [
// // [1, 0, 0, 0],
// // [0, 1, 0, 0],
// // [0, 0, 1, 0],
// // [0, 0, 0, 1]
// // ]
function ident(n) = [
for (i = [0:1:n-1]) [
for (j = [0:1:n-1]) (i==j)? 1 : 0
]
];
// Function: diagonal_matrix()
// Usage:
// mat = diagonal_matrix(diag, [offdiag]);
// Topics: Matrices
// See Also: column(), submatrix()
// Description:
// Creates a square matrix with the items in the list `diag` on
// its diagonal. The off diagonal entries are set to offdiag,
// which is zero by default.
// Arguments:
// diag = A list of items to put in the diagnal cells of the matrix.
// offdiag = Value to put in non-diagonal matrix cells.
function diagonal_matrix(diag, offdiag=0) =
assert(is_list(diag) && len(diag)>0)
[for(i=[0:1:len(diag)-1]) [for(j=[0:len(diag)-1]) i==j?diag[i] : offdiag]];
// Function: transpose()
// Usage:
// M = transpose(M, [reverse]);
// Topics: Matrices
// See Also: submatrix(), block_matrix(), hstack(), flatten()
// Description:
// Returns the transpose of the given input matrix. The input can be a matrix with arbitrary entries or
// a numerical vector. If you give a vector then transpose returns it unchanged.
// When reverse=true, the transpose is done across to the secondary diagonal. (See example below.)
// By default, reverse=false.
// Example:
// M = [
// [1, 2, 3],
// [4, 5, 6],
// [7, 8, 9]
// ];
// t = transpose(M);
// // Returns:
// // [
// // [1, 4, 7],
// // [2, 5, 8],
// // [3, 6, 9]
// // ]
// Example:
// M = [
// [1, 2, 3],
// [4, 5, 6]
// ];
// t = transpose(M);
// // Returns:
// // [
// // [1, 4],
// // [2, 5],
// // [3, 6],
// // ]
// Example:
// M = [
// [1, 2, 3],
// [4, 5, 6],
// [7, 8, 9]
// ];
// t = transpose(M, reverse=true);
// // Returns:
// // [
// // [9, 6, 3],
// // [8, 5, 2],
// // [7, 4, 1]
// // ]
// Example: Transpose on a list of numbers returns the list unchanged
// transpose([3,4,5]); // Returns: [3,4,5]
// Example: Transpose on non-numeric input
// arr = [
// [ "a", "b", "c"],
// [ "d", "e", "f"],
// [[1,2],[3,4],[5,6]]
// ];
// t = transpose(arr);
// // Returns:
// // [
// // ["a", "d", [1,2]],
// // ["b", "e", [3,4]],
// // ["c", "f", [5,6]],
// // ]
function transpose(M, reverse=false) =
assert( is_list(M) && len(M)>0, "Input to transpose must be a nonempty list.")
is_list(M[0])
? let( len0 = len(M[0]) )
assert([for(a=M) if(!is_list(a) || len(a)!=len0) 1 ]==[], "Input to transpose has inconsistent row lengths." )
reverse
? [for (i=[0:1:len0-1])
[ for (j=[0:1:len(M)-1]) M[len(M)-1-j][len0-1-i] ] ]
: [for (i=[0:1:len0-1])
[ for (j=[0:1:len(M)-1]) M[j][i] ] ]
: assert( is_vector(M), "Input to transpose must be a vector or list of lists.")
M;
// Function: outer_product()
// Usage:
// x = outer_product(u,v);
// Description:
// Compute the outer product of two vectors, a matrix.
// Usage:
// M = outer_product(u,v);
function outer_product(u,v) =
assert(is_vector(u) && is_vector(v), "The inputs must be vectors.")
[for(ui=u) ui*v];
// Function: submatrix_set()
// Usage:
// mat = submatrix_set(M, A, [m], [n]);
// Topics: Matrices
// See Also: column(), submatrix()
// Description:
// Sets a submatrix of M equal to the matrix A. By default the top left corner of M is set to A, but
// you can specify offset coordinates m and n. If A (as adjusted by m and n) extends beyond the bounds
// of M then the extra entries are ignored. You can pass in A=[[]], a null matrix, and M will be
// returned unchanged. This function works on arbitrary lists of lists and the input M need not be rectangular in shape.
// Arguments:
// M = Original matrix.
// A = Submatrix of new values to write into M
// m = Row number of upper-left corner to place A at. Default: 0
// n = Column number of upper-left corner to place A at. Default: 0
function submatrix_set(M,A,m=0,n=0) =
assert(is_list(M))
assert(is_list(A))
assert(is_int(m))
assert(is_int(n))
let( badrows = [for(i=idx(A)) if (!is_list(A[i])) i])
assert(badrows==[], str("Input submatrix malformed rows: ",badrows))
[for(i=[0:1:len(M)-1])
assert(is_list(M[i]), str("Row ",i," of input matrix is not a list"))
[for(j=[0:1:len(M[i])-1])
i>=m && i <len(A)+m && j>=n && j<len(A[0])+n ? A[i-m][j-n] : M[i][j]]];
// Function: hstack()
// Usage:
// A = hstack(M1, M2)
// A = hstack(M1, M2, M3)
// A = hstack([M1, M2, M3, ...])
// Topics: Matrices
// See Also: column(), submatrix(), block_matrix()
// Description:
// Constructs a matrix by horizontally "stacking" together compatible matrices or vectors. Vectors are treated as columsn in the stack.
// This command is the inverse of `column`. Note: strings given in vectors are broken apart into lists of characters. Strings given
// in matrices are preserved as strings. If you need to combine vectors of strings use {{list_to_matrix()}} as shown below to convert the
// vector into a column matrix. Also note that vertical stacking can be done directly with concat.
// Arguments:
// M1 = If given with other arguments, the first matrix (or vector) to stack. If given alone, a list of matrices/vectors to stack.
// M2 = Second matrix/vector to stack
// M3 = Third matrix/vector to stack.
// Example:
// M = ident(3);
// v1 = [2,3,4];
// v2 = [5,6,7];
// v3 = [8,9,10];
// a = hstack(v1,v2); // Returns [[2, 5], [3, 6], [4, 7]]
// b = hstack(v1,v2,v3); // Returns [[2, 5, 8],
// // [3, 6, 9],
// // [4, 7, 10]]
// c = hstack([M,v1,M]); // Returns [[1, 0, 0, 2, 1, 0, 0],
// // [0, 1, 0, 3, 0, 1, 0],
// // [0, 0, 1, 4, 0, 0, 1]]
// d = hstack(column(M,0), submatrix(M,idx(M),[1 2])); // Returns M
// strvec = ["one","two"];
// strmat = [["three","four"], ["five","six"]];
// e = hstack(strvec,strvec); // Returns [["o", "n", "e", "o", "n", "e"],
// // ["t", "w", "o", "t", "w", "o"]]
// f = hstack(list_to_matrix(strvec,1), list_to_matrix(strvec,1));
// // Returns [["one", "one"],
// // ["two", "two"]]
// g = hstack(strmat,strmat); // Returns: [["three", "four", "three", "four"],
// // [ "five", "six", "five", "six"]]
function hstack(M1, M2, M3) =
(M3!=undef)? hstack([M1,M2,M3]) :
(M2!=undef)? hstack([M1,M2]) :
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()
// Usage:
// bmat = block_matrix([[M11, M12,...],[M21, M22,...], ... ]);
// Topics: Matrices
// See Also: column(), submatrix()
// 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()
// Usage:
// solv = linear_solve(A,b)
// 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.
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: matrix_inverse()
// 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()
// 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.
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()
// 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()
// 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()
// 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()
// 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()
// 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()
// 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()
// 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];
// Function: determinant()
// Usage:
// d = determinant(M);
// Description:
// Returns the determinant for the given square matrix.
// Arguments:
// M = The NxN square matrix to get the determinant of.
// Example:
// M = [ [6,4,-2,9], [1,-2,8,3], [1,5,7,6], [4,2,5,1] ];
// det = determinant(M); // Returns: 2267
function determinant(M) =
assert(is_list(M), "Input must be a square matrix." )
len(M)==1? M[0][0] :
len(M)==2? det2(M) :
len(M)==3? det3(M) :
len(M)==4? det4(M) :
assert(is_matrix(M, square=true), "Input must be a square matrix." )
sum(
[for (col=[0:1:len(M)-1])
((col%2==0)? 1 : -1) *
M[col][0] *
determinant(
[for (r=[1:1:len(M)-1])
[for (c=[0:1:len(M)-1])
if (c!=col) M[c][r]
]
]
)
]
);
// Function: norm_fro()
// Usage:
// norm_fro(A)
// Description:
// Computes frobenius norm of input matrix. The frobenius norm is the square root of the sum of the
// squares of all of the entries of the matrix. On vectors it is the same as the usual 2-norm.
// This is an easily computed norm that is convenient for comparing two matrices.
function norm_fro(A) =
assert(is_matrix(A) || is_vector(A))
norm(flatten(A));
// Function: matrix_trace()
// Usage:
// matrix_trace(M)
// Description:
// Computes the trace of a square matrix, the sum of the entries on the diagonal.
function matrix_trace(M) =
assert(is_matrix(M,square=true), "Input to trace must be a square matrix")
[for(i=[0:1:len(M)-1])1] * [for(i=[0:1:len(M)-1]) M[i][i]];