Nearly equal with relative and absolute tolerance

docs/core/configuration.md

@@ -8,7 +8,8 @@ import { create, all } from 'mathjs'
 
 // create a mathjs instance with configuration
 const config = {
-  epsilon: 1e-12,
+  relTol: 1e-12,
+  absTol: 1e-15,
   matrix: 'Matrix',
   number: 'number',
   precision: 64,
@@ -28,10 +29,14 @@ math.config({
 
 The following configuration options are available:
 
-- `epsilon`. The minimum relative difference used to test equality between two
+- `relTol`. The minimum relative difference used to test equality between two
   compared values. This value is used by all relational functions.
   Default value is `1e-12`.
 
+- `absTol`. The minimum absolute difference used to test equality between two
+  compared values. This value is used by all relational functions.
+  Default value is `1e-15`.
+
 - `matrix`. The default type of matrix output for functions.
   Available values are: `'Matrix'` (default) or `'Array'`.
   Where possible, the type of matrix output from functions is determined from

docs/datatypes/bignumbers.md

@@ -24,7 +24,8 @@ math.config({
   number: 'BigNumber',      // Default type of number:
                             // 'number' (default), 'BigNumber', or 'Fraction'
   precision: 64,            // Number of significant digits for BigNumbers
-  epsilon: 1e-60
+  relTol: 1e-60,
+  absTol: 1e-63
 })
 
 // use math
@@ -34,12 +35,12 @@ math.evaluate('0.1 + 0.2')  // BigNumber, 0.3
 The default precision for BigNumber is 64 digits, and can be configured with
 the option `precision`. 
 
-Note that we also change the configuration of `epsilon` 
-to be close to the precision limit of our BigNumbers. `epsilon` is used for 
+Note that we also change the configuration of `relTol` and `absTol` 
+to be close to the precision limit of our BigNumbers. `relTol` and `absTol` are used for 
 example in relational and rounding functions (`equal`, `larger`, `smaller`, 
 `round`, `floor`, etc) to determine when a value is nearly equal, 
-see [Equality](numbers.md#equality). If we would leave `epsilon` unchanged, 
-having the default value of `1e-12`, we could get inaccurate and misleading 
+see [Equality](numbers.md#equality). If we would leave `relTol` and `absTol`  unchanged, 
+having the default value of `1e-12` and `1e-15` respectively, we could get inaccurate and misleading 
 results since we're now working with a higher precision.
 
 

docs/datatypes/numbers.md

@@ -73,16 +73,14 @@ false, as the addition `0.1 + 0.2` introduces a round-off error and does not
 return exactly `0.3`.
 
 To solve this problem, the relational functions of math.js check whether the
-relative difference between the compared values is smaller than the configured
-option `epsilon`. In pseudo code (without exceptions for 0, Infinity and NaN):
+relative and absolute differences between the compared values is smaller than the configured
+option `relTol` and `absTol`. In pseudo code (without exceptions for 0, Infinity and NaN):
 
-    diff = abs(x - y)
-    nearlyEqual = (diff <= max(abs(x), abs(y)) * EPSILON) OR (diff < DBL_EPSILON)
+    abs(a-b) <= max(relTol * max(abs(a), abs(b)), absTol)
 
 where:
 
- - `EPSILON` is the relative difference between x and y. Epsilon is configurable
-   and is `1e-12` by default. See [Configuration](../core/configuration.md).
+ - `relTol` is the relative tolerance between x and y and `absTol` the absolute tolerance. Relative tolerance and absolute tolerance are configurable and are `1e-12` and `1e-15` respectively by default. See [Configuration](../core/configuration.md).
  - `DBL_EPSILON` is the minimum positive floating point number such that
    `1.0 + DBL_EPSILON !== 1.0`. This is a constant with a value of approximately
    `2.2204460492503130808472633361816e-16`.

src/core/config.js

@@ -1,7 +1,11 @@
 export const DEFAULT_CONFIG = {
   // minimum relative difference between two compared values,
   // used by all comparison functions
-  epsilon: 1e-12,
+  relTol: 1e-12,
+
+  // minimum absolute difference between two compared values,
+  // used by all comparison functions
+  absTol: 1e-15,
 
   // type of default matrix output. Choose 'matrix' (default) or 'array'
   matrix: 'Matrix',

src/core/create.js

@@ -63,9 +63,12 @@ import { DEFAULT_CONFIG } from './config.js'
  *                             The object can contain nested objects,
  *                             all nested objects will be flattened.
  * @param {Object} [config]    Available options:
- *                            {number} epsilon
+ *                            {number} relTol
  *                              Minimum relative difference between two
  *                              compared values, used by all comparison functions.
+ *                            {number} absTol
+ *                              Minimum absolute difference between two
+ *                              compared values, used by all comparison functions.
  *                            {string} matrix
  *                              A string 'Matrix' (default) or 'Array'.
  *                            {string} number

src/core/function/config.js

@@ -1,4 +1,4 @@
-import { clone, mapObject, deepExtend } from '../../utils/object.js'
+import { clone, deepExtend } from '../../utils/object.js'
 import { DEFAULT_CONFIG } from '../config.js'
 
 export const MATRIX_OPTIONS = ['Matrix', 'Array'] // valid values for option matrix
@@ -29,9 +29,12 @@ export function configFactory (config, emit) {
    *     math.evaluate('0.4')                // outputs Fraction 2/5
    *
    * @param {Object} [options] Available options:
-   *                            {number} epsilon
+   *                            {number} relTol
    *                              Minimum relative difference between two
    *                              compared values, used by all comparison functions.
+   *                            {number} absTol
+   *                              Minimum absolute difference between two
+   *                              compared values, used by all comparison functions.
    *                            {string} matrix
    *                              A string 'Matrix' (default) or 'Array'.
    *                            {string} number
@@ -49,7 +52,16 @@ export function configFactory (config, emit) {
    */
   function _config (options) {
     if (options) {
-      const prev = mapObject(config, clone)
+      if (options.epsilon !== undefined) {
+        // this if is only for backwards compatibility, it can be removed in the future.
+        console.warn('Warning: The configuration option "epsilon" is deprecated. Use "relTol" and "absTol" instead.')
+        const optionsFix = clone(options)
+        optionsFix.relTol = options.epsilon
+        optionsFix.absTol = options.epsilon * 1e-3
+        delete optionsFix.epsilon
+        return _config(optionsFix)
+      }
+      const prev = clone(config)
 
       // validate some of the options
       validateOption(options, 'matrix', MATRIX_OPTIONS)
@@ -58,16 +70,16 @@ export function configFactory (config, emit) {
       // merge options
       deepExtend(config, options)
 
-      const curr = mapObject(config, clone)
+      const curr = clone(config)
 
-      const changes = mapObject(options, clone)
+      const changes = clone(options)
 
       // emit 'config' event
       emit('config', curr, prev, changes)
 
       return curr
     } else {
-      return mapObject(config, clone)
+      return clone(config)
     }
   }
 

src/function/arithmetic/ceil.js

@@ -14,15 +14,15 @@ export const createCeilNumber = /* #__PURE__ */ factory(
   name, ['typed', 'config', 'round'], ({ typed, config, round }) => {
     return typed(name, {
       number: function (x) {
-        if (nearlyEqual(x, round(x), config.epsilon)) {
+        if (nearlyEqual(x, round(x), config.relTol, config.absTol)) {
           return round(x)
         } else {
           return Math.ceil(x)
         }
       },
 
       'number, number': function (x, n) {
-        if (nearlyEqual(x, round(x, n), config.epsilon)) {
+        if (nearlyEqual(x, round(x, n), config.relTol, config.absTol)) {
           return round(x, n)
         } else {
           let [number, exponent] = `${x}e`.split('e')
@@ -95,15 +95,15 @@ export const createCeil = /* #__PURE__ */ factory(name, dependencies, ({ typed,
     },
 
     BigNumber: function (x) {
-      if (bigNearlyEqual(x, round(x), config.epsilon)) {
+      if (bigNearlyEqual(x, round(x), config.relTol, config.absTol)) {
         return round(x)
       } else {
         return x.ceil()
       }
     },
 
     'BigNumber, BigNumber': function (x, n) {
-      if (bigNearlyEqual(x, round(x, n), config.epsilon)) {
+      if (bigNearlyEqual(x, round(x, n), config.relTol, config.absTol)) {
         return round(x, n)
       } else {
         return x.toDecimalPlaces(n.toNumber(), Decimal.ROUND_CEIL)

src/function/arithmetic/floor.js

@@ -14,15 +14,15 @@ export const createFloorNumber = /* #__PURE__ */ factory(
   name, ['typed', 'config', 'round'], ({ typed, config, round }) => {
     return typed(name, {
       number: function (x) {
-        if (nearlyEqual(x, round(x), config.epsilon)) {
+        if (nearlyEqual(x, round(x), config.relTol, config.absTol)) {
           return round(x)
         } else {
           return Math.floor(x)
         }
       },
 
       'number, number': function (x, n) {
-        if (nearlyEqual(x, round(x, n), config.epsilon)) {
+        if (nearlyEqual(x, round(x, n), config.relTol, config.absTol)) {
           return round(x, n)
         } else {
           let [number, exponent] = `${x}e`.split('e')
@@ -98,15 +98,15 @@ export const createFloor = /* #__PURE__ */ factory(name, dependencies, ({ typed,
     },
 
     BigNumber: function (x) {
-      if (bigNearlyEqual(x, round(x), config.epsilon)) {
+      if (bigNearlyEqual(x, round(x), config.relTol, config.absTol)) {
         return round(x)
       } else {
         return x.floor()
       }
     },
 
     'BigNumber, BigNumber': function (x, n) {
-      if (bigNearlyEqual(x, round(x, n), config.epsilon)) {
+      if (bigNearlyEqual(x, round(x, n), config.relTol, config.absTol)) {
         return round(x, n)
       } else {
         return x.toDecimalPlaces(n.toNumber(), Decimal.ROUND_FLOOR)

src/function/arithmetic/round.js

@@ -75,19 +75,19 @@ export const createRound = /* #__PURE__ */ factory(name, dependencies, ({ typed,
    */
   return typed(name, {
     number: function (x) {
-      // Handle round off errors by first rounding to epsilon precision
-      const xEpsilon = roundNumber(x, toExponent(config.epsilon))
-      const xSelected = nearlyEqual(x, xEpsilon, config.epsilon) ? xEpsilon : x
+      // Handle round off errors by first rounding to relTol precision
+      const xEpsilon = roundNumber(x, toExponent(config.relTol))
+      const xSelected = nearlyEqual(x, xEpsilon, config.relTol, config.absTol) ? xEpsilon : x
       return roundNumber(xSelected)
     },
 
     'number, number': function (x, n) {
-      // Same as number: unless user specifies more decimals than epsilon
-      const epsilonExponent = toExponent(config.epsilon)
+      // Same as number: unless user specifies more decimals than relTol
+      const epsilonExponent = toExponent(config.relTol)
       if (n >= epsilonExponent) { return roundNumber(x, n) }
 
       const xEpsilon = roundNumber(x, epsilonExponent)
-      const xSelected = nearlyEqual(x, xEpsilon, config.epsilon) ? xEpsilon : x
+      const xSelected = nearlyEqual(x, xEpsilon, config.relTol, config.absTol) ? xEpsilon : x
       return roundNumber(xSelected, n)
     },
 
@@ -115,21 +115,21 @@ export const createRound = /* #__PURE__ */ factory(name, dependencies, ({ typed,
     },
 
     BigNumber: function (x) {
-      // Handle round off errors by first rounding to epsilon precision
-      const xEpsilon = new BigNumber(x).toDecimalPlaces(toExponent(config.epsilon))
-      const xSelected = bigNearlyEqual(x, xEpsilon, config.epsilon) ? xEpsilon : x
+      // Handle round off errors by first rounding to relTol precision
+      const xEpsilon = new BigNumber(x).toDecimalPlaces(toExponent(config.relTol))
+      const xSelected = bigNearlyEqual(x, xEpsilon, config.relTol, config.absTol) ? xEpsilon : x
       return xSelected.toDecimalPlaces(0)
     },
 
     'BigNumber, BigNumber': function (x, n) {
       if (!n.isInteger()) { throw new TypeError(NO_INT) }
 
-      // Same as BigNumber: unless user specifies more decimals than epsilon
-      const epsilonExponent = toExponent(config.epsilon)
+      // Same as BigNumber: unless user specifies more decimals than relTol
+      const epsilonExponent = toExponent(config.relTol)
       if (n >= epsilonExponent) { return x.toDecimalPlaces(n.toNumber()) }
 
       const xEpsilon = x.toDecimalPlaces(epsilonExponent)
-      const xSelected = bigNearlyEqual(x, xEpsilon, config.epsilon) ? xEpsilon : x
+      const xSelected = bigNearlyEqual(x, xEpsilon, config.relTol, config.absTol) ? xEpsilon : x
       return xSelected.toDecimalPlaces(n.toNumber())
     },
 

src/function/geometry/intersect.js

@@ -119,7 +119,7 @@ export const createIntersect = /* #__PURE__ */ factory(name, dependencies, ({ ty
     const d2 = subtract(o2, p2b)
     const det = subtract(multiplyScalar(d1[0], d2[1]), multiplyScalar(d2[0], d1[1]))
     if (isZero(det)) return null
-    if (smaller(abs(det), config.epsilon)) {
+    if (smaller(abs(det), config.relTol)) {
       return null
     }
     const d20o11 = multiplyScalar(d2[0], o1[1])

src/function/matrix/eigs.js

@@ -68,7 +68,7 @@ export const createEigs = /* #__PURE__ */ factory(name, dependencies, ({ config,
    *
    * @param {Array | Matrix} x  Matrix to be diagonalized
    *
-   * @param {number | BigNumber | OptsObject} [opts] Object with keys `precision`, defaulting to config.epsilon, and `eigenvectors`, defaulting to true and specifying whether to compute eigenvectors. If just a number, specifies precision.
+   * @param {number | BigNumber | OptsObject} [opts] Object with keys `precision`, defaulting to config.relTol, and `eigenvectors`, defaulting to true and specifying whether to compute eigenvectors. If just a number, specifies precision.
    * @return {{values: Array|Matrix, eigenvectors?: Array<EVobj>}} Object containing an array of eigenvalues and an array of {value: number|BigNumber, vector: Array|Matrix} objects. The eigenvectors property is undefined if eigenvectors were not requested.
    *
    */
@@ -100,7 +100,7 @@ export const createEigs = /* #__PURE__ */ factory(name, dependencies, ({ config,
 
   function doEigs (mat, opts = {}) {
     const computeVectors = 'eigenvectors' in opts ? opts.eigenvectors : true
-    const prec = opts.precision ?? config.epsilon
+    const prec = opts.precision ?? config.relTol
     const result = computeValuesAndVectors(mat, prec, computeVectors)
     if (opts.matricize) {
       result.values = matrix(result.values)

src/function/matrix/eigs/realSymmetric.js

@@ -7,7 +7,7 @@ export function createRealSymmetric ({ config, addScalar, subtract, abs, atan, c
    * @param {number} prec
    * @param {'number' | 'BigNumber'} type
    */
-  function main (arr, N, prec = config.epsilon, type, computeVectors) {
+  function main (arr, N, prec = config.relTol, type, computeVectors) {
     if (type === 'number') {
       return diag(arr, prec, computeVectors)
     }
@@ -85,7 +85,7 @@ export function createRealSymmetric ({ config, addScalar, subtract, abs, atan, c
   // get angle
   function getTheta (aii, ajj, aij) {
     const denom = (ajj - aii)
-    if (Math.abs(denom) <= config.epsilon) {
+    if (Math.abs(denom) <= config.relTol) {
       return Math.PI / 4.0
     } else {
       return 0.5 * Math.atan(2.0 * aij / (ajj - aii))
@@ -95,7 +95,7 @@ export function createRealSymmetric ({ config, addScalar, subtract, abs, atan, c
   // get angle
   function getThetaBig (aii, ajj, aij) {
     const denom = subtract(ajj, aii)
-    if (abs(denom) <= config.epsilon) {
+    if (abs(denom) <= config.relTol) {
       return bignumber(-1).acos().div(4)
     } else {
       return multiplyScalar(0.5, atan(multiply(2.0, aij, inv(denom))))

src/function/relational/compare.js

@@ -30,7 +30,7 @@ export const createCompare = /* #__PURE__ */ factory(name, dependencies, ({ type
    * Compare two values. Returns 1 when x > y, -1 when x < y, and 0 when x == y.
    *
    * x and y are considered equal when the relative difference between x and y
-   * is smaller than the configured epsilon. The function cannot be used to
+   * is smaller than the configured absTol and relTol. The function cannot be used to
    * compare values smaller than approximately 2.22e-16.
    *
    * For matrices, the function is evaluated element wise.
@@ -72,7 +72,7 @@ export const createCompare = /* #__PURE__ */ factory(name, dependencies, ({ type
       },
 
       'BigNumber, BigNumber': function (x, y) {
-        return bigNearlyEqual(x, y, config.epsilon)
+        return bigNearlyEqual(x, y, config.relTol, config.absTol)
           ? new BigNumber(0)
           : new BigNumber(x.cmp(y))
       },
@@ -97,7 +97,7 @@ export const createCompare = /* #__PURE__ */ factory(name, dependencies, ({ type
 export const createCompareNumber = /* #__PURE__ */ factory(name, ['typed', 'config'], ({ typed, config }) => {
   return typed(name, {
     'number, number': function (x, y) {
-      return nearlyEqual(x, y, config.epsilon)
+      return nearlyEqual(x, y, config.relTol, config.absTol)
         ? 0
         : (x > y ? 1 : -1)
     }

src/function/relational/compareNatural.js

@@ -19,7 +19,7 @@ export const createCompareNatural = /* #__PURE__ */ factory(name, dependencies,
    * the function compares in a natural way.
    *
    * For numeric values, x and y are considered equal when the relative
-   * difference between x and y is smaller than the configured epsilon.
+   * difference between x and y is smaller than the configured relTol and absTol.
    * The function cannot be used to compare values smaller than
    * approximately 2.22e-16.
    *

src/function/relational/equal.js

@@ -23,7 +23,7 @@ export const createEqual = /* #__PURE__ */ factory(name, dependencies, ({ typed,
    * Test whether two values are equal.
    *
    * The function tests whether the relative difference between x and y is
-   * smaller than the configured epsilon. The function cannot be used to
+   * smaller than the configured relTol and absTol. The function cannot be used to
    * compare values smaller than approximately 2.22e-16.
    *
    * For matrices, the function is evaluated element wise.