1 /* 2 Copyright 2008-2023 3 Matthias Ehmann, 4 Michael Gerhaeuser, 5 Carsten Miller, 6 Bianca Valentin, 7 Alfred Wassermann, 8 Peter Wilfahrt 9 10 This file is part of JSXGraph. 11 12 JSXGraph is free software dual licensed under the GNU LGPL or MIT License. 13 14 You can redistribute it and/or modify it under the terms of the 15 16 * GNU Lesser General Public License as published by 17 the Free Software Foundation, either version 3 of the License, or 18 (at your option) any later version 19 OR 20 * MIT License: https://github.com/jsxgraph/jsxgraph/blob/master/LICENSE.MIT 21 22 JSXGraph is distributed in the hope that it will be useful, 23 but WITHOUT ANY WARRANTY; without even the implied warranty of 24 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 25 GNU Lesser General Public License for more details. 26 27 You should have received a copy of the GNU Lesser General Public License and 28 the MIT License along with JSXGraph. If not, see <https://www.gnu.org/licenses/> 29 and <https://opensource.org/licenses/MIT/>. 30 */ 31 32 /*global JXG: true, define: true, Float32Array: true */ 33 /*jslint nomen: true, plusplus: true, bitwise: true*/ 34 35 /** 36 * @fileoverview In this file the namespace JXG.Math is defined, which is the base namespace 37 * for namespaces like JXG.Math.Numerics, JXG.Math.Plot, JXG.Math.Statistics, JXG.Math.Clip etc. 38 */ 39 import JXG from "../jxg"; 40 import Type from "../utils/type"; 41 42 var undef, 43 /* 44 * Dynamic programming approach for recursive functions. 45 * From "Speed up your JavaScript, Part 3" by Nicholas C. Zakas. 46 * @see JXG.Math.factorial 47 * @see JXG.Math.binomial 48 * http://blog.thejit.org/2008/09/05/memoization-in-javascript/ 49 * 50 * This method is hidden, because it is only used in JXG.Math. If someone wants 51 * to use it in JSXGraph outside of JXG.Math, it should be moved to jsxgraph.js 52 */ 53 memoizer = function (f) { 54 var cache, join; 55 56 if (f.memo) { 57 return f.memo; 58 } 59 60 cache = {}; 61 join = Array.prototype.join; 62 63 f.memo = function () { 64 var key = join.call(arguments); 65 66 // Seems to be a bit faster than "if (a in b)" 67 return cache[key] !== undef ? cache[key] : (cache[key] = f.apply(this, arguments)); 68 }; 69 70 return f.memo; 71 }; 72 73 /** 74 * Math namespace. 75 * @namespace 76 */ 77 JXG.Math = { 78 /** 79 * eps defines the closeness to zero. If the absolute value of a given number is smaller 80 * than eps, it is considered to be equal to zero. 81 * @type Number 82 */ 83 eps: 0.000001, 84 85 /** 86 * Determine the relative difference between two numbers. 87 * @param {Number} a First number 88 * @param {Number} b Second number 89 * @returns {Number} Relative difference between a and b: |a-b| / max(|a|, |b|) 90 */ 91 relDif: function (a, b) { 92 var c = Math.abs(a), 93 d = Math.abs(b); 94 95 d = Math.max(c, d); 96 97 return d === 0.0 ? 0.0 : Math.abs(a - b) / d; 98 }, 99 100 /** 101 * The JavaScript implementation of the % operator returns the symmetric modulo. 102 * mod and "%" are both identical if a >= 0 and m >= 0 but the results differ if a or m < 0. 103 * @param {Number} a 104 * @param {Number} m 105 * @returns {Number} Mathematical modulo <tt>a mod m</tt> 106 */ 107 mod: function (a, m) { 108 return a - Math.floor(a / m) * m; 109 }, 110 111 /** 112 * Initializes a vector as an array with the coefficients set to the given value resp. zero. 113 * @param {Number} n Length of the vector 114 * @param {Number} [init=0] Initial value for each coefficient 115 * @returns {Array} A <tt>n</tt> times <tt>m</tt>-matrix represented by a 116 * two-dimensional array. The inner arrays hold the columns, the outer array holds the rows. 117 */ 118 vector: function (n, init) { 119 var r, i; 120 121 init = init || 0; 122 r = []; 123 124 for (i = 0; i < n; i++) { 125 r[i] = init; 126 } 127 128 return r; 129 }, 130 131 /** 132 * Initializes a matrix as an array of rows with the given value. 133 * @param {Number} n Number of rows 134 * @param {Number} [m=n] Number of columns 135 * @param {Number} [init=0] Initial value for each coefficient 136 * @returns {Array} A <tt>n</tt> times <tt>m</tt>-matrix represented by a 137 * two-dimensional array. The inner arrays hold the columns, the outer array holds the rows. 138 */ 139 matrix: function (n, m, init) { 140 var r, i, j; 141 142 init = init || 0; 143 m = m || n; 144 r = []; 145 146 for (i = 0; i < n; i++) { 147 r[i] = []; 148 149 for (j = 0; j < m; j++) { 150 r[i][j] = init; 151 } 152 } 153 154 return r; 155 }, 156 157 /** 158 * Generates an identity matrix. If n is a number and m is undefined or not a number, a square matrix is generated, 159 * if n and m are both numbers, an nxm matrix is generated. 160 * @param {Number} n Number of rows 161 * @param {Number} [m=n] Number of columns 162 * @returns {Array} A square matrix of length <tt>n</tt> with all coefficients equal to 0 except a_(i,i), i out of (1, ..., n), if <tt>m</tt> is undefined or not a number 163 * or a <tt>n</tt> times <tt>m</tt>-matrix with a_(i,j) = 0 and a_(i,i) = 1 if m is a number. 164 */ 165 identity: function (n, m) { 166 var r, i; 167 168 if (m === undef && typeof m !== "number") { 169 m = n; 170 } 171 172 r = this.matrix(n, m); 173 174 for (i = 0; i < Math.min(n, m); i++) { 175 r[i][i] = 1; 176 } 177 178 return r; 179 }, 180 181 /** 182 * Generates a 4x4 matrix for 3D to 2D projections. 183 * @param {Number} l Left 184 * @param {Number} r Right 185 * @param {Number} t Top 186 * @param {Number} b Bottom 187 * @param {Number} n Near 188 * @param {Number} f Far 189 * @returns {Array} 4x4 Matrix 190 */ 191 frustum: function (l, r, b, t, n, f) { 192 var ret = this.matrix(4, 4); 193 194 ret[0][0] = (n * 2) / (r - l); 195 ret[0][1] = 0; 196 ret[0][2] = (r + l) / (r - l); 197 ret[0][3] = 0; 198 199 ret[1][0] = 0; 200 ret[1][1] = (n * 2) / (t - b); 201 ret[1][2] = (t + b) / (t - b); 202 ret[1][3] = 0; 203 204 ret[2][0] = 0; 205 ret[2][1] = 0; 206 ret[2][2] = -(f + n) / (f - n); 207 ret[2][3] = -(f * n * 2) / (f - n); 208 209 ret[3][0] = 0; 210 ret[3][1] = 0; 211 ret[3][2] = -1; 212 ret[3][3] = 0; 213 214 return ret; 215 }, 216 217 /** 218 * Generates a 4x4 matrix for 3D to 2D projections. 219 * @param {Number} fov Field of view in vertical direction, given in rad. 220 * @param {Number} ratio Aspect ratio of the projection plane. 221 * @param {Number} n Near 222 * @param {Number} f Far 223 * @returns {Array} 4x4 Projection Matrix 224 */ 225 projection: function (fov, ratio, n, f) { 226 var t = n * Math.tan(fov / 2), 227 r = t * ratio; 228 229 return this.frustum(-r, r, -t, t, n, f); 230 }, 231 232 /** 233 * Multiplies a vector vec to a matrix mat: mat * vec. The matrix is interpreted by this function as an array of rows. Please note: This 234 * function does not check if the dimensions match. 235 * @param {Array} mat Two dimensional array of numbers. The inner arrays describe the columns, the outer ones the matrix' rows. 236 * @param {Array} vec Array of numbers 237 * @returns {Array} Array of numbers containing the result 238 * @example 239 * var A = [[2, 1], 240 * [1, 3]], 241 * b = [4, 5], 242 * c; 243 * c = JXG.Math.matVecMult(A, b) 244 * // c === [13, 19]; 245 */ 246 matVecMult: function (mat, vec) { 247 var i, 248 s, 249 k, 250 m = mat.length, 251 n = vec.length, 252 res = []; 253 254 if (n === 3) { 255 for (i = 0; i < m; i++) { 256 res[i] = mat[i][0] * vec[0] + mat[i][1] * vec[1] + mat[i][2] * vec[2]; 257 } 258 } else { 259 for (i = 0; i < m; i++) { 260 s = 0; 261 for (k = 0; k < n; k++) { 262 s += mat[i][k] * vec[k]; 263 } 264 res[i] = s; 265 } 266 } 267 return res; 268 }, 269 270 /** 271 * Computes the product of the two matrices mat1*mat2. 272 * @param {Array} mat1 Two dimensional array of numbers 273 * @param {Array} mat2 Two dimensional array of numbers 274 * @returns {Array} Two dimensional Array of numbers containing result 275 */ 276 matMatMult: function (mat1, mat2) { 277 var i, 278 j, 279 s, 280 k, 281 m = mat1.length, 282 n = m > 0 ? mat2[0].length : 0, 283 m2 = mat2.length, 284 res = this.matrix(m, n); 285 286 for (i = 0; i < m; i++) { 287 for (j = 0; j < n; j++) { 288 s = 0; 289 for (k = 0; k < m2; k++) { 290 s += mat1[i][k] * mat2[k][j]; 291 } 292 res[i][j] = s; 293 } 294 } 295 return res; 296 }, 297 298 /** 299 * Transposes a matrix given as a two dimensional array. 300 * @param {Array} M The matrix to be transposed 301 * @returns {Array} The transpose of M 302 */ 303 transpose: function (M) { 304 var MT, i, j, m, n; 305 306 // number of rows of M 307 m = M.length; 308 // number of columns of M 309 n = M.length > 0 ? M[0].length : 0; 310 MT = this.matrix(n, m); 311 312 for (i = 0; i < n; i++) { 313 for (j = 0; j < m; j++) { 314 MT[i][j] = M[j][i]; 315 } 316 } 317 318 return MT; 319 }, 320 321 /** 322 * Compute the inverse of an nxn matrix with Gauss elimination. 323 * @param {Array} Ain 324 * @returns {Array} Inverse matrix of Ain 325 */ 326 inverse: function (Ain) { 327 var i, 328 j, 329 k, 330 s, 331 ma, 332 r, 333 swp, 334 n = Ain.length, 335 A = [], 336 p = [], 337 hv = []; 338 339 for (i = 0; i < n; i++) { 340 A[i] = []; 341 for (j = 0; j < n; j++) { 342 A[i][j] = Ain[i][j]; 343 } 344 p[i] = i; 345 } 346 347 for (j = 0; j < n; j++) { 348 // pivot search: 349 ma = Math.abs(A[j][j]); 350 r = j; 351 352 for (i = j + 1; i < n; i++) { 353 if (Math.abs(A[i][j]) > ma) { 354 ma = Math.abs(A[i][j]); 355 r = i; 356 } 357 } 358 359 // Singular matrix 360 if (ma <= this.eps) { 361 return []; 362 } 363 364 // swap rows: 365 if (r > j) { 366 for (k = 0; k < n; k++) { 367 swp = A[j][k]; 368 A[j][k] = A[r][k]; 369 A[r][k] = swp; 370 } 371 372 swp = p[j]; 373 p[j] = p[r]; 374 p[r] = swp; 375 } 376 377 // transformation: 378 s = 1.0 / A[j][j]; 379 for (i = 0; i < n; i++) { 380 A[i][j] *= s; 381 } 382 A[j][j] = s; 383 384 for (k = 0; k < n; k++) { 385 if (k !== j) { 386 for (i = 0; i < n; i++) { 387 if (i !== j) { 388 A[i][k] -= A[i][j] * A[j][k]; 389 } 390 } 391 A[j][k] = -s * A[j][k]; 392 } 393 } 394 } 395 396 // swap columns: 397 for (i = 0; i < n; i++) { 398 for (k = 0; k < n; k++) { 399 hv[p[k]] = A[i][k]; 400 } 401 for (k = 0; k < n; k++) { 402 A[i][k] = hv[k]; 403 } 404 } 405 406 return A; 407 }, 408 409 /** 410 * Inner product of two vectors a and b. n is the length of the vectors. 411 * @param {Array} a Vector 412 * @param {Array} b Vector 413 * @param {Number} [n] Length of the Vectors. If not given the length of the first vector is taken. 414 * @returns {Number} The inner product of a and b. 415 */ 416 innerProduct: function (a, b, n) { 417 var i, 418 s = 0; 419 420 if (n === undef || !Type.isNumber(n)) { 421 n = a.length; 422 } 423 424 for (i = 0; i < n; i++) { 425 s += a[i] * b[i]; 426 } 427 428 return s; 429 }, 430 431 /** 432 * Calculates the cross product of two vectors both of length three. 433 * In case of homogeneous coordinates this is either 434 * <ul> 435 * <li>the intersection of two lines</li> 436 * <li>the line through two points</li> 437 * </ul> 438 * @param {Array} c1 Homogeneous coordinates of line or point 1 439 * @param {Array} c2 Homogeneous coordinates of line or point 2 440 * @returns {Array} vector of length 3: homogeneous coordinates of the resulting point / line. 441 */ 442 crossProduct: function (c1, c2) { 443 return [ 444 c1[1] * c2[2] - c1[2] * c2[1], 445 c1[2] * c2[0] - c1[0] * c2[2], 446 c1[0] * c2[1] - c1[1] * c2[0] 447 ]; 448 }, 449 450 /** 451 * Euclidean norm of a vector. 452 * 453 * @param {Array} a Array containing a vector. 454 * @param {Number} n (Optional) length of the array. 455 * @returns {Number} Euclidean norm of the vector. 456 */ 457 norm: function (a, n) { 458 var i, 459 sum = 0.0; 460 461 if (n === undef || !Type.isNumber(n)) { 462 n = a.length; 463 } 464 465 for (i = 0; i < n; i++) { 466 sum += a[i] * a[i]; 467 } 468 469 return Math.sqrt(sum); 470 }, 471 472 axpy: function (a, x, y) { 473 var i, 474 le = x.length, 475 p = []; 476 for (i = 0; i < le; i++) { 477 p[i] = a * x[i] + y[i]; 478 } 479 return p; 480 }, 481 482 /** 483 * Compute the factorial of a positive integer. If a non-integer value 484 * is given, the fraction will be ignored. 485 * @function 486 * @param {Number} n 487 * @returns {Number} n! = n*(n-1)*...*2*1 488 */ 489 factorial: memoizer(function (n) { 490 if (n < 0) { 491 return NaN; 492 } 493 494 n = Math.floor(n); 495 496 if (n === 0 || n === 1) { 497 return 1; 498 } 499 500 return n * this.factorial(n - 1); 501 }), 502 503 /** 504 * Computes the binomial coefficient n over k. 505 * @function 506 * @param {Number} n Fraction will be ignored 507 * @param {Number} k Fraction will be ignored 508 * @returns {Number} The binomial coefficient n over k 509 */ 510 binomial: memoizer(function (n, k) { 511 var b, i; 512 513 if (k > n || k < 0) { 514 return NaN; 515 } 516 517 k = Math.round(k); 518 n = Math.round(n); 519 520 if (k === 0 || k === n) { 521 return 1; 522 } 523 524 b = 1; 525 526 for (i = 0; i < k; i++) { 527 b *= n - i; 528 b /= i + 1; 529 } 530 531 return b; 532 }), 533 534 /** 535 * Calculates the cosine hyperbolicus of x. 536 * @function 537 * @param {Number} x The number the cosine hyperbolicus will be calculated of. 538 * @returns {Number} Cosine hyperbolicus of the given value. 539 */ 540 cosh: 541 Math.cosh || 542 function (x) { 543 return (Math.exp(x) + Math.exp(-x)) * 0.5; 544 }, 545 546 /** 547 * Sine hyperbolicus of x. 548 * @function 549 * @param {Number} x The number the sine hyperbolicus will be calculated of. 550 * @returns {Number} Sine hyperbolicus of the given value. 551 */ 552 sinh: 553 Math.sinh || 554 function (x) { 555 return (Math.exp(x) - Math.exp(-x)) * 0.5; 556 }, 557 558 /** 559 * Hyperbolic arc-cosine of a number. 560 * 561 * @param {Number} x 562 * @returns {Number} 563 */ 564 acosh: 565 Math.acosh || 566 function (x) { 567 return Math.log(x + Math.sqrt(x * x - 1)); 568 }, 569 570 /** 571 * Hyperbolic arcsine of a number 572 * @param {Number} x 573 * @returns {Number} 574 */ 575 asinh: 576 Math.asinh || 577 function (x) { 578 if (x === -Infinity) { 579 return x; 580 } 581 return Math.log(x + Math.sqrt(x * x + 1)); 582 }, 583 584 /** 585 * Computes the cotangent of x. 586 * @function 587 * @param {Number} x The number the cotangent will be calculated of. 588 * @returns {Number} Cotangent of the given value. 589 */ 590 cot: function (x) { 591 return 1 / Math.tan(x); 592 }, 593 594 /** 595 * Computes the inverse cotangent of x. 596 * @param {Number} x The number the inverse cotangent will be calculated of. 597 * @returns {Number} Inverse cotangent of the given value. 598 */ 599 acot: function (x) { 600 return (x >= 0 ? 0.5 : -0.5) * Math.PI - Math.atan(x); 601 }, 602 603 /** 604 * Compute n-th real root of a real number. n must be strictly positive integer. 605 * If n is odd, the real n-th root exists and is negative. 606 * For n even, for negative valuees of x NaN is returned 607 * @param {Number} x radicand. Must be non-negative, if n even. 608 * @param {Number} n index of the root. must be strictly positive integer. 609 * @returns {Number} returns real root or NaN 610 * 611 * @example 612 * nthroot(16, 4): 2 613 * nthroot(-27, 3): -3 614 * nthroot(-4, 2): NaN 615 */ 616 nthroot: function (x, n) { 617 var inv = 1 / n; 618 619 if (n <= 0 || Math.floor(n) !== n) { 620 return NaN; 621 } 622 623 if (x === 0.0) { 624 return 0.0; 625 } 626 627 if (x > 0) { 628 return Math.exp(inv * Math.log(x)); 629 } 630 631 // From here on, x is negative 632 if (n % 2 === 1) { 633 return -Math.exp(inv * Math.log(-x)); 634 } 635 636 // x negative, even root 637 return NaN; 638 }, 639 640 /** 641 * Computes cube root of real number 642 * Polyfill for Math.cbrt(). 643 * 644 * @function 645 * @param {Number} x Radicand 646 * @returns {Number} Cube root of x. 647 */ 648 cbrt: 649 Math.cbrt || 650 function (x) { 651 return this.nthroot(x, 3); 652 }, 653 654 /** 655 * Compute base to the power of exponent. 656 * @param {Number} base 657 * @param {Number} exponent 658 * @returns {Number} base to the power of exponent. 659 */ 660 pow: function (base, exponent) { 661 if (base === 0) { 662 if (exponent === 0) { 663 return 1; 664 } 665 return 0; 666 } 667 668 // exponent is an integer 669 if (Math.floor(exponent) === exponent) { 670 return Math.pow(base, exponent); 671 } 672 673 // exponent is not an integer 674 if (base > 0) { 675 return Math.exp(exponent * Math.log(base)); 676 } 677 678 return NaN; 679 }, 680 681 /** 682 * Compute base to the power of the rational exponent m / n. 683 * This function first reduces the fraction m/n and then computes 684 * JXG.Math.pow(base, m/n). 685 * 686 * This function is necessary to have the same results for e.g. 687 * (-8)^(1/3) = (-8)^(2/6) = -2 688 * @param {Number} base 689 * @param {Number} m numerator of exponent 690 * @param {Number} n denominator of exponent 691 * @returns {Number} base to the power of exponent. 692 */ 693 ratpow: function (base, m, n) { 694 var g; 695 if (m === 0) { 696 return 1; 697 } 698 if (n === 0) { 699 return NaN; 700 } 701 702 g = this.gcd(m, n); 703 return this.nthroot(this.pow(base, m / g), n / g); 704 }, 705 706 /** 707 * Logarithm to base 10. 708 * @param {Number} x 709 * @returns {Number} log10(x) Logarithm of x to base 10. 710 */ 711 log10: function (x) { 712 return Math.log(x) / Math.log(10.0); 713 }, 714 715 /** 716 * Logarithm to base 2. 717 * @param {Number} x 718 * @returns {Number} log2(x) Logarithm of x to base 2. 719 */ 720 log2: function (x) { 721 return Math.log(x) / Math.log(2.0); 722 }, 723 724 /** 725 * Logarithm to arbitrary base b. If b is not given, natural log is taken, i.e. b = e. 726 * @param {Number} x 727 * @param {Number} b base 728 * @returns {Number} log(x, b) Logarithm of x to base b, that is log(x)/log(b). 729 */ 730 log: function (x, b) { 731 if (b !== undefined && Type.isNumber(b)) { 732 return Math.log(x) / Math.log(b); 733 } 734 735 return Math.log(x); 736 }, 737 738 /** 739 * The sign() function returns the sign of a number, indicating whether the number is positive, negative or zero. 740 * 741 * @function 742 * @param {Number} x A Number 743 * @returns {[type]} This function has 5 kinds of return values, 744 * 1, -1, 0, -0, NaN, which represent "positive number", "negative number", "positive zero", "negative zero" 745 * and NaN respectively. 746 */ 747 sign: 748 Math.sign || 749 function (x) { 750 x = +x; // convert to a number 751 if (x === 0 || isNaN(x)) { 752 return x; 753 } 754 return x > 0 ? 1 : -1; 755 }, 756 757 /** 758 * A square & multiply algorithm to compute base to the power of exponent. 759 * Implementated by Wolfgang Riedl. 760 * 761 * @param {Number} base 762 * @param {Number} exponent 763 * @returns {Number} Base to the power of exponent 764 */ 765 squampow: function (base, exponent) { 766 var result; 767 768 if (Math.floor(exponent) === exponent) { 769 // exponent is integer (could be zero) 770 result = 1; 771 772 if (exponent < 0) { 773 // invert: base 774 base = 1.0 / base; 775 exponent *= -1; 776 } 777 778 while (exponent !== 0) { 779 if (exponent & 1) { 780 result *= base; 781 } 782 783 exponent >>= 1; 784 base *= base; 785 } 786 return result; 787 } 788 789 return this.pow(base, exponent); 790 }, 791 792 /** 793 * Greatest common divisor (gcd) of two numbers. 794 * @see <a href="https://rosettacode.org/wiki/Greatest_common_divisor#JavaScript">rosettacode.org</a> 795 * 796 * @param {Number} a First number 797 * @param {Number} b Second number 798 * @returns {Number} gcd(a, b) if a and b are numbers, NaN else. 799 */ 800 gcd: function (a, b) { 801 var tmp, 802 endless = true; 803 804 a = Math.abs(a); 805 b = Math.abs(b); 806 807 if (!(Type.isNumber(a) && Type.isNumber(b))) { 808 return NaN; 809 } 810 if (b > a) { 811 tmp = a; 812 a = b; 813 b = tmp; 814 } 815 816 while (endless) { 817 a %= b; 818 if (a === 0) { 819 return b; 820 } 821 b %= a; 822 if (b === 0) { 823 return a; 824 } 825 } 826 }, 827 828 /** 829 * Least common multiple (lcm) of two numbers. 830 * 831 * @param {Number} a First number 832 * @param {Number} b Second number 833 * @returns {Number} lcm(a, b) if a and b are numbers, NaN else. 834 */ 835 lcm: function (a, b) { 836 var ret; 837 838 if (!(Type.isNumber(a) && Type.isNumber(b))) { 839 return NaN; 840 } 841 842 ret = a * b; 843 if (ret !== 0) { 844 return ret / this.gcd(a, b); 845 } 846 847 return 0; 848 }, 849 850 /** 851 * Error function, see {@link https://en.wikipedia.org/wiki/Error_function}. 852 * 853 * @see JXG.Math.PropFunc.erf 854 * @param {Number} x 855 * @returns {Number} 856 */ 857 erf: function (x) { 858 return this.ProbFuncs.erf(x); 859 }, 860 861 /** 862 * Complementary error function, i.e. 1 - erf(x). 863 * 864 * @see JXG.Math.erf 865 * @see JXG.Math.PropFunc.erfc 866 * @param {Number} x 867 * @returns {Number} 868 */ 869 erfc: function (x) { 870 return this.ProbFuncs.erfc(x); 871 }, 872 873 /** 874 * Inverse of error function 875 * 876 * @see JXG.Math.erf 877 * @see JXG.Math.PropFunc.erfi 878 * @param {Number} x 879 * @returns {Number} 880 */ 881 erfi: function (x) { 882 return this.ProbFuncs.erfi(x); 883 }, 884 885 /** 886 * Normal distribution function 887 * 888 * @see JXG.Math.PropFunc.ndtr 889 * @param {Number} x 890 * @returns {Number} 891 */ 892 ndtr: function (x) { 893 return this.ProbFuncs.ndtr(x); 894 }, 895 896 /** 897 * Inverse of normal distribution function 898 * 899 * @see JXG.Math.ndtr 900 * @see JXG.Math.PropFunc.ndtri 901 * @param {Number} x 902 * @returns {Number} 903 */ 904 ndtri: function (x) { 905 return this.ProbFuncs.ndtri(x); 906 }, 907 908 /* ******************** Comparisons and logical operators ************** */ 909 910 /** 911 * Logical test: a < b? 912 * 913 * @param {Number} a 914 * @param {Number} b 915 * @returns {Boolean} 916 */ 917 lt: function (a, b) { 918 return a < b; 919 }, 920 921 /** 922 * Logical test: a <= b? 923 * 924 * @param {Number} a 925 * @param {Number} b 926 * @returns {Boolean} 927 */ 928 leq: function (a, b) { 929 return a <= b; 930 }, 931 932 /** 933 * Logical test: a > b? 934 * 935 * @param {Number} a 936 * @param {Number} b 937 * @returns {Boolean} 938 */ 939 gt: function (a, b) { 940 return a > b; 941 }, 942 943 /** 944 * Logical test: a >= b? 945 * 946 * @param {Number} a 947 * @param {Number} b 948 * @returns {Boolean} 949 */ 950 geq: function (a, b) { 951 return a >= b; 952 }, 953 954 /** 955 * Logical test: a === b? 956 * 957 * @param {Number} a 958 * @param {Number} b 959 * @returns {Boolean} 960 */ 961 eq: function (a, b) { 962 return a === b; 963 }, 964 965 /** 966 * Logical test: a !== b? 967 * 968 * @param {Number} a 969 * @param {Number} b 970 * @returns {Boolean} 971 */ 972 neq: function (a, b) { 973 return a !== b; 974 }, 975 976 /** 977 * Logical operator: a && b? 978 * 979 * @param {Boolean} a 980 * @param {Boolean} b 981 * @returns {Boolean} 982 */ 983 and: function (a, b) { 984 return a && b; 985 }, 986 987 /** 988 * Logical operator: !a? 989 * 990 * @param {Boolean} a 991 * @returns {Boolean} 992 */ 993 not: function (a) { 994 return !a; 995 }, 996 997 /** 998 * Logical operator: a || b? 999 * 1000 * @param {Boolean} a 1001 * @param {Boolean} b 1002 * @returns {Boolean} 1003 */ 1004 or: function (a, b) { 1005 return a || b; 1006 }, 1007 1008 /** 1009 * Logical operator: either a or b? 1010 * 1011 * @param {Boolean} a 1012 * @param {Boolean} b 1013 * @returns {Boolean} 1014 */ 1015 xor: function (a, b) { 1016 return (a || b) && !(a && b); 1017 }, 1018 1019 /* *************************** Normalize *************************** */ 1020 1021 /** 1022 * Normalize the standard form [c, b0, b1, a, k, r, q0, q1]. 1023 * @private 1024 * @param {Array} stdform The standard form to be normalized. 1025 * @returns {Array} The normalized standard form. 1026 */ 1027 normalize: function (stdform) { 1028 var n, 1029 signr, 1030 a2 = 2 * stdform[3], 1031 r = stdform[4] / a2; 1032 1033 stdform[5] = r; 1034 stdform[6] = -stdform[1] / a2; 1035 stdform[7] = -stdform[2] / a2; 1036 1037 if (!isFinite(r)) { 1038 n = Math.sqrt(stdform[1] * stdform[1] + stdform[2] * stdform[2]); 1039 1040 stdform[0] /= n; 1041 stdform[1] /= n; 1042 stdform[2] /= n; 1043 stdform[3] = 0; 1044 stdform[4] = 1; 1045 } else if (Math.abs(r) >= 1) { 1046 stdform[0] = (stdform[6] * stdform[6] + stdform[7] * stdform[7] - r * r) / (2 * r); 1047 stdform[1] = -stdform[6] / r; 1048 stdform[2] = -stdform[7] / r; 1049 stdform[3] = 1 / (2 * r); 1050 stdform[4] = 1; 1051 } else { 1052 signr = r <= 0 ? -1 : 1; 1053 stdform[0] = 1054 signr * (stdform[6] * stdform[6] + stdform[7] * stdform[7] - r * r) * 0.5; 1055 stdform[1] = -signr * stdform[6]; 1056 stdform[2] = -signr * stdform[7]; 1057 stdform[3] = signr / 2; 1058 stdform[4] = signr * r; 1059 } 1060 1061 return stdform; 1062 }, 1063 1064 /** 1065 * Converts a two dimensional array to a one dimensional Float32Array that can be processed by WebGL. 1066 * @param {Array} m A matrix in a two dimensional array. 1067 * @returns {Float32Array} A one dimensional array containing the matrix in column wise notation. Provides a fall 1068 * back to the default JavaScript Array if Float32Array is not available. 1069 */ 1070 toGL: function (m) { 1071 var v, i, j; 1072 1073 if (typeof Float32Array === "function") { 1074 v = new Float32Array(16); 1075 } else { 1076 v = new Array(16); 1077 } 1078 1079 if (m.length !== 4 && m[0].length !== 4) { 1080 return v; 1081 } 1082 1083 for (i = 0; i < 4; i++) { 1084 for (j = 0; j < 4; j++) { 1085 v[i + 4 * j] = m[i][j]; 1086 } 1087 } 1088 1089 return v; 1090 }, 1091 1092 /** 1093 * Theorem of Vieta: Given a set of simple zeroes x_0, ..., x_n 1094 * of a polynomial f, compute the coefficients s_k, (k=0,...,n-1) 1095 * of the polynomial of the form. See {@link https://de.wikipedia.org/wiki/Elementarsymmetrisches_Polynom}. 1096 * <p> 1097 * f(x) = (x-x_0)*...*(x-x_n) = 1098 * x^n + sum_{k=1}^{n} (-1)^(k) s_{k-1} x^(n-k) 1099 * </p> 1100 * @param {Array} x Simple zeroes of the polynomial. 1101 * @returns {Array} Coefficients of the polynomial. 1102 * 1103 */ 1104 Vieta: function (x) { 1105 var n = x.length, 1106 s = [], 1107 m, 1108 k, 1109 y; 1110 1111 s = x.slice(); 1112 for (m = 1; m < n; ++m) { 1113 y = s[m]; 1114 s[m] *= s[m - 1]; 1115 for (k = m - 1; k >= 1; --k) { 1116 s[k] += s[k - 1] * y; 1117 } 1118 s[0] += y; 1119 } 1120 return s; 1121 } 1122 }; 1123 1124 export default JXG.Math; 1125