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changes to hide recursion args and avoid repetitive arg validations
cumsum broken in two to hide recursion args and avoid repetitive arg validations. back_substitute changed to avoid repetitive arg validations in the recursion. minor change in deriv2 and deriv3 to avoid an unecessary call to is_matrix. change in is_matrix for better performance any() and all() broken in two to avoid repetitive arg validation in the recursion and to hide recursion args. change in polymult to call convolve break of poly_div in two to avoid repetitive arg validations in the recursion.
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RonaldoCMP
parent
0e7517ecfe
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e06519bbfb
2 changed files with 82 additions and 73 deletions
143
math.scad
143
math.scad
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@ -84,7 +84,7 @@ function hypot(x,y,z=0) =
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// y = factorial(6); // Returns: 720
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// z = factorial(9); // Returns: 362880
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function factorial(n,d=0) =
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assert(is_int(n) && is_int(d) && n>=0 && d>=0, "Factorial is not defined for negative numbers")
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assert(is_int(n) && is_int(d) && n>=0 && d>=0, "Factorial is defined only for non negative integers")
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assert(d<=n, "d cannot be larger than n")
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product([1,for (i=[n:-1:d+1]) i]);
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@ -164,7 +164,7 @@ function binomial_coefficient(n,k) =
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function lerp(a,b,u) =
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assert(same_shape(a,b), "Bad or inconsistent inputs to lerp")
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is_finite(u)? (1-u)*a + u*b :
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assert(is_finite(u) || is_vector(u) || valid_range(u), "Input u to lerp must be a number, vector, or range.")
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assert(is_finite(u) || is_vector(u) || valid_range(u), "Input u to lerp must be a number, vector, or valid range.")
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[for (v = u) (1-v)*a + v*b ];
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@ -387,12 +387,13 @@ function modang(x) =
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// modrange(90,270,360, step=-45); // Returns: [90,45,0,315,270]
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// modrange(270,90,360, step=-45); // Returns: [270,225,180,135,90]
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function modrange(x, y, m, step=1) =
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assert( is_finite(x+y+step+m) && !approx(m,0), "Input must be finite numbers. The module value cannot be zero.")
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assert( is_finite(x+y+step+m) && !approx(m,0), "Input must be finite numbers and the module value cannot be zero." )
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let(
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a = posmod(x, m),
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b = posmod(y, m),
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c = step>0? (a>b? b+m : b) : (a<b? b-m : b)
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) [for (i=[a:step:c]) (i%m+m)%m];
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c = step>0? (a>b? b+m : b)
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: (a<b? b-m : b)
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) [for (i=[a:step:c]) (i%m+m)%m ];
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@ -536,9 +537,13 @@ function _sum(v,_total,_i=0) = _i>=len(v) ? _total : _sum(v,_total+v[_i], _i+1);
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// cumsum([2,2,2]); // returns [2,4,6]
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// cumsum([1,2,3]); // returns [1,3,6]
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// cumsum([[1,2,3], [3,4,5], [5,6,7]]); // returns [[1,2,3], [4,6,8], [9,12,15]]
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function cumsum(v,_i=0,_acc=[]) =
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function cumsum(v) =
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assert(is_consistent(v), "The input is not consistent." )
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_cumsum(v,_i=0,_acc=[]);
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function _cumsum(v,_i=0,_acc=[]) =
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_i==len(v) ? _acc :
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cumsum(
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_cumsum(
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v, _i+1,
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concat(
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_acc,
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@ -598,7 +603,7 @@ function deltas(v) =
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// Description:
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// Returns the product of all entries in the given list.
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// If passed a list of vectors of same dimension, returns a vector of products of each part.
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// If passed a list of square matrices, returns a the resulting product matrix.
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// If passed a list of square matrices, returns the resulting product matrix.
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// Arguments:
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// v = The list to get the product of.
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// Example:
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@ -606,7 +611,7 @@ function deltas(v) =
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// product([[1,2,3], [3,4,5], [5,6,7]]); // returns [15, 48, 105]
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function product(v) =
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assert( is_vector(v) || is_matrix(v) || ( is_matrix(v[0],square=true) && is_consistent(v)),
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"Invalid input.")
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"Invalid input.")
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_product(v, 1, v[0]);
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function _product(v, i=0, _tot) =
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@ -691,9 +696,12 @@ function linear_solve(A,b) =
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zeros = [for(i=[0:mindim-1]) if (approx(R[i][i],0)) i]
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)
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zeros != [] ? [] :
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m<n ? Q*back_substitute(R,b,transpose=true) :
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back_substitute(R, transpose(Q)*b);
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m<n
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// avoiding input validation in back_substitute
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? let( n = len(R),
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Rt = [for(i=[0:n-1]) [for(j=[0:n-1]) R[n-1-j][n-1-i]]] )
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Q*reverse(_back_substitute(Rt,reverse(b)))
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: _back_substitute(R, transpose(Q)*b);
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// Function: matrix_inverse()
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// Usage:
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@ -784,7 +792,6 @@ function _back_substitute(R, b, x=[]) =
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_back_substitute(R, b, concat([newvalue],x));
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// Function: det2()
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// Description:
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// Optimized function that returns the determinant for the given 2x2 square matrix.
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@ -794,7 +801,7 @@ function _back_substitute(R, b, x=[]) =
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// M = [ [6,-2], [1,8] ];
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// det = det2(M); // Returns: 50
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function det2(M) =
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assert( is_matrix(M,2,2), "Matrix should be 2x2." )
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assert( 0*M==[[0,0],[0,0]], "Matrix should be 2x2." )
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M[0][0] * M[1][1] - M[0][1]*M[1][0];
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@ -807,7 +814,7 @@ function det2(M) =
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// M = [ [6,4,-2], [1,-2,8], [1,5,7] ];
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// det = det3(M); // Returns: -334
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function det3(M) =
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assert( is_matrix(M,3,3), "Matrix should be 3x3." )
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assert( 0*M==[[0,0,0],[0,0,0],[0,0,0]], "Matrix should be 3x3." )
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M[0][0] * (M[1][1]*M[2][2]-M[2][1]*M[1][2]) -
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M[1][0] * (M[0][1]*M[2][2]-M[2][1]*M[0][2]) +
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M[2][0] * (M[0][1]*M[1][2]-M[1][1]*M[0][2]);
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@ -856,10 +863,10 @@ function determinant(M) =
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// square = set to true to require a square matrix. Default: false
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function is_matrix(A,m,n,square=false) =
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is_list(A[0])
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&& ( let(v = A*A[0]) is_num(0*(v*v)) ) // a matrix of finite numbers
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&& (is_undef(n) || len(A[0])==n )
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&& (is_undef(m) || len(A)==m )
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&& ( !square || len(A)==len(A[0]));
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&& ( let(v = A*A[0]) is_num(0*(v*v)) ) // a matrix of finite numbers
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&& (is_undef(n) || len(A[0])==n )
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&& (is_undef(m) || len(A)==m )
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&& ( !square || len(A)==len(A[0]));
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// Section: Comparisons and Logic
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@ -949,13 +956,16 @@ function compare_lists(a, b) =
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// any([1,5,true]); // Returns true.
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// any([[0,0], [0,0]]); // Returns false.
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// any([[0,0], [1,0]]); // Returns true.
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function any(l, i=0, succ=false) =
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(i>=len(l) || succ)? succ :
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any( l,
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i+1,
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succ = is_list(l[i]) ? any(l[i]) : !(!l[i])
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);
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function any(l) =
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assert(is_list(l), "The input is not a list." )
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_any(l, i=0, succ=false);
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function _any(l, i=0, succ=false) =
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(i>=len(l) || succ)? succ :
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_any( l,
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i+1,
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succ = is_list(l[i]) ? _any(l[i]) : !(!l[i])
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);
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// Function: all()
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@ -972,12 +982,15 @@ function any(l, i=0, succ=false) =
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// all([[0,0], [1,0]]); // Returns false.
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// all([[1,1], [1,1]]); // Returns true.
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function all(l, i=0, fail=false) =
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(i>=len(l) || fail)? !fail :
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all( l,
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i+1,
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fail = is_list(l[i]) ? !all(l[i]) : !l[i]
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) ;
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assert( is_list(l), "The input is not a list." )
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_all(l, i=0, fail=false);
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function _all(l, i=0, fail=false) =
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(i>=len(l) || fail)? !fail :
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_all( l,
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i+1,
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fail = is_list(l[i]) ? !_all(l[i]) : !l[i]
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) ;
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// Function: count_true()
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@ -1195,12 +1208,12 @@ function C_div(z1,z2) =
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// where a_n is the z^n coefficient. Polynomial coefficients are real.
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// The result is a number if `z` is a number and a complex number otherwise.
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function polynomial(p,z,k,total) =
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is_undef(k)
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? assert( is_vector(p) , "Input polynomial coefficients must be a vector." )
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assert( is_finite(z) || is_vector(z,2), "The value of `z` must be a real or a complex number." )
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polynomial( _poly_trim(p), z, 0, is_num(z) ? 0 : [0,0])
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: k==len(p) ? total
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: polynomial(p,z,k+1, is_num(z) ? total*z+p[k] : C_times(total,z)+[p[k],0]);
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is_undef(k)
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? assert( is_vector(p) , "Input polynomial coefficients must be a vector." )
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assert( is_finite(z) || is_vector(z,2), "The value of `z` must be a real or a complex number." )
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polynomial( _poly_trim(p), z, 0, is_num(z) ? 0 : [0,0])
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: k==len(p) ? total
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: polynomial(p,z,k+1, is_num(z) ? total*z+p[k] : C_times(total,z)+[p[k],0]);
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// Function: poly_mult()
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// Usage:
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@ -1210,22 +1223,16 @@ function polynomial(p,z,k,total) =
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// Given a list of polynomials represented as real coefficient lists, with the highest degree coefficient first,
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// computes the coefficient list of the product polynomial.
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function poly_mult(p,q) =
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is_undef(q) ?
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assert( is_list(p)
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&& []==[for(pi=p) if( !is_vector(pi) && pi!=[]) 0],
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"Invalid arguments to poly_mult")
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len(p)==2 ? poly_mult(p[0],p[1])
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: poly_mult(p[0], poly_mult(select(p,1,-1)))
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:
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_poly_trim(
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[
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for(n = [len(p)+len(q)-2:-1:0])
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sum( [for(i=[0:1:len(p)-1])
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let(j = len(p)+len(q)- 2 - n - i)
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if (j>=0 && j<len(q)) p[i]*q[j]
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])
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]);
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is_undef(q) ?
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len(p)==2
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? poly_mult(p[0],p[1])
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: poly_mult(p[0], poly_mult(select(p,1,-1)))
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:
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assert( is_vector(p) && is_vector(q),"Invalid arguments to poly_mult")
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p*p==0 || q*q==0
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? [0]
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: _poly_trim(convolve(p,q));
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// Function: poly_div()
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// Usage:
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@ -1235,19 +1242,23 @@ function poly_mult(p,q) =
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// a list of two polynomials, [quotient, remainder]. If the division has no remainder then
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// the zero polynomial [] is returned for the remainder. Similarly if the quotient is zero
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// the returned quotient will be [].
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function poly_div(n,d,q) =
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is_undef(q)
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? assert( is_vector(n) && is_vector(d) , "Invalid polynomials." )
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let( d = _poly_trim(d) )
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assert( d!=[0] , "Denominator cannot be a zero polynomial." )
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poly_div(n,d,q=[])
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: len(n)<len(d) ? [q,_poly_trim(n)] :
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let(
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t = n[0] / d[0],
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newq = concat(q,[t]),
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newn = [for(i=[1:1:len(n)-1]) i<len(d) ? n[i] - t*d[i] : n[i]]
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)
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poly_div(newn,d,newq);
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function poly_div(n,d) =
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assert( is_vector(n) && is_vector(d) , "Invalid polynomials." )
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let( d = _poly_trim(d),
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n = _poly_trim(n) )
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assert( d!=[0] , "Denominator cannot be a zero polynomial." )
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n==[0]
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? [[0],[0]]
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: _poly_div(n,d,q=[]);
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function _poly_div(n,d,q) =
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len(n)<len(d) ? [q,_poly_trim(n)] :
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let(
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t = n[0] / d[0],
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newq = concat(q,[t]),
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newn = [for(i=[1:1:len(n)-1]) i<len(d) ? n[i] - t*d[i] : n[i]]
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)
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_poly_div(newn,d,newq);
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// Internal Function: _poly_trim()
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@ -1378,4 +1389,4 @@ function real_roots(p,eps=undef,tol=1e-14) =
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? [for(z=roots) if (abs(z.y)/(1+norm(z))<eps) z.x]
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: [for(i=idx(roots)) if (abs(roots[i].y)<=err[i]) roots[i].x];
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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@ -913,23 +913,21 @@ test_qr_factor();
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module test_poly_mult(){
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assert_equal(poly_mult([3,2,1],[4,5,6,7]),[12,23,32,38,20,7]);
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assert_equal(poly_mult([3,2,1],[0]),[0]);
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// assert_equal(poly_mult([3,2,1],[]),[]);
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assert_equal(poly_mult([[1,2],[3,4],[5,6]]), [15,68,100,48]);
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assert_equal(poly_mult([3,2,1],[0]),[0]);
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assert_equal(poly_mult([[1,2],[0],[5,6]]), [0]);
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// assert_equal(poly_mult([[1,2],[],[5,6]]), []);
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assert_equal(poly_mult([[3,4,5],[0,0,0]]),[0]);
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// assert_equal(poly_mult([[3,4,5],[0,0,0]]),[]);
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assert_equal(poly_mult([[3,4,5],[0,0,0]]), [0]);
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assert_equal(poly_mult([[0],[0,0,0]]),[0]);
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}
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test_poly_mult();
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module test_poly_div(){
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assert_equal(poly_div(poly_mult([4,3,3,2],[2,1,3]), [2,1,3]),[[4,3,3,2],[0]]);
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// assert_equal(poly_div(poly_mult([4,3,3,2],[2,1,3]), [2,1,3]),[[4,3,3,2],[]]);
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assert_equal(poly_div([1,2,3,4],[1,2,3,4,5]), [[], [1,2,3,4]]);
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assert_equal(poly_div(poly_add(poly_mult([1,2,3,4],[2,0,2]), [1,1,2]), [1,2,3,4]), [[2,0,2],[1,1,2]]);
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assert_equal(poly_div([1,2,3,4], [1,-3]), [[1,5,18],[58]]);
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assert_equal(poly_div([0], [1,-3]), [[0],[0]]);
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}
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test_poly_div();
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@ -942,4 +940,4 @@ module test_poly_add(){
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}
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test_poly_add();
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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