interval
The interval package for real-valued interval arithmetic allows one to evaluate functions over subsets of their domain. All results are verified, because interval computations automatically keep track of any errors. These concepts can be used to handle uncertainties, estimate arithmetic errors and produce reliable results. Also it can be applied to computer-assisted proofs, constraint programming, and verified computing. The implementation is based on interval boundaries represented by binary64 numbers and is conforming to IEEE Std 1788-2015, IEEE standard for interval arithmetic.
Select category:
Interval constant
Interval constructor
Interval function (most with tightest accuracy)
Interval matrix operation
Interval comparison
Set operation
Interval reverse operation
Interval numeric function
Interval input and output
Interval solver or optimizer
Interval contractor arithmetic
Verified solver or optimizer
Utility function
API function to low-level libraries
Internal utility function (use at one's own risk)
Return the empty interval.
Return the entire set of real numbers.
Return the ill-formed decorated interval, called NaI (Not an Interval).
Create an interval enclosure for a list of parameters.
Create an interval enclosure I for [M-R, M+R].
Create an interval (from boundaries).
Create a decorated interval from a bare interval.
Create a decorated interval from boundaries.
Compute the absolute value of numbers.
Compute the inverse cosine in radians (arccosine).
Compute the inverse hyperbolic cosine.
Compute the inverse sine in radians (arcsine).
Compute the inverse hyperbolic sine.
Compute the inverse tangent in radians.
Compute the inverse tangent with two arguments.
Compute the inverse hyperbolic tangent.
Recover interval Z from intervals X and Y, given that one knows X was obtained as the sum Y + Z.
Recover interval Z from intervals X and Y, given that one knows X was obtained as the difference Z - Y.
Compute the cube root.
Round each number in interval X towards +Inf.
Compute the cosine in radians.
Compute the hyperbolic cosine.
Compute the cotangent in radians, that is the reciprocal tangent.
Compute the hyperbolic cotangent, that is the reciprocal hyperbolic tangent.
Compute the cosecant in radians, that is the reciprocal sine.
Compute the hyperbolic cosecant, that is the reciprocal hyperbolic sine.
Compute the real part of the dilogarithm function.
Compute the exponential integral for positive arguments.
Compute the error function.
Compute the complementary error function ‘1 - erf (X)’.
Compute the exponential function.
Compute ‘exp (X) - 1’ accurately in the neighborhood of zero.
Compute the factorial of N where N is a non-negative integer.
Truncate fractional portion of each number in interval X.
Round each number in interval X towards -Inf.
Fused multiply and add ‘X * Y + Z’.
Compute the gamma function.
Compute the logarithm of the gamma function for positive arguments.
Compute the euclidean norm.
Divide all numbers of interval Y by all numbers of X.
Return a row vector of N linearly spaced members between BASE and LIMIT.
Compute the natural logarithm.
Compute the decimal (base-10) logarithm.
Compute ‘log (1 + X)’ accurately in the neighborhood of zero.
Compute the binary (base-2) logarithm.
Compute the maximum value chosen from intervals.
Compute the minimum value chosen from intervals.
Subtract all numbers of interval Y from all numbers of X.
Compute the modulus of X and Y.
Compute the real n-th root of X.
Add all numbers of interval X to all numbers of Y.
Compute the simple power function on intervals defined by ‘exp (Y * log (X))’.
Compute ‘10^x’ for all numbers in X.
Compute ‘2^x’ for all numbers in X.
Compute the general power function on intervals, which is defined for (1) any positive base X; (2) ‘X = 0’ when Y is positive; (3) negative base X together with integral exponent Y.
Compute the monomial ‘x^P’ for all numbers in X.
Compute the digamma function, also known as the psi function.
Divide all numbers of interval X by all numbers of Y.
Compute the square root (for all non-negative numbers).
Compute the remainder of the division X by Y.
Round each number in interval X to the nearest integer.
Round each number in interval X to the nearest integer.
Compute the reciprocal square root (for all positive numbers).
Compute the secant in radians, that is the reciprocal cosine.
Compute the hyperbolic secant, that is the reciprocal hyperbolic cosine.
Compute the signum function for each number in interval X.
Compute the sine in radians.
Compute the hyperbolic sine.
Compute the square root (for all non-negative numbers).
Compute the tangent in radians.
Compute the hyperbolic tangent.
Multiply all numbers of interval X by all numbers of Y.
Negate all numbers in the interval.
Return the interval itself.
Compute the Cholesky factor, R, of each symmetric positive definite matrix in A.
Compute the determinant of matrix A.
Compute the dot product of two interval vectors.
Compute the matrix exponential of square matrix A.
Compute the inverse of the square matrix A.
Compute the LU decomposition of A.
Return the interval matrix left division of X and Y.
Return the matrix power operation of X raised to the Y power.
Return the interval matrix right division of X and Y.
Compute the interval matrix multiplication.
Compute the p-norm (or p,q-norm) of the matrix A.
Product of elements along dimension DIM.
Compute the QR decomposition of A.
Sum of elements along dimension DIM.
Sum of absolute values along dimension DIM.
Sum of squares along dimension DIM.
Evaluate disjoint comparison on intervals.
Compare intervals A and B for equality.
Compare intervals A and B for weakly greater.
Compare intervals A and B for strict greater.
Evaluate interior comparison on intervals.
Check if the interval is a common interval, that is a nonemty, closed bounded real interval.
Check if the interval represents the empty set.
Check if the interval represents the entire set of real numbers.
Check if the interval X contains the number M.
Check if the interval represents a set that contains a single real only.
Compare intervals A and B for weakly less.
Compare intervals A and B for strictly less.
Compare intervals A and B for inequality.
Extensively compare the positions of intervals A and B on the real number line.
Evaluate precedes comparison on intervals.
Evaluate strict precedes comparison on intervals.
Evaluate strict subset comparison on intervals.
Evaluate subset comparison on intervals.
Check if the interval is the result of a failed interval construction.
Bisect an interval into two intervals, which contain half the amount of binary64 numbers each.
Intersect intervals.
Mince interval X into a row vector of N sub-intervals of equal size.
Increases the interval’s boundaries in each direction to the next number.
Build the relative complement of interval B in interval A.
Build the symmetric difference of intervals A and B.
Build the interval hull of the union of intervals.
Compute the reverse absolute value function.
Compute the reverse atan2 function for the first parameter.
Compute the reverse atan2 function for the second parameter.
Compute the reverse hyperbolic cosine function.
Compute the reverse cosine function.
Compute the reverse multiplication function or the two-output division.
Compute the reverse monomial ‘x^P’.
Compute the reverse power function for the first parameter.
Compute the reverse power function for the second parameter.
Compute the reverse sine function.
Compute the reverse square function.
Compute the reverse tangent function.
Compute the Hausdorff distance between two intervals as sets.
Compute the inner distance between two intervals as sets.
Get the (greatest) lower boundary for all numbers of interval X.
Get the magnitude of numbers in interval X, that is the maximum of absolute values for each element.
Get the midpoint of interval X.
Get the mignitude of numbers in interval X, that is the minimum of absolute values for each element.
Get the radius (and midpoint) of interval X.
Compute the signed distance between two intervals as sets.
Get the signed mignitude of numbers in interval X, that is the unique number closest to zero for each element.
Get the (least) upper boundary for all numbers of interval X.
Get the width of interval X.
Decode an interval from its interchange format.
Convert interval X to an interval literal string S, which contains X.
Create a bare interval.
Encode bare interval X in interchange format.
Display the value of interval X.
Display the variable name and value of interval X.
Write interval X under the control of a template string TEMPLATE to the file descriptor ‘fid’ and return the number of characters printed.
Build an exact representation of the interval X in hexadecimal-significand form.
Create a 2D-plot of intervals.
Create a 3D-plot of intervals.
Print interval X under the control of a template string TEMPLATE to the stream ‘stdout’ and return the number of characters printed.
Convert interval X under the control of a template string TEMPLATE to a string with interval literals.
Minimize the function F over the interval box X0 and return rigorous bounds.
Compute the preimage of the set Y under function F.
Compute the enclosure of all roots of function F in interval X0.
Solve a linear interval system A * X = B using Gaussian elimination.
Evaluate polynomial P with argument X.
Return a contractor function for the intersection of two sets.
Return a contractor function for the union of two sets.
Verified backward error analysis of eigenpairs.
Verified real eigenvector of an interval matrix.
Verified strong solution of interval linear inequalities.
Verified nonnegative invertibility of an interval matrix.
Verified nonnegative solution of a system of linear inequalities.
Verified linear programming.
Return the concatenation of N-D interval arrays ARRAY1, ARRAY2, … along dimension DIM.
Return the number of columns of A.
Return the complex conjugate transpose of X.
Create a diagonal matrix M with vector V on diagonal K or extract a vector V from the K-th diagonal of matrix M.
The magic index ‘end’ refers to the last valid entry in an indexing operation.
Return the horizontal concatenation of interval array objects along dimension 2.
Return true if A is an interval column vector.
Return true if A is an interval row vector.
Return true if A is an interval scalar.
Return true if A is a square interval matrix.
Return true if A is an interval vector.
Return the length of interval object A.
Return the number of dimensions of A.
Return the number of elements in the interval object A.
Append the scalar interval value C to the interval vector X until it is of length L.
Prepend the scalar interval value C to the interval vector X until it is of length L.
Return an interval matrix with the specified dimensions (M, N, ...) whose elements are taken from the interval matrix A.
Resize interval array X cutting off elements as necessary.
Return the number of rows of A.
Return a row vector with the size (number of elements) of each dimension for the interval array A.
Perform the subscripted assignment operation according to the subscript specified by IDX.
Select property P or elements I from interval array A.
Return the transpose of interval matrix or vector X.
Return a new matrix formed by extracting the lower triangular part of the matrix A, and setting all other elements to zero.
Return a new matrix formed by extracting the upper triangular part of the matrix A, and setting all other elements to zero.
Return the vertical concatenation of interval array objects along dimension 1.
Return the decoration of the decorated interval X.
Return the bare interval for the decorated interval X.
Evaluate a function in binary64 with correctly rounded result.
Evaluate a function in binary64 with correctly rounded result.
Return a row vector with N linearly spaced elements between BASE and LIMIT.
Compute the matrix product with binary64 numbers and correctly rounded result.
Compute the lower and upper boundary of the matrix square of interval matrix [XL, XU].
Convert binary64 numbers X to string representation, either exact or correctly rounded.
Compute the sum S of all numbers in a binary64 array X along dimension DIM with correctly rounded result.
Compute the dot product of arrays of binary 64 numbers along dimensionDIM with correctly rounded result.
Changes the floating-point rounding direction for the current thread and any new threads which will be spawned from the current thread.
Check whether crlibm is available and working.
Split string S into a cell array of interval literals.
Package: interval