////////////////////////////////////////////////////////////////////// // LibFile: math.scad // Assorted math functions, including linear interpolation, list operations (sums, mean, products), // convolution, quantization, log2, hyperbolic trig functions, random numbers, derivatives, // polynomials, and root finding. // Includes: // include // FileGroup: Math // FileSummary: Math on lists, special functions, quantization, random numbers, calculus, root finding // // FileFootnotes: STD=Included in std.scad ////////////////////////////////////////////////////////////////////// // Section: Math Constants // Constant: PHI // Synopsis: The golden ratio φ (phi). Approximately 1.6180339887 // Topics: Constants, Math // See Also: EPSILON, INF, NAN // Description: The golden ratio φ (phi). Approximately 1.6180339887 PHI = (1+sqrt(5))/2; // Constant: EPSILON // Synopsis: A tiny value to compare floating point values. `1e-9` // Topics: Constants, Math // See Also: PHI, EPSILON, INF, NAN // Description: A really small value useful in comparing floating point numbers. ie: abs(a-b)0 ? // Function: hypot() // Synopsis: Returns the hypotenuse length of a 2D or 3D triangle. // Topics: Math // See Also: hypot(), sqr(), log2() // Usage: // l = hypot(x, y, [z]); // Description: // Calculate hypotenuse length of a 2D or 3D triangle. // Arguments: // x = Length on the X axis. // y = Length on the Y axis. // z = Length on the Z axis. Optional. // Example: // l = hypot(3,4); // Returns: 5 // l = hypot(3,4,5); // Returns: ~7.0710678119 function hypot(x,y,z=0) = assert( is_vector([x,y,z]), "Improper number(s).") norm([x,y,z]); // Function: factorial() // Synopsis: Returns the factorial of the given integer. // Topics: Math // See Also: hypot(), sqr(), log2() // Usage: // x = factorial(n, [d]); // Description: // Returns the factorial of the given integer value, or n!/d! if d is given. // Arguments: // n = The integer number to get the factorial of. (n!) // d = If given, the returned value will be (n! / d!) // Example: // x = factorial(4); // Returns: 24 // y = factorial(6); // Returns: 720 // z = factorial(9); // Returns: 362880 function factorial(n,d=0) = assert(is_int(n) && is_int(d) && n>=0 && d>=0, "Factorial is defined only for non negative integers") assert(d<=n, "d cannot be larger than n") product([1,for (i=[n:-1:d+1]) i]); // Function: binomial() // Synopsis: Returns the binomial coefficients of the integer `n`. // Topics: Math // See Also: hypot(), sqr(), log2(), factorial() // Usage: // x = binomial(n); // Description: // Returns the binomial coefficients of the integer `n`. // Arguments: // n = The integer to get the binomial coefficients of // Example: // x = binomial(3); // Returns: [1,3,3,1] // y = binomial(4); // Returns: [1,4,6,4,1] // z = binomial(6); // Returns: [1,6,15,20,15,6,1] function binomial(n) = assert( is_int(n) && n>0, "Input is not an integer greater than 0.") [for( c = 1, i = 0; i<=n; c = c*(n-i)/(i+1), i = i+1 ) c ] ; // Function: binomial_coefficient() // Synopsis: Returns the `k`-th binomial coefficient of the integer `n`. // Topics: Math // See Also: hypot(), sqr(), log2(), factorial() // Usage: // x = binomial_coefficient(n, k); // Description: // Returns the `k`-th binomial coefficient of the integer `n`. // Arguments: // n = The integer to get the binomial coefficient of // k = The binomial coefficient index // Example: // x = binomial_coefficient(3,2); // Returns: 3 // y = binomial_coefficient(10,6); // Returns: 210 function binomial_coefficient(n,k) = assert( is_int(n) && is_int(k), "Some input is not a number.") k < 0 || k > n ? 0 : k ==0 || k ==n ? 1 : let( k = min(k, n-k), b = [for( c = 1, i = 0; i<=k; c = c*(n-i)/(i+1), i = i+1 ) c] ) b[len(b)-1]; // Function: gcd() // Synopsis: Returns the Greatest Common Divisor/Factor of two integers. // Topics: Math // See Also: hypot(), sqr(), log2(), factorial(), binomial(), gcd(), lcm() // Usage: // x = gcd(a,b) // Description: // Computes the Greatest Common Divisor/Factor of `a` and `b`. function gcd(a,b) = assert(is_int(a) && is_int(b),"Arguments to gcd must be integers") b==0 ? abs(a) : gcd(b,a % b); // Computes lcm for two integers function _lcm(a,b) = assert(is_int(a) && is_int(b), "Invalid non-integer parameters to lcm") assert(a!=0 && b!=0, "Arguments to lcm should not be zero") abs(a*b) / gcd(a,b); // Computes lcm for a list of values function _lcmlist(a) = len(a)==1 ? a[0] : _lcmlist(concat(lcm(a[0],a[1]),list_tail(a,2))); // Function: lcm() // Synopsis: Returns the Least Common Multiple of two or more integers. // Topics: Math // See Also: hypot(), sqr(), log2(), factorial(), binomial(), gcd(), lcm() // Usage: // div = lcm(a, b); // divs = lcm(list); // Description: // Computes the Least Common Multiple of the two arguments or a list of arguments. Inputs should // be non-zero integers. The output is always a positive integer. It is an error to pass zero // as an argument. function lcm(a,b=[]) = !is_list(a) && !is_list(b) ? _lcm(a,b) : let( arglist = concat(force_list(a),force_list(b)) ) assert(len(arglist)>0, "Invalid call to lcm with empty list(s)") _lcmlist(arglist); // Function rational_approx() // Usage: // pq = rational_approx(x, maxq); // Description: // Finds the best rational approximation p/q to the number x so that q<=maxq. Returns // the result as `[p,q]`. If the input is zero, then returns `[0,1]`. // Example: // pq1 = rational_approx(PI,10); // Returns: [22,7] // pq2 = rational_approx(PI,10000); // Returns: [355, 113] // pq3 = rational_approx(221/323,500); // Returns: [13,19] // pq4 = rational_approx(0,50); // Returns: [0,1] function rational_approx(x, maxq, cfrac=[], p, q) = let( next = floor(x), fracpart = x-next, cfrac = [each cfrac, next], pq = _cfrac_to_pq(cfrac) ) approx(fracpart,0) ? pq : pq[1]>maxq ? [p,q] : rational_approx(1/fracpart,maxq,cfrac, pq[0], pq[1]); // Converts a continued fraction given as a list with leading integer term // into a fraction in the form p / q, returning [p,q]. function _cfrac_to_pq(cfrac,p=0,q=1,ind) = is_undef(ind) ? _cfrac_to_pq(cfrac,p,q,len(cfrac)-1) : ind==0 ? [p+q*cfrac[0], q] : _cfrac_to_pq(cfrac, q, cfrac[ind]*q+p, ind-1); // Section: Hyperbolic Trigonometry // Function: sinh() // Synopsis: Returns the hyperbolic sine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = sinh(x); // Description: Takes a value `x`, and returns the hyperbolic sine of it. function sinh(x) = assert(is_finite(x), "The input must be a finite number.") (exp(x)-exp(-x))/2; // Function: cosh() // Synopsis: Returns the hyperbolic cosine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = cosh(x); // Description: Takes a value `x`, and returns the hyperbolic cosine of it. function cosh(x) = assert(is_finite(x), "The input must be a finite number.") (exp(x)+exp(-x))/2; // Function: tanh() // Synopsis: Returns the hyperbolic tangent of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = tanh(x); // Description: Takes a value `x`, and returns the hyperbolic tangent of it. function tanh(x) = assert(is_finite(x), "The input must be a finite number.") sinh(x)/cosh(x); // Function: asinh() // Synopsis: Returns the hyperbolic arc-sine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = asinh(x); // Description: Takes a value `x`, and returns the inverse hyperbolic sine of it. function asinh(x) = assert(is_finite(x), "The input must be a finite number.") ln(x+sqrt(x*x+1)); // Function: acosh() // Synopsis: Returns the hyperbolic arc-cosine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = acosh(x); // Description: Takes a value `x`, and returns the inverse hyperbolic cosine of it. function acosh(x) = assert(is_finite(x), "The input must be a finite number.") ln(x+sqrt(x*x-1)); // Function: atanh() // Synopsis: Returns the hyperbolic arc-tangent of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = atanh(x); // Description: Takes a value `x`, and returns the inverse hyperbolic tangent of it. function atanh(x) = assert(is_finite(x), "The input must be a finite number.") ln((1+x)/(1-x))/2; // Section: Quantization // Function: quant() // Synopsis: Returns `x` quantized to the nearest integer multiple of `y`. // Topics: Math, Quantization // See Also: quant(), quantdn(), quantup() // Usage: // num = quant(x, y); // Description: // Quantize a value `x` to an integer multiple of `y`, rounding to the nearest multiple. // The value of `y` does NOT have to be an integer. If `x` is a list, then every item // in that list will be recursively quantized. // Arguments: // x = The value or list to quantize. // y = Positive quantum to quantize to // Example: // a = quant(12,4); // Returns: 12 // b = quant(13,4); // Returns: 12 // c = quant(13.1,4); // Returns: 12 // d = quant(14,4); // Returns: 16 // e = quant(14.1,4); // Returns: 16 // f = quant(15,4); // Returns: 16 // g = quant(16,4); // Returns: 16 // h = quant(9,3); // Returns: 9 // i = quant(10,3); // Returns: 9 // j = quant(10.4,3); // Returns: 9 // k = quant(10.5,3); // Returns: 12 // l = quant(11,3); // Returns: 12 // m = quant(12,3); // Returns: 12 // n = quant(11,2.5); // Returns: 10 // o = quant(12,2.5); // Returns: 12.5 // p = quant([12,13,13.1,14,14.1,15,16],4); // Returns: [12,12,12,16,16,16,16] // q = quant([9,10,10.4,10.5,11,12],3); // Returns: [9,9,9,12,12,12] // r = quant([[9,10,10.4],[10.5,11,12]],3); // Returns: [[9,9,9],[12,12,12]] function quant(x,y) = assert( is_finite(y) && y>0, "The quantum `y` must be a positive value.") is_num(x) ? round(x/y)*y : _roundall(x/y)*y; function _roundall(data) = [for(x=data) is_list(x) ? _roundall(x) : round(x)]; // Function: quantdn() // Synopsis: Returns `x` quantized down to an integer multiple of `y`. // Topics: Math, Quantization // See Also: quant(), quantdn(), quantup() // Usage: // num = quantdn(x, y); // Description: // Quantize a value `x` to an integer multiple of `y`, rounding down to the previous multiple. // The value of `y` does NOT have to be an integer. If `x` is a list, then every item in that // list will be recursively quantized down. // Arguments: // x = The value or list to quantize. // y = Postive quantum to quantize to. // Example: // a = quantdn(12,4); // Returns: 12 // b = quantdn(13,4); // Returns: 12 // c = quantdn(13.1,4); // Returns: 12 // d = quantdn(14,4); // Returns: 12 // e = quantdn(14.1,4); // Returns: 12 // f = quantdn(15,4); // Returns: 12 // g = quantdn(16,4); // Returns: 16 // h = quantdn(9,3); // Returns: 9 // i = quantdn(10,3); // Returns: 9 // j = quantdn(10.4,3); // Returns: 9 // k = quantdn(10.5,3); // Returns: 9 // l = quantdn(11,3); // Returns: 9 // m = quantdn(12,3); // Returns: 12 // n = quantdn(11,2.5); // Returns: 10 // o = quantdn(12,2.5); // Returns: 10 // p = quantdn([12,13,13.1,14,14.1,15,16],4); // Returns: [12,12,12,12,12,12,16] // q = quantdn([9,10,10.4,10.5,11,12],3); // Returns: [9,9,9,9,9,12] // r = quantdn([[9,10,10.4],[10.5,11,12]],3); // Returns: [[9,9,9],[9,9,12]] function quantdn(x,y) = assert( is_finite(y) && y>0, "The quantum `y` must be a positive value.") is_num(x) ? floor(x/y)*y : _floorall(x/y)*y; function _floorall(data) = [for(x=data) is_list(x) ? _floorall(x) : floor(x)]; // Function: quantup() // Synopsis: Returns `x` quantized uo to an integer multiple of `y`. // Topics: Math, Quantization // See Also: quant(), quantdn(), quantup() // Usage: // num = quantup(x, y); // Description: // Quantize a value `x` to an integer multiple of `y`, rounding up to the next multiple. // The value of `y` does NOT have to be an integer. If `x` is a list, then every item in // that list will be recursively quantized up. // Arguments: // x = The value or list to quantize. // y = Positive quantum to quantize to. // Example: // a = quantup(12,4); // Returns: 12 // b = quantup(13,4); // Returns: 16 // c = quantup(13.1,4); // Returns: 16 // d = quantup(14,4); // Returns: 16 // e = quantup(14.1,4); // Returns: 16 // f = quantup(15,4); // Returns: 16 // g = quantup(16,4); // Returns: 16 // h = quantup(9,3); // Returns: 9 // i = quantup(10,3); // Returns: 12 // j = quantup(10.4,3); // Returns: 12 // k = quantup(10.5,3); // Returns: 12 // l = quantup(11,3); // Returns: 12 // m = quantup(12,3); // Returns: 12 // n = quantdn(11,2.5); // Returns: 12.5 // o = quantdn(12,2.5); // Returns: 12.5 // p = quantup([12,13,13.1,14,14.1,15,16],4); // Returns: [12,16,16,16,16,16,16] // q = quantup([9,10,10.4,10.5,11,12],3); // Returns: [9,12,12,12,12,12] // r = quantup([[9,10,10.4],[10.5,11,12]],3); // Returns: [[9,12,12],[12,12,12]] function quantup(x,y) = assert( is_finite(y) && y>0, "The quantum `y` must be a positive value.") is_num(x) ? ceil(x/y)*y : _ceilall(x/y)*y; function _ceilall(data) = [for(x=data) is_list(x) ? _ceilall(x) : ceil(x)]; // Section: Constraints and Modulos // Function: constrain() // Synopsis: Returns a value constrained between `minval` and `maxval`, inclusive. // Topics: Math // See Also: posmod(), modang() // Usage: // val = constrain(v, minval, maxval); // Description: // Constrains value to a range of values between minval and maxval, inclusive. // Arguments: // v = value to constrain. // minval = minimum value to return, if out of range. // maxval = maximum value to return, if out of range. // Example: // a = constrain(-5, -1, 1); // Returns: -1 // b = constrain(5, -1, 1); // Returns: 1 // c = constrain(0.3, -1, 1); // Returns: 0.3 // d = constrain(9.1, 0, 9); // Returns: 9 // e = constrain(-0.1, 0, 9); // Returns: 0 function constrain(v, minval, maxval) = assert( is_finite(v+minval+maxval), "Input must be finite number(s).") min(maxval, max(minval, v)); // Function: posmod() // Synopsis: Returns the positive modulo of a value. // Topics: Math // See Also: constrain(), posmod(), modang() // Usage: // mod = posmod(x, m) // Description: // Returns the positive modulo `m` of `x`. Value returned will be in the range 0 ... `m`-1. // Arguments: // x = The value to constrain. // m = Modulo value. // Example: // a = posmod(-700,360); // Returns: 340 // b = posmod(-270,360); // Returns: 90 // c = posmod(-120,360); // Returns: 240 // d = posmod(120,360); // Returns: 120 // e = posmod(270,360); // Returns: 270 // f = posmod(700,360); // Returns: 340 // g = posmod(3,2.5); // Returns: 0.5 function posmod(x,m) = assert( is_finite(x) && is_finite(m) && !approx(m,0) , "Input must be finite numbers. The divisor cannot be zero.") (x%m+m)%m; // Function: modang() // Synopsis: Returns an angle normalized to between -180º and 180º. // Topics: Math // See Also: constrain(), posmod(), modang() // Usage: // ang = modang(x); // Description: // Takes an angle in degrees and normalizes it to an equivalent angle value between -180 and 180. // Example: // a1 = modang(-700); // Returns: 20 // a2 = modang(-270); // Returns: 90 // a3 = modang(-120); // Returns: -120 // a4 = modang(120); // Returns: 120 // a5 = modang(270); // Returns: -90 // a6 = modang(700); // Returns: -20 function modang(x) = assert( is_finite(x), "Input must be a finite number.") let(xx = posmod(x,360)) xx<180? xx : xx-360; // Section: Operations on Lists (Sums, Mean, Products) // Function: sum() // Synopsis: Returns the sum of a list of values. // Topics: Math // See Also: mean(), median(), product(), cumsum() // Usage: // x = sum(v, [dflt]); // Description: // Returns the sum of all entries in the given consistent list. // If passed an array of vectors, returns the sum the vectors. // If passed an array of matrices, returns the sum of the matrices. // If passed an empty list, the value of `dflt` will be returned. // Arguments: // v = The list to get the sum of. // dflt = The default value to return if `v` is an empty list. Default: 0 // Example: // sum([1,2,3]); // returns 6. // sum([[1,2,3], [3,4,5], [5,6,7]]); // returns [9, 12, 15] function sum(v, dflt=0) = v==[]? dflt : assert(is_consistent(v), "Input to sum is non-numeric or inconsistent") is_finite(v[0]) || is_vector(v[0]) ? [for(i=v) 1]*v : _sum(v,v[0]*0); function _sum(v,_total,_i=0) = _i>=len(v) ? _total : _sum(v,_total+v[_i], _i+1); // Function: mean() // Synopsis: Returns the mean value of a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), median(), product() // Usage: // x = mean(v); // Description: // Returns the arithmetic mean/average of all entries in the given array. // If passed a list of vectors, returns a vector of the mean of each part. // Arguments: // v = The list of values to get the mean of. // Example: // mean([2,3,4]); // returns 3. // mean([[1,2,3], [3,4,5], [5,6,7]]); // returns [3, 4, 5] function mean(v) = assert(is_list(v) && len(v)>0, "Invalid list.") sum(v)/len(v); // Function: median() // Synopsis: Returns the median value of a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), median(), product() // Usage: // middle = median(v) // Description: // Returns the median of the given vector. function median(v) = assert(is_vector(v), "Input to median must be a vector") len(v)%2 ? max( list_smallest(v, ceil(len(v)/2)) ) : let( lowest = list_smallest(v, len(v)/2 + 1), max = max(lowest), imax = search(max,lowest,1), max2 = max([for(i=idx(lowest)) if(i!=imax[0]) lowest[i] ]) ) (max+max2)/2; // Function: deltas() // Synopsis: Returns the deltas between a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), median(), product() // Usage: // delts = deltas(v,[wrap]); // Description: // Returns a list with the deltas of adjacent entries in the given list, optionally wrapping back to the front. // The list should be a consistent list of numeric components (numbers, vectors, matrix, etc). // Given [a,b,c,d], returns [b-a,c-b,d-c]. // Arguments: // v = The list to get the deltas of. // wrap = If true, wrap back to the start from the end. ie: return the difference between the last and first items as the last delta. Default: false // Example: // deltas([2,5,9,17]); // returns [3,4,8]. // deltas([[1,2,3], [3,6,8], [4,8,11]]); // returns [[2,4,5], [1,2,3]] function deltas(v, wrap=false) = assert( is_consistent(v) && len(v)>1 , "Inconsistent list or with length<=1.") [for (p=pair(v,wrap)) p[1]-p[0]] ; // Function: cumsum() // Synopsis: Returns the running cumulative sum of a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), median(), product() // Usage: // sums = cumsum(v); // Description: // Returns a list where each item is the cumulative sum of all items up to and including the corresponding entry in the input list. // If passed an array of vectors, returns a list of cumulative vectors sums. // Arguments: // v = The list to get the sum of. // Example: // cumsum([1,1,1]); // returns [1,2,3] // cumsum([2,2,2]); // returns [2,4,6] // cumsum([1,2,3]); // returns [1,3,6] // cumsum([[1,2,3], [3,4,5], [5,6,7]]); // returns [[1,2,3], [4,6,8], [9,12,15]] function cumsum(v) = v==[] ? [] : assert(is_consistent(v), "The input is not consistent." ) [for (a = v[0], i = 1 ; i <= len(v) ; a = i= minval, "Max value cannot be smaller than minval") let (rvect = is_def(seed) ? rands(minval,maxval+1,n,seed) : rands(minval,maxval+1,n)) [for(entry = rvect) floor(entry)]; // Function: random_points() // Synopsis: Returns a list of random points. // Topics: Random, Points // See Also: rand_int(), random_points(), random_polygon(), spherical_random_points() // Usage: // points = random_points(n, dim, [scale], [seed]); // Description: // Generate `n` uniform random points of dimension `dim` with data ranging from -scale to +scale. // The `scale` may be a number, in which case the random data lies in a cube, // or a vector with dimension `dim`, in which case each dimension has its own scale. // Arguments: // n = number of points to generate. Default: 1 // dim = dimension of the points. Default: 2 // scale = the scale of the point coordinates. Default: 1 // seed = an optional seed for the random generation. function random_points(n, dim, scale=1, seed) = assert( is_int(n) && n>=0, "The number of points should be a non-negative integer.") assert( is_int(dim) && dim>=1, "The point dimensions should be an integer greater than 1.") assert( is_finite(scale) || is_vector(scale,dim), "The scale should be a number or a vector with length equal to d.") let( rnds = is_undef(seed) ? rands(-1,1,n*dim) : rands(-1,1,n*dim, seed) ) is_num(scale) ? scale*[for(i=[0:1:n-1]) [for(j=[0:dim-1]) rnds[i*dim+j] ] ] : [for(i=[0:1:n-1]) [for(j=[0:dim-1]) scale[j]*rnds[i*dim+j] ] ]; // Function: gaussian_rands() // Synopsis: Returns a list of random numbers with a gaussian distribution. // Topics: Random, Statistics // See Also: rand_int(), random_points(), gaussian_rands(), random_polygon(), spherical_random_points(), exponential_rands() // Usage: // arr = gaussian_rands([n],[mean], [cov], [seed]); // Description: // Returns a random number or vector with a Gaussian/normal distribution. // Arguments: // n = the number of points to return. Default: 1 // mean = The average of the random value (a number or vector). Default: 0 // cov = covariance matrix of the random numbers, or variance in the 1D case. Default: 1 // seed = If given, sets the random number seed. function gaussian_rands(n=1, mean=0, cov=1, seed=undef) = assert(is_num(mean) || is_vector(mean)) let( dim = is_num(mean) ? 1 : len(mean) ) assert((dim==1 && is_num(cov)) || is_matrix(cov,dim,dim),"mean and covariance matrix not compatible") assert(is_undef(seed) || is_finite(seed)) let( nums = is_undef(seed)? rands(0,1,dim*n*2) : rands(0,1,dim*n*2,seed), rdata = [for (i = count(dim*n,0,2)) sqrt(-2*ln(nums[i]))*cos(360*nums[i+1])] ) dim==1 ? add_scalar(sqrt(cov)*rdata,mean) : assert(is_matrix_symmetric(cov),"Supplied covariance matrix is not symmetric") let( L = cholesky(cov) ) assert(is_def(L), "Supplied covariance matrix is not positive definite") move(mean,list_to_matrix(rdata,dim)*transpose(L)); // Function: exponential_rands() // Synopsis: Returns a list of random numbers with an exponential distribution. // Topics: Random, Statistics // See Also: rand_int(), random_points(), gaussian_rands(), random_polygon(), spherical_random_points() // Usage: // arr = exponential_rands([n], [lambda], [seed]) // Description: // Returns random numbers with an exponential distribution with parameter lambda, and hence mean 1/lambda. // Arguments: // n = number of points to return. Default: 1 // lambda = distribution parameter. The mean will be 1/lambda. Default: 1 function exponential_rands(n=1, lambda=1, seed) = assert( is_int(n) && n>=1, "The number of points should be an integer greater than zero.") assert( is_num(lambda) && lambda>0, "The lambda parameter must be a positive number.") let( unif = is_def(seed) ? rands(0,1,n,seed=seed) : rands(0,1,n) ) -(1/lambda) * [for(x=unif) x==1 ? 708.3964185322641 : ln(1-x)]; // Use ln(min_float) when x is 1 // Function: spherical_random_points() // Synopsis: Returns a list of random points on the surface of a sphere. // Topics: Random, Points // See Also: rand_int(), random_points(), gaussian_rands(), random_polygon(), spherical_random_points() // Usage: // points = spherical_random_points([n], [radius], [seed]); // Description: // Generate `n` 3D uniformly distributed random points lying on a sphere centered at the origin with radius equal to `radius`. // Arguments: // n = number of points to generate. Default: 1 // radius = the sphere radius. Default: 1 // seed = an optional seed for the random generation. // See https://mathworld.wolfram.com/SpherePointPicking.html function spherical_random_points(n=1, radius=1, seed) = assert( is_int(n) && n>=1, "The number of points should be an integer greater than zero.") assert( is_num(radius) && radius>0, "The radius should be a non-negative number.") let( theta = is_undef(seed) ? rands(0,360,n) : rands(0,360,n, seed), cosphi = rands(-1,1,n)) [for(i=[0:1:n-1]) let( sin_phi=sqrt(1-cosphi[i]*cosphi[i]) ) radius*[sin_phi*cos(theta[i]),sin_phi*sin(theta[i]), cosphi[i]]]; // Function: random_polygon() // Synopsis: Returns the CCW path of a simple random polygon. // Topics: Random, Polygon // See Also: random_points(), spherical_random_points() // Usage: // points = random_polygon([n], [size], [seed]); // Description: // Generate the `n` vertices of a random counter-clockwise simple 2d polygon // inside a circle centered at the origin with radius `size`. // Arguments: // n = number of vertices of the polygon. Default: 3 // size = the radius of a circle centered at the origin containing the polygon. Default: 1 // seed = an optional seed for the random generation. function random_polygon(n=3,size=1, seed) = assert( is_int(n) && n>2, "Improper number of polygon vertices.") assert( is_num(size) && size>0, "Improper size.") let( seed = is_undef(seed) ? rands(0,1,1)[0] : seed, cumm = cumsum(rands(0.1,10,n+1,seed)), angs = 360*cumm/cumm[n-1], rads = rands(.01,size,n,seed+cumm[0]) ) [for(i=count(n)) rads[i]*[cos(angs[i]), sin(angs[i])] ]; // Section: Calculus // Function: deriv() // Synopsis: Returns the first derivative estimate of a list of data. // Topics: Math, Calculus // See Also: deriv(), deriv2(), deriv3() // Usage: // x = deriv(data, [h], [closed]) // Description: // Computes a numerical derivative estimate of the data, which may be scalar or vector valued. // The `h` parameter gives the step size of your sampling so the derivative can be scaled correctly. // If the `closed` parameter is true the data is assumed to be defined on a loop with data[0] adjacent to // data[len(data)-1]. This function uses a symetric derivative approximation // for internal points, f'(t) = (f(t+h)-f(t-h))/2h. For the endpoints (when closed=false) the algorithm // uses a two point method if sufficient points are available: f'(t) = (3*(f(t+h)-f(t)) - (f(t+2*h)-f(t+h)))/2h. // . // If `h` is a vector then it is assumed to be nonuniform, with h[i] giving the sampling distance // between data[i+1] and data[i], and the data values will be linearly resampled at each corner // to produce a uniform spacing for the derivative estimate. At the endpoints a single point method // is used: f'(t) = (f(t+h)-f(t))/h. // Arguments: // data = the list of the elements to compute the derivative of. // h = the parametric sampling of the data. // closed = boolean to indicate if the data set should be wrapped around from the end to the start. function deriv(data, h=1, closed=false) = assert( is_consistent(data) , "Input list is not consistent or not numerical.") assert( len(data)>=2, "Input `data` should have at least 2 elements.") assert( is_finite(h) || is_vector(h), "The sampling `h` must be a number or a list of numbers." ) assert( is_num(h) || len(h) == len(data)-(closed?0:1), str("Vector valued `h` must have length ",len(data)-(closed?0:1))) is_vector(h) ? _deriv_nonuniform(data, h, closed=closed) : let( L = len(data) ) closed ? [ for(i=[0:1:L-1]) (data[(i+1)%L]-data[(L+i-1)%L])/2/h ] : let( first = L<3 ? data[1]-data[0] : 3*(data[1]-data[0]) - (data[2]-data[1]), last = L<3 ? data[L-1]-data[L-2]: (data[L-3]-data[L-2])-3*(data[L-2]-data[L-1]) ) [ first/2/h, for(i=[1:1:L-2]) (data[i+1]-data[i-1])/2/h, last/2/h ]; function _dnu_calc(f1,fc,f2,h1,h2) = let( f1 = h2

=3, "Input list has less than 3 elements.") closed ? [ for(i=[0:1:L-1]) (data[(i+1)%L]-2*data[i]+data[(L+i-1)%L])/h/h ] : let( first = L==3? data[0] - 2*data[1] + data[2] : L==4? 2*data[0] - 5*data[1] + 4*data[2] - data[3] : (35*data[0] - 104*data[1] + 114*data[2] - 56*data[3] + 11*data[4])/12, last = L==3? data[L-1] - 2*data[L-2] + data[L-3] : L==4? -2*data[L-1] + 5*data[L-2] - 4*data[L-3] + data[L-4] : (35*data[L-1] - 104*data[L-2] + 114*data[L-3] - 56*data[L-4] + 11*data[L-5])/12 ) [ first/h/h, for(i=[1:1:L-2]) (data[i+1]-2*data[i]+data[i-1])/h/h, last/h/h ]; // Function: deriv3() // Synopsis: Returns the third derivative estimate of a list of data. // Topics: Math, Calculus // See Also: deriv(), deriv2(), deriv3() // Usage: // x = deriv3(data, [h], [closed]) // Description: // Computes a numerical third derivative estimate of the data, which may be scalar or vector valued. // The `h` parameter gives the step size of your sampling so the derivative can be scaled correctly. // If the `closed` parameter is true the data is assumed to be defined on a loop with data[0] adjacent to // data[len(data)-1]. This function uses a five point derivative estimate, so the input data must include // at least five points: // f'''(t) = (-f(t-2*h)+2*f(t-h)-2*f(t+h)+f(t+2*h)) / 2h^3. At the first and second points from the end // the estimates are f'''(t) = (-5*f(t)+18*f(t+h)-24*f(t+2*h)+14*f(t+3*h)-3*f(t+4*h)) / 2h^3 and // f'''(t) = (-3*f(t-h)+10*f(t)-12*f(t+h)+6*f(t+2*h)-f(t+3*h)) / 2h^3. // Arguments: // data = the list of the elements to compute the derivative of. // h = the constant parametric sampling of the data. // closed = boolean to indicate if the data set should be wrapped around from the end to the start. function deriv3(data, h=1, closed=false) = assert( is_consistent(data) , "Input list is not consistent or not numerical.") assert( len(data)>=5, "Input list has less than 5 elements.") assert( is_finite(h), "The sampling `h` must be a number." ) let( L = len(data), h3 = h*h*h ) closed? [ for(i=[0:1:L-1]) (-data[(L+i-2)%L]+2*data[(L+i-1)%L]-2*data[(i+1)%L]+data[(i+2)%L])/2/h3 ] : let( first=(-5*data[0]+18*data[1]-24*data[2]+14*data[3]-3*data[4])/2, second=(-3*data[0]+10*data[1]-12*data[2]+6*data[3]-data[4])/2, last=(5*data[L-1]-18*data[L-2]+24*data[L-3]-14*data[L-4]+3*data[L-5])/2, prelast=(3*data[L-1]-10*data[L-2]+12*data[L-3]-6*data[L-4]+data[L-5])/2 ) [ first/h3, second/h3, for(i=[2:1:L-3]) (-data[i-2]+2*data[i-1]-2*data[i+1]+data[i+2])/2/h3, prelast/h3, last/h3 ]; // Section: Complex Numbers // Function: complex() // Synopsis: Replaces scalars in a list or matrix with complex number 2-vectors. // Topics: Math, Complex Numbers // See Also: c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // z = complex(list) // Description: // Converts a real valued number, vector or matrix into its complex analog // by replacing all entries with a 2-vector that has zero imaginary part. function complex(list) = is_num(list) ? [list,0] : [for(entry=list) is_num(entry) ? [entry,0] : complex(entry)]; // Function: c_mul() // Synopsis: Multiplies two complex numbers. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // c = c_mul(z1,z2) // Description: // Multiplies two complex numbers, vectors or matrices, where complex numbers // or entries are represented as vectors: [REAL, IMAGINARY]. Note that all // entries in both arguments must be complex. // Arguments: // z1 = First complex number, vector or matrix // z2 = Second complex number, vector or matrix function c_mul(z1,z2) = is_matrix([z1,z2],2,2) ? _c_mul(z1,z2) : _combine_complex(_c_mul(_split_complex(z1), _split_complex(z2))); function _split_complex(data) = is_vector(data,2) ? data : is_num(data[0][0]) ? [data*[1,0], data*[0,1]] : [ [for(vec=data) vec * [1,0]], [for(vec=data) vec * [0,1]] ]; function _combine_complex(data) = is_vector(data,2) ? data : is_num(data[0][0]) ? [for(i=[0:len(data[0])-1]) [data[0][i],data[1][i]]] : [for(i=[0:1:len(data[0])-1]) [for(j=[0:1:len(data[0][0])-1]) [data[0][i][j], data[1][i][j]]]]; function _c_mul(z1,z2) = [ z1.x*z2.x - z1.y*z2.y, z1.x*z2.y + z1.y*z2.x ]; // Function: c_div() // Synopsis: Divides two complex numbers. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // x = c_div(z1,z2) // Description: // Divides two complex numbers represented by 2D vectors. // Returns a complex number as a 2D vector [REAL, IMAGINARY]. // Arguments: // z1 = First complex number, given as a 2D vector [REAL, IMAGINARY] // z2 = Second complex number, given as a 2D vector [REAL, IMAGINARY] function c_div(z1,z2) = assert( is_vector(z1,2) && is_vector(z2), "Complex numbers should be represented by 2D vectors." ) assert( !approx(z2,0), "The divisor `z2` cannot be zero." ) let(den = z2.x*z2.x + z2.y*z2.y) [(z1.x*z2.x + z1.y*z2.y)/den, (z1.y*z2.x - z1.x*z2.y)/den]; // Function: c_conj() // Synopsis: Returns the complex conjugate of the input. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // w = c_conj(z) // Description: // Computes the complex conjugate of the input, which can be a complex number, // complex vector or complex matrix. function c_conj(z) = is_vector(z,2) ? [z.x,-z.y] : [for(entry=z) c_conj(entry)]; // Function: c_real() // Synopsis: Returns the real part of a complex number, vector or matrix.. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // x = c_real(z) // Description: // Returns real part of a complex number, vector or matrix. function c_real(z) = is_vector(z,2) ? z.x : is_num(z[0][0]) ? z*[1,0] : [for(vec=z) vec * [1,0]]; // Function: c_imag() // Synopsis: Returns the imaginary part of a complex number, vector or matrix.. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // x = c_imag(z) // Description: // Returns imaginary part of a complex number, vector or matrix. function c_imag(z) = is_vector(z,2) ? z.y : is_num(z[0][0]) ? z*[0,1] : [for(vec=z) vec * [0,1]]; // Function: c_ident() // Synopsis: Returns an n by n complex identity matrix. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // I = c_ident(n) // Description: // Produce an n by n complex identity matrix function c_ident(n) = [for (i = [0:1:n-1]) [for (j = [0:1:n-1]) (i==j)?[1,0]:[0,0]]]; // Function: c_norm() // Synopsis: Returns the norm of a complex number or vector. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm() // Usage: // n = c_norm(z) // Description: // Compute the norm of a complex number or vector. function c_norm(z) = norm_fro(z); // Section: Polynomials // Function: quadratic_roots() // Synopsis: Computes roots for the quadratic equation. // Topics: Math, Geometry, Complex Numbers // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add() // Usage: // roots = quadratic_roots(a, b, c, [real]) // Description: // Computes roots of the quadratic equation a*x^2+b*x+c==0, where the // coefficients are real numbers. If real is true then returns only the // real roots. Otherwise returns a pair of complex values. This method // may be more reliable than the general root finder at distinguishing // real roots from complex roots. // Algorithm from: https://people.csail.mit.edu/bkph/articles/Quadratics.pdf function quadratic_roots(a,b,c,real=false) = real ? [for(root = quadratic_roots(a,b,c,real=false)) if (root.y==0) root.x] : is_undef(b) && is_undef(c) && is_vector(a,3) ? quadratic_roots(a[0],a[1],a[2]) : assert(is_num(a) && is_num(b) && is_num(c)) assert(a!=0 || b!=0 || c!=0, "Quadratic must have a nonzero coefficient") a==0 && b==0 ? [] : // No solutions a==0 ? [[-c/b,0]] : let( descrim = b*b-4*a*c, sqrt_des = sqrt(abs(descrim)) ) descrim < 0 ? // Complex case [[-b, sqrt_des], [-b, -sqrt_des]]/2/a : b<0 ? // b positive [[2*c/(-b+sqrt_des),0], [(-b+sqrt_des)/a/2,0]] : // b negative [[(-b-sqrt_des)/2/a, 0], [2*c/(-b-sqrt_des),0]]; // Function: polynomial() // Synopsis: Evaluate a polynomial at a real or complex value. // Topics: Math, Complex Numbers // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots() // Usage: // x = polynomial(p, z) // Description: // Evaluates specified real polynomial, p, at the complex or real input value, z. // The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0] // where a_n is the z^n coefficient. Polynomial coefficients are real. // The result is a number if `z` is a number and a complex number otherwise. function polynomial(p,z,k,total) = is_undef(k) ? assert( is_vector(p) , "Input polynomial coefficients must be a vector." ) assert( is_finite(z) || is_vector(z,2), "The value of `z` must be a real or a complex number." ) polynomial( _poly_trim(p), z, 0, is_num(z) ? 0 : [0,0]) : k==len(p) ? total : polynomial(p,z,k+1, is_num(z) ? total*z+p[k] : c_mul(total,z)+[p[k],0]); // Function: poly_mult() // Synopsis: Compute product of two polynomials, returning a polynomial. // Topics: Math // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots() // Usage: // x = polymult(p,q) // x = polymult([p1,p2,p3,...]) // Description: // Given a list of polynomials represented as real algebraic coefficient lists, with the highest degree coefficient first, // computes the coefficient list of the product polynomial. function poly_mult(p,q) = is_undef(q) ? len(p)==2 ? poly_mult(p[0],p[1]) : poly_mult(p[0], poly_mult(list_tail(p))) : assert( is_vector(p) && is_vector(q),"Invalid arguments to poly_mult") p*p==0 || q*q==0 ? [0] : _poly_trim(convolve(p,q)); // Function: poly_div() // Synopsis: Returns the polynomial quotient and remainder results of dividing two polynomials. // Topics: Math // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots() // Usage: // [quotient,remainder] = poly_div(n,d) // Description: // Computes division of the numerator polynomial by the denominator polynomial and returns // a list of two polynomials, [quotient, remainder]. If the division has no remainder then // the zero polynomial [0] is returned for the remainder. Similarly if the quotient is zero // the returned quotient will be [0]. function poly_div(n,d) = assert( is_vector(n) && is_vector(d) , "Invalid polynomials." ) let( d = _poly_trim(d), n = _poly_trim(n) ) assert( d!=[0] , "Denominator cannot be a zero polynomial." ) n==[0] ? [[0],[0]] : _poly_div(n,d,q=[]); function _poly_div(n,d,q) = len(n)qlen ? p : q, short = plen>qlen ? q : p ) _poly_trim(long + concat(repeat(0,len(long)-len(short)),short)); // Function: poly_roots() // Synopsis: Returns all complex valued roots of the given real polynomial. // Topics: Math, Complex Numbers // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots() // Usage: // roots = poly_roots(p, [tol]); // Description: // Returns all complex roots of the specified real polynomial p. // The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0] // where a_n is the z^n coefficient. The tol parameter gives // the stopping tolerance for the iteration. The polynomial // must have at least one non-zero coefficient. Convergence is poor // if the polynomial has any repeated roots other than zero. // Arguments: // p = polynomial coefficients with higest power coefficient first // tol = tolerance for iteration. Default: 1e-14 // Uses the Aberth method https://en.wikipedia.org/wiki/Aberth_method // // Dario Bini. "Numerical computation of polynomial zeros by means of Aberth's Method", Numerical Algorithms, Feb 1996. // https://www.researchgate.net/publication/225654837_Numerical_computation_of_polynomial_zeros_by_means_of_Aberth's_method function poly_roots(p,tol=1e-14,error_bound=false) = assert( is_vector(p), "Invalid polynomial." ) let( p = _poly_trim(p,eps=0) ) assert( p!=[0], "Input polynomial cannot be zero." ) p[len(p)-1] == 0 ? // Strip trailing zero coefficients let( solutions = poly_roots(list_head(p),tol=tol, error_bound=error_bound)) (error_bound ? [ [[0,0], each solutions[0]], [0, each solutions[1]]] : [[0,0], each solutions]) : len(p)==1 ? (error_bound ? [[],[]] : []) : // Nonzero constant case has no solutions len(p)==2 ? let( solution = [[-p[1]/p[0],0]]) // Linear case needs special handling (error_bound ? [solution,[0]] : solution) : let( n = len(p)-1, // polynomial degree pderiv = [for(i=[0:n-1]) p[i]*(n-i)], s = [for(i=[0:1:n]) abs(p[i])*(4*(n-i)+1)], // Error bound polynomial from Bini // Using method from: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0915-24.pdf beta = -p[1]/p[0]/n, r = 1+pow(abs(polynomial(p,beta)/p[0]),1/n), init = [for(i=[0:1:n-1]) // Initial guess for roots let(angle = 360*i/n+270/n/PI) [beta,0]+r*[cos(angle),sin(angle)] ], roots = _poly_roots(p,pderiv,s,init,tol=tol), error = error_bound ? [for(xi=roots) n * (norm(polynomial(p,xi))+tol*polynomial(s,norm(xi))) / abs(norm(polynomial(pderiv,xi))-tol*polynomial(s,norm(xi)))] : 0 ) error_bound ? [roots, error] : roots; // Internal function // p = polynomial // pderiv = derivative polynomial of p // z = current guess for the roots // tol = root tolerance // i=iteration counter function _poly_roots(p, pderiv, s, z, tol, i=0) = assert(i<45, str("Polyroot exceeded iteration limit. Current solution:", z)) let( n = len(z), svals = [for(zk=z) tol*polynomial(s,norm(zk))], p_of_z = [for(zk=z) polynomial(p,zk)], done = [for(k=[0:n-1]) norm(p_of_z[k])<=svals[k]], newton = [for(k=[0:n-1]) c_div(p_of_z[k], polynomial(pderiv,z[k]))], zdiff = [for(k=[0:n-1]) sum([for(j=[0:n-1]) if (j!=k) c_div([1,0], z[k]-z[j])])], w = [for(k=[0:n-1]) done[k] ? [0,0] : c_div( newton[k], [1,0] - c_mul(newton[k], zdiff[k]))] ) all(done) ? z : _poly_roots(p,pderiv,s,z-w,tol,i+1); // Function: real_roots() // Synopsis: Returns all real roots of the given real polynomial. // Topics: Math, Complex Numbers // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots() // Usage: // roots = real_roots(p, [eps], [tol]) // Description: // Returns the real roots of the specified real polynomial p. // The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0] // where a_n is the x^n coefficient. This function works by // computing the complex roots and returning those roots where // the imaginary part is closed to zero. By default it uses a computed // error bound from the polynomial solver to decide whether imaginary // parts are zero. You can specify eps, in which case the test is // z.y/(1+norm(z)) < eps. Because // of poor convergence and higher error for repeated roots, such roots may // be missed by the algorithm because error can make their imaginary parts // large enough to appear non-zero. // Arguments: // p = polynomial to solve as coefficient list, highest power term first // eps = used to determine whether imaginary parts of roots are zero // tol = tolerance for the complex polynomial root finder function real_roots(p,eps=undef,tol=1e-14) = assert( is_vector(p), "Invalid polynomial." ) let( p = _poly_trim(p,eps=0) ) assert( p!=[0], "Input polynomial cannot be zero." ) let( roots_err = poly_roots(p,error_bound=true), roots = roots_err[0], err = roots_err[1] ) is_def(eps) ? [for(z=roots) if (abs(z.y)/(1+norm(z))