# R8LIB A Double Precision Real Arithmetic Utility Library

R8LIB is a FORTRAN90 library which contains a number of utility routines for "R8" or "double precision real" arithmetic.

### Languages:

R8LIB is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

C4LIB, a FORTRAN90 library which implements certain elementary functions for "C4" or single precision complex variables;

C8LIB, a FORTRAN90 library which implements certain elementary functions for "C8" or double precision complex variables;

I4LIB, a FORTRAN90 library which contains many utility routines, using "I4" or "single precision integer" arithmetic.

I8LIB, a FORTRAN90 library which contains many utility routines, using "I8" or "double precision integer" arithmetic.

L4LIB, a FORTRAN90 library which contains many utility routines, using one byte logical (L4) variables.

R4LIB, a FORTRAN90 library which contains many utility routines, using "R4" or "single precision real" arithmetic.

R8COL, a FORTRAN90 library which contains utility routines for R8COL's, that is, double precision real MxN arrays, considered as N column vectors, each of length M. The data may be thought of as a matrix of multiple columns, and many operations will be carried out columnwise.

R8ROW, a FORTRAN90 library which contains utility routines for R8ROW's, that is, double precision real MxN arrays, considered as M row vectors, each of length N. The data may be thought of as a matrix of multiple rows, and many operations will be carried out rowwise.

### References:

1. Thomas Cormen, Charles Leiserson, Ronald Rivest,
Introduction to Algorithms,
MIT Press, 2001,
ISBN: 0262032937,
LC: QA76.C662.
2. Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
ISBN: 0-12-519260-6,
LC: QA164.N54.

### List of Routines:

• GAMMA_VALUES returns some values of the Gamma function.
• GAMMA_LOG_VALUES returns some values of the Log Gamma function.
• GET_UNIT returns a free FORTRAN unit number.
• I4_LOG_10 returns the integer part of the logarithm base 10 of an I4.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_UNIFORM_AB returns a scaled pseudorandom I4.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• I4INT_TO_R8INT maps an I4INT to an R8INT.
• I4MAT_PRINT prints an I4MAT.
• I4MAT_PRINT_SOME prints some of an I4MAT.
• I4VEC_INDICATOR0 sets an I4VEC to the indicator vector (0,1,2,...)
• I4VEC_INDICATOR1 sets an I4VEC to the indicator vector (1,2,3,...)
• I4VEC_PERMUTE permutes an I4VEC in place.
• I4VEC_PRINT prints an I4VEC.
• I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed".
• LEGENDRE_ZEROS computes the zeros of the Legendre polynomial of degree N.
• PERM0_CHECK checks a 0-based permutation.
• PERM0_UNIFORM selects a random permutation of objects 0, ..., N-1.
• PERM1_CHECK checks a 1-based permutation.
• PERM1_UNIFORM selects a random permutation of objects 1, ..., N.
• R8_ABS returns the absolute value of an R8.
• R8_ACOS computes the arc cosine function, with argument truncation.
• R8_ACOSH returns the inverse hyperbolic cosine of a number.
• R8_ADD returns the sum of two R8's.
• R8_AGM computes the arithmetic-geometric mean of A and B.
• R8_AINT truncates an R8 argument to an integer.
• R8_ASIN computes the arc sine function, with argument truncation.
• R8_ASINH returns the inverse hyperbolic sine of a number.
• R8_ATAN computes the inverse tangent of the ratio Y / X.
• R8_ATANH returns the inverse hyperbolic tangent of a number.
• R8_BIG returns a big R8.
• R8_CAS returns the "casine" of an R8.
• R8_CEILING rounds an R8 "up" (towards +oo) to an integral R8.
• R8_CHOOSE computes the combinatorial coefficient C(N,K).
• R8_CHOP chops an R8 to a given number of binary places.
• R8_COSD returns the cosine of an angle given in degrees.
• R8_COT returns the cotangent of an angle.
• R8_COTD returns the cotangent of an angle given in degrees.
• R8_CSC returns the cosecant of X.
• R8_CSCD returns the cosecant of an angle given in degrees.
• R8_CSQRT returns the complex square root of an R8.
• R8_CUBE_ROOT returns the cube root of an R8.
• R8_DEGREES converts an angle from radian to degree measure.
• R8_DIFF computes the difference of two R8's to a specified accuracy.
• R8_DIGIT returns a particular decimal digit of an R8.
• R8_DIVIDE_I4 returns an I4 fraction as an R8.
• R8_E returns the value of the base of the natural logarithm system.
• R8_EPSILON returns the R8 roundoff unit.
• R8_EPSILON_COMPUTE computes the R8 roundoff unit.
• R8_EXP computes the exponential of an R8, avoiding overflow and underflow.
• R8_FACTORIAL computes the factorial of N.
• R8_FACTORIAL_VALUES returns values of the real factorial function.
• R8_FACTORIAL2 computes the double factorial function.
• R8_FACTORIAL2_VALUES returns values of the double factorial function.
• R8_FALL computes the falling factorial function [X]_N.
• R8_FALL_VALUES returns some values of the falling factorial function.
• R8_FLOOR rounds an R8 "down" (towards -oo) to the nearest integral R8.
• R8_FRACTION uses real arithmetic on an integer ratio.
• R8_FRACTIONAL returns the fractional part of an R8.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_GAMMA_LOG evaluates the logarithm of the gamma function.
• R8_HUGE returns a very large R8.
• R8_HYPOT returns the value of sqrt ( X^2 + Y^2 ).
• R8_IN_01 is TRUE if an R8 is in the range [0,1].
• R8_INSIGNIFICANT determines if an R8 is insignificant.
• R8_IS_INT determines if an R8 represents an integer value.
• R8_LOG_2 returns the logarithm base 2 of an R8.
• R8_LOG_10 returns the logarithm base 10 of an R8.
• R8_LOG_B returns the logarithm base B of an R8.
• R8_MANT computes the "mantissa" or "fraction part" of an R8.
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_MOD returns the remainder of R8 division.
• R8_MODP returns the nonnegative remainder of R8 division.
• R8_MOP returns the I-th power of -1 as an R8.
• R8_NINT returns the nearest integer to an R8.
• R8_NORMAL_01 returns a unit pseudonormal R8.
• R8_NORMAL_AB returns a scaled pseudonormal R8.
• R8_PI returns the value of pi as an R8.
• R8_PI_SQRT returns the square root of pi as an R8.
• R8_POWER computes the P-th power of an R8.
• R8_POWER_FAST computes an integer power of an R8.
• R8_PRINT prints an R8.
• R8_PYTHAG computes sqrt ( A * A + B * B ), avoiding overflow and underflow.
• R8_RISE computes the rising factorial function [X]^N.
• R8_RISE_VALUES returns some values of the rising factorial function.
• R8_ROUND sets an R8 to the nearest integral value.
• R8_ROUND_I4 sets an R8 to the nearest integral value, returning an I4
• R8_ROUND2 rounds an R8 in base 2.
• R8_ROUNDB rounds an R8 in a given base.
• R8_ROUNDX rounds an R8 in base 10.
• R8_SECD returns the secant of an angle given in degrees.
• R8_SECH evaluates the hyperbolic secant, while avoiding COSH overflow.
• R8_SIGN returns the sign of an R8.
• R8_SIGN3 returns the three-way sign of an R8.
• R8_SIGN_CHAR returns a character indicating the sign of an R8.
• R8_SIGN_MATCH is TRUE if two R8's are of the same sign.
• R8_SIGN_MATCH_STRICT is TRUE if two R8's are of the same strict sign.
• R8_SIGN_OPPOSITE is TRUE if two R8's are not of the same sign.
• R8_SIGN_OPPOSITE_STRICT is TRUE if two R8's are strictly of opposite sign.
• R8_SINCOS_SUM simplifies a*sin(cx)+b*cos(cx).
• R8_SIND returns the sine of an angle given in degrees.
• R8_SQRT_I4 returns the square root of an I4 as an R8.
• R8_SWAP swaps two R8's.
• R8_SWAP3 swaps three R8's.
• R8_TAND returns the tangent of an angle given in degrees.
• R8_TINY returns a very small but positive R8.
• R8_TO_R8_DISCRETE maps R to RD in [RMIN, RMAX] with NR possible values.
• R8_TO_DHMS converts decimal days into days, hours, minutes, seconds.
• R8_TO_I4 maps X in [XMIN, XMAX] to integer IX in [IXMIN, IXMAX].
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8_UNIFORM_AB returns a scaled pseudorandom R8.
• R8_UNSWAP3 unswaps three R8's.
• R8_WALSH_1D evaluates the Walsh function.
• R8_WRAP forces an R8 to lie between given limits by wrapping.
• R82_DIST_L2 returns the L2 distance between a pair of R82's.
• R82_EQ == ( A1 == A2 ) for two R82's.
• R82_GE == ( A1 >= A2 ) for two R82's.
• R82_GT == ( A1 > A2 ) for two R82's.
• R82_LE == ( A1 <= A2 ) for two R82's.
• R82_LT == ( A1 < A2 ) for two R82's.
• R82_NE == ( A1 /= A2 ) for two R82's.
• R82_NORM returns the Euclidean norm of an R82.
• R82_NORMALIZE Euclidean normalizes an R82.
• R82_PRINT prints an R82.
• R82_SWAP swaps two R82 values.
• R82_UNIFORM_AB returns a random R82 value in a given range.
• R82COL_PRINT_PART prints "part" of an R82COL.
• R82POLY2_PRINT prints a second order polynomial in two variables.
• R82POLY2_TYPE analyzes a second order polynomial in two variables.
• R82POLY2_TYPE_PRINT prints the meaning of the output from R82POLY2_TYPE.
• R82ROW_MAX returns the maximum value in an R82ROW.
• R82ROW_MIN returns the minimum value in an R82ROW.
• R82ROW_ORDER_TYPE finds the order type of an R82ROW.
• R82ROW_PART_QUICK_A reorders an R82ROW as part of a quick sort.
• R82ROW_PERMUTE permutes an R82ROW in place.
• R82ROW_PRINT prints an R82ROW.
• R82ROW_PRINT_PART prints "part" of an R82ROW.
• R82ROW_SORT_HEAP_INDEX_A ascending index heaps an R82ROW.
• R82ROW_SORT_QUICK_A ascending sorts an R82ROW using quick sort.
• R83_NORM returns the Euclidean norm of an R83.
• R83_NORMALIZE normalizes an R83.
• R83_PRINT prints an R83.
• R83_SWAP swaps two R83's.
• R83COL_PRINT_PART prints "part" of an R83COL.
• R83ROW_MAX returns the maximum value in an R83ROW.
• R83ROW_MIN returns the minimum value in an R83ROW.
• R83ROW_NORMALIZE normalizes each R83 in an R83ROW.
• R83ROW_PRINT_PART prints "part" of an R83ROW.
• R84_NORMALIZE normalizes an R84.
• R8BLOCK_EXPAND_LINEAR linearly interpolates new data into an R8BLOCK.
• R8BLOCK_PRINT prints an R8BLOCK.
• R8CMAT_PRINT prints an R8CMAT.
• R8CMAT_PRINT_SOME prints some of an R8CMAT.
• R8CMAT_TO_R8MAT transfers data from an R8CMAT to an R8MAT.
• R8INT_TO_R8INT maps one R8INT to another.
• R8INT_TO_I4INT maps an R8INT to an integer interval.
• R8MAT_ADD computes C = alpha * A + beta * B for R8MAT's.
• R8MAT_AMAX returns the maximum absolute value entry of an R8MAT.
• R8MAT_BORDER_CUT cuts the "border" of an R8MAT.
• R8MAT_CHOLESKY_FACTOR computes the Cholesky factor of a symmetric matrix.
• R8MAT_CHOLESKY_FACTOR_UPPER: upper Cholesky factor of a symmetric matrix.
• R8MAT_CHOLESKY_INVERSE computes the inverse of a symmetric matrix.
• R8MAT_CHOLESKY_SOLVE solves a Cholesky factored linear system A * x = b.
• R8MAT_CHOLESKY_SOLVE_UPPER solves a Cholesky factored system A * x = b.
• R8MAT_COPY copies an R8MAT.
• R8MAT_COVARIANCE computes the sample covariance of a set of vector data.
• R8MAT_DET computes the determinant of an R8MAT.
• R8MAT_DET_2D computes the determinant of a 2 by 2 R8MAT.
• R8MAT_DET_3D computes the determinant of a 3 by 3 R8MAT.
• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• R8MAT_DET_5D computes the determinant of a 5 by 5 R8MAT.
• R8MAT_DIAG_GET_VECTOR gets the value of the diagonal of an R8MAT.
• R8MAT_DIAG_SET_SCALAR sets the diagonal of an R8MAT to a scalar value.
• R8MAT_DIAG_SET_VECTOR sets the diagonal of an R8MAT to a vector.
• R8MAT_DIAGONAL returns a diagonal matrix as an R8MAT.
• R8MAT_DIFF_FROBENIUS returns the Frobenius norm of an R8MAT difference.
• R8MAT_EXPAND_LINEAR linearly interpolates new data into an R8MAT.
• R8MAT_EXPAND_LINEAR2 expands an R8MAT by linear interpolation.
• R8MAT_FLIP_COLS reverses the column order of an R8MAT.
• R8MAT_FLIP_ROWS reverses the row order of an R8MAT.
• R8MAT_FS factors and solves a system with one right hand side.
• R8MAT_FSS factors and solves a system with multiple right hand sides.
• R8MAT_GIVENS_POST computes the Givens postmultiplier rotation matrix.
• R8MAT_GIVENS_PRE computes the Givens premultiplier rotation matrix.
• R8MAT_HESS approximates a Hessian matrix via finite differences.
• R8MAT_HOUSE_AXH computes A*H where H is a compact Householder matrix.
• R8MAT_HOUSE_FORM constructs a Householder matrix from its compact form.
• R8MAT_HOUSE_HXA computes H*A where H is a compact Householder matrix.
• R8MAT_HOUSE_POST computes a Householder post-multiplier matrix.
• R8MAT_HOUSE_PRE computes a Householder pre-multiplier matrix.
• R8MAT_IDENTITY stores the identity matrix in an R8MAT.
• R8MAT_IN_01 is TRUE if the entries of an R8MAT are in the range [0,1].
• R8MAT_INDICATOR sets up an "indicator" R8MAT.
• R8MAT_INSIGNIFICANT determines if an R8MAT is insignificant.
• R8MAT_INVERSE_2D inverts a 2 by 2 R8MAT using Cramer's rule.
• R8MAT_INVERSE_3D inverts a 3 by 3 R8MAT using Cramer's rule.
• R8MAT_INVERSE_4D inverts a 4 by 4 R8MAT using Cramer's rule.
• R8MAT_IS_IDENTITY determines if an R8MAT is the identity.
• R8MAT_IS_NONNEGATIVE checks whether an R8MAT is nonnegative.
• R8MAT_IS_SYMMETRIC checks an R8MAT for symmetry.
• R8MAT_JAC estimates a dense jacobian matrix of the function FX.
• R8MAT_KRONECKER computes the Kronecker product of two R8MAT's.
• R8MAT_L_INVERSE inverts a lower triangular R8MAT.
• R8MAT_L_PRINT prints a lower triangular R8MAT.
• R8MAT_L_SOLVE solves a lower triangular linear system.
• R8MAT_L1_INVERSE inverts a unit lower triangular R8MAT.
• R8MAT_LT_SOLVE solves a transposed lower triangular linear system.
• R8MAT_LU computes the LU factorization of a rectangular R8MAT.
• R8MAT_MAX returns the maximum entry of an R8MAT.
• R8MAT_MAX_INDEX returns the location of the maximum entry of an R8MAT.
• R8MAT_MAXCOL_MINROW gets the maximum column minimum row of an M by N R8MAT.
• R8MAT_MAXROW_MINCOL gets the maximum row minimum column of an M by N R8MAT.
• R8MAT_MEAN returns the mean of an M by N R8MAT.
• R8MAT_MIN returns the minimum entry of an M by N R8MAT.
• R8MAT_MIN_INDEX returns the location of the minimum entry of an R8MAT.
• R8MAT_MINCOL_MAXROW gets the minimum column maximum row of an M by N R8MAT.
• R8MAT_MINROW_MAXCOL gets the minimum row maximum column of an M by N R8MAT.
• R8MAT_MINVM computes inverse(A) * B for R8MAT's.
• R8MAT_MM multiplies two R8MAT's.
• R8MAT_MMT computes C = A * B' for two R8MAT's.
• R8MAT_MTM computes C = A' * B for two R8MAT's.
• R8MAT_MTV multiplies a transposed matrix times a vector
• R8MAT_MV multiplies a matrix times a vector.
• R8MAT_NINT rounds the entries of an R8MAT.
• R8MAT_NONZEROS counts the nonzeros in an R8MAT.
• R8MAT_NORM_EIS returns the EISPACK norm of an R8MAT.
• R8MAT_NORM_FRO returns the Frobenius norm of an R8MAT.
• R8MAT_NORM_FRO_AFFINE returns the Frobenius norm of an R8MAT difference.
• R8MAT_NORM_L1 returns the matrix L1 norm of an R8MAT.
• R8MAT_NORM_L2 returns the matrix L2 norm of an R8MAT.
• R8MAT_NORM_LI returns the matrix L-oo norm of an R8MAT.
• R8MAT_NORMAL_01 returns a unit pseudonormal R8MAT.
• R8MAT_NULLSPACE computes the nullspace of a matrix.
• R8MAT_NULLSPACE_SIZE computes the size of the nullspace of a matrix.
• R8MAT_ORTH_UNIFORM returns a random orthogonal R8MAT.
• R8MAT_PLOT "plots" an R8MAT, with an optional title.
• R8MAT_PLOT_SYMBOL returns a symbol for an element of an R8MAT.
• R8MAT_POLY_CHAR computes the characteristic polynomial of an R8MAT.
• R8MAT_POWER computes a nonnegative power of an R8MAT.
• R8MAT_POWER_METHOD applies the power method to an R8MAT.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_PRINT2 prints an R8MAT.
• R8MAT_REF computes the row echelon form of a matrix.
• R8MAT_RMS returns the RMS norm of an R8MAT.
• R8MAT_ROW_COPY copies a vector into a row of an R8MAT.
• R8MAT_ROW_SET copies a vector into a row of an R8MAT.
• R8MAT_RREF computes the reduced row echelon form of a matrix.
• R8MAT_SCALE multiplies an R8MAT by a scalar.
• R8MAT_SOLVE uses Gauss-Jordan elimination to solve an N by N linear system.
• R8MAT_SOLVE_2D solves a 2 by 2 linear system using Cramer's rule.
• R8MAT_SOLVE_3D solves a 3 by 3 linear system using Cramer's rule.
• R8MAT_SOLVE2 computes the solution of an N by N linear system.
• R8MAT_SUB computes the difference of two R8MAT's.
• R8MAT_SUM returns the sum of the entries of an R8MAT.
• R8MAT_SYMM_EIGEN returns a symmetric matrix with given eigensystem.
• R8MAT_SYMM_JACOBI applies Jacobi eigenvalue iteration to a symmetric matrix.
• R8MAT_TO_R8CMAT transfers data from an R8MAT to an R8CMAT.
• R8MAT_TO_R8PLU factors a general R8MAT.
• R8MAT_TRACE computes the trace of an R8MAT.
• R8MAT_TRANSPOSE makes a transposed copy of an R8MAT.
• R8MAT_TRANSPOSE_IN_PLACE transposes an R8MAT in place.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_U_INVERSE inverts an upper triangular R8MAT.
• R8MAT_U_SOLVE solves an upper triangular linear system.
• R8MAT_U1_INVERSE inverts a unit upper triangular R8MAT.
• R8MAT_UNIFORM_01 fills an R8MAT with unit pseudorandom numbers.
• R8MAT_UNIFORM_AB returns a scaled pseudorandom R8MAT.
• R8MAT_UNIFORM_ABVEC returns a scaled pseudorandom R8MAT.
• R8MAT_UT_SOLVE solves a transposed upper triangular linear system.
• R8MAT_VAND2 returns the N by N row Vandermonde matrix A.
• R8MAT_VTMV multiplies computes the scalar x' * A * y.
• R8MAT_ZEROS zeroes an R8MAT.
• R8PLU_DET computes the determinant of an R8PLU matrix.
• R8PLU_INVERSE computes the inverse of an R8PLU matrix.
• R8PLU_MUL computes A * x using the PLU factors of A.
• R8PLU_SOLVE solves a linear system A*x=b from the PLU factors.
• R8PLU_TO_R8MAT recovers the matrix A that was factored by R8MAT_TO_R8PLU.
• R8POLY_DEGREE returns the degree of a polynomial.
• R8POLY_DERIV returns the derivative of a polynomial.
• R8POLY_LAGRANGE_0 evaluates the Lagrange factor at a point.
• R8POLY_LAGRANGE_1 evaluates the first derivative of the Lagrange factor.
• R8POLY_LAGRANGE_2 evaluates the second derivative of the Lagrange factor.
• R8POLY_LAGRANGE_COEF returns the coefficients of a Lagrange polynomial.
• R8POLY_LAGRANGE_FACTOR evaluates the polynomial Lagrange factor at a point.
• R8POLY_LAGRANGE_VAL evaluates the IPOL-th Lagrange polynomial.
• R8POLY_ORDER returns the order of a polynomial.
• R8POLY_PRINT prints out a polynomial.
• R8POLY_SHIFT adjusts the coefficients of a polynomial for a new argument.
• R8POLY_VALUE_HORNER evaluates a polynomial using Horner's method.
• R8POLY_VALUES_HORNER evaluates a polynomial using Horner's method.
• R8POLY_VALUE_2D evaluates a polynomial in 2 variables, X and Y.
• R8POLY2_EX finds the extremal point of a parabola determined by three points.
• R8POLY2_EX2 finds extremal point of a parabola determined by three points.
• R8POLY2_ROOT returns the two roots of a quadratic polynomial.
• R8POLY2_RROOT returns the real parts of the roots of a quadratic polynomial.
• R8POLY2_VAL evaluates a parabola defined by three data values.
• R8POLY2_VAL2 evaluates a parabolic interpolant through tabular data.
• R8POLY3_ROOT returns the three roots of a cubic polynomial.
• R8POLY4_ROOT returns the four roots of a quartic polynomial.
• R8R8_COMPARE compares two R8R8's.
• R8R8_PRINT prints an R8R8.
• R8R8R8_COMPARE compares two R8R8R8's.
• R8R8R8VEC_INDEX_INSERT_UNIQUE inserts unique R8R8R in an indexed sorted list.
• R8R8R8VEC_INDEX_SEARCH searches for R8R8R8 value in an indexed sorted list.
• R8R8VEC_INDEX_INSERT_UNIQUE inserts a unique R8R8 in an indexed sorted list.
• R8R8VEC_INDEX_SEARCH searches for an R8R8 in an indexed sorted list.
• R8SLMAT_PRINT prints a strict lower triangular R8MAT.
• R8VEC_01_TO_AB shifts and rescales an R8VEC to lie within given bounds.
• R8VEC_AB_TO_01 shifts and rescales an R8VEC to lie within [0,1].
• R8VEC_AB_TO_CD shifts and rescales an R8VEC from one interval to another.
• R8VEC_ALL_NONPOSITIVE: ( all ( A <= 0 ) ) for R8VEC's.
• R8VEC_AMAX returns the maximum absolute value in an R8VEC.
• R8VEC_AMAX_INDEX returns the index of the maximum absolute value in an R8VEC.
• R8VEC_AMIN returns the minimum absolute value in an R8VEC.
• R8VEC_AMIN_INDEX returns the index of the minimum absolute value in an R8VEC.
• R8VEC_ANY_NEGATIVE: ( any A < 0 ) for R8VEC's.
• R8VEC_ANY_NONZERO: ( any A nonzero ) for R8VEC's.
• R8VEC_ANY_NORMAL returns some normal vector to V1.
• R8VEC_ASCENDS determines if an R8VEC is (weakly) ascending.
• R8VEC_ASCENDS_STRICTLY determines if an R8VEC is strictly ascending.
• R8VEC_ASUM returns the sum of the absolute values of the entries of an R8VEC.
• R8VEC_BIN computes bins based on a given R8VEC.
• R8VEC_BLEND performs weighted interpolation of two R8VEC's.
• R8VEC_BRACKET searches a sorted R8VEC for successive brackets of a value.
• R8VEC_BRACKET2 searches a sorted R8VEC for successive brackets of a value.
• R8VEC_BRACKET3 finds the interval containing or nearest a given value.
• R8VEC_BRACKET4 finds the nearest interval to each of a vector of values.
• R8VEC_BRACKET5 brackets data between successive entries of a sorted R8VEC.
• R8VEC_BRACKET6 brackets data between successive entries of a sorted R8VEC.
• R8VEC_CEILING rounds "up" (towards +oo) entries of an R8VEC.
• R8VEC_CHEBYSPACE creates a vector of Chebyshev spaced values in [A,B].
• R8VEC_CHEBY1SPACE creates Type 1 Chebyshev spaced values in [A,B].
• R8VEC_CHEBY2SPACE creates Type 2 Chebyshev spaced values in [A,B].
• R8VEC_CIRCULAR_VARIANCE returns the circular variance of an R8VEC.
• R8VEC_COMPARE compares two R8VEC's.
• R8VEC_CONCATENATE concatenates two R8VEC's.
• R8VEC_CONVOLUTION returns the convolution of two R8VEC's.
• R8VEC_CONVOLUTION_CIRC: discrete circular convolution of two R8VEC's.
• R8VEC_COPY copies an R8VEC.
• R8VEC_CORRELATION returns the correlation of two R8VEC's.
• R8VEC_COVAR computes the covariance of two vectors.
• R8VEC_CROSS_PRODUCT_2D finds the cross product of a pair of vectors in 2D.
• R8VEC_CROSS_PRODUCT_AFFINE_2D finds the affine cross product in 2D.
• R8VEC_CROSS_PRODUCT_3D computes the cross product of two R8VEC's in 3D.
• R8VEC_CROSS_PRODUCT_AFFINE_3D computes the affine cross product in 3D.
• R8VEC_CUM computes the cumulutive sums of an R8VEC.
• R8VEC_CUM0 computes the cumulutive sums of an R8VEC.
• R8VEC_DIF computes coefficients for estimating the N-th derivative.
• R8VEC_DIFF_DOT_PRODUCT: dot product of a pair of R8VEC differences.
• R8VEC_DIFF_NORM returns the L2 norm of the difference of R8VEC's.
• R8VEC_DIFF_NORM_L1 returns the L1 norm of the difference of R8VEC's.
• R8VEC_DIFF_NORM_L2 returns the L2 norm of the difference of R8VEC's.
• R8VEC_DIFF_NORM_LI returns the L-oo norm of the difference of R8VEC's.
• R8VEC_DIFF_NORM_SQUARED: square of the L2 norm of the difference of R8VEC's.
• R8VEC_DIRECT_PRODUCT creates a direct product of R8VEC's.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• R8VEC_DISTANCE returns the Euclidean distance between two R8VEC's.
• R8VEC_DISTINCT is true if the entries in an R8VEC are distinct.
• R8VEC_DOT_PRODUCT finds the dot product of a pair of R8VEC's.
• R8VEC_DOT_PRODUCT_AFFINE computes the affine dot product V1-V0 * V2-V0.
• R8VEC_ENTROPY computes the entropy of an R8VEC.
• R8VEC_EQ is true if two R8VECs are equal.
• R8VEC_EVEN returns an R8VEC of evenly spaced values.
• R8VEC_EVEN_SELECT returns the I-th of N evenly spaced values in [ XLO, XHI ].
• R8VEC_EVEN2 linearly interpolates new numbers into an R8VEC.
• R8VEC_EVEN2_SELECT returns the I-th of N evenly spaced midpoint values.
• R8VEC_EVEN3 evenly interpolates new data into an R8VEC.
• R8VEC_EXPAND_LINEAR linearly interpolates new data into an R8VEC.
• R8VEC_EXPAND_LINEAR2 linearly interpolates new data into an R8VEC.
• R8VEC_FIRST_INDEX indexes the first occurrence of values in an R8VEC.
• R8VEC_FLOOR rounds "down" (towards -oo) entries of an R8VEC.
• R8VEC_FRAC searches for the K-th smallest entry in an R8VEC.
• R8VEC_FRACTION returns the fraction parts of an R8VEC.
• R8VEC_GT == ( A1 > A2 ) for R8VEC's.
• R8VEC_HEAP_A reorders an R8VEC into an ascending heap.
• R8VEC_HEAP_D reorders an R8VEC into an descending heap.
• R8VEC_HEAP_D_EXTRACT: extract maximum from a heap descending sorted R8VEC.
• R8VEC_HEAP_D_INSERT inserts a value into a heap descending sorted R8VEC.
• R8VEC_HEAP_D_MAX returns the maximum value in a heap descending sorted R8VEC.
• R8VEC_HISTOGRAM histograms an R8VEC.
• R8VEC_HOUSE_COLUMN defines a Householder premultiplier that "packs" a column.
• R8VEC_I4VEC_DOT_PRODUCT finds the dot product of an R8VEC and an I4VEC.
• R8VEC_IN_01 is TRUE if the entries of an R8VEC are in the range [0,1].
• R8VEC_IN_AB is TRUE if the entries of an R8VEC are in the range [A,B].
• R8VEC_INDEX_DELETE_ALL deletes a value from an indexed sorted R8VEC.
• R8VEC_INDEX_DELETE_DUPES deletes duplicates from an indexed sorted R8VEC.
• R8VEC_INDEX_DELETE_ONE deletes one copy of a value from indexed sorted R8VEC.
• R8VEC_INDEX_INSERT inserts a value in an indexed sorted R8VEC.
• R8VEC_INDEX_INSERT_UNIQUE inserts a unique value in an indexed sorted R8VEC.
• R8VEC_INDEX_ORDER sorts an R8VEC using an index vector.
• R8VEC_INDEX_SEARCH searches for a value in an indexed sorted R8VEC.
• R8VEC_INDEX_SORT_UNIQUE creates a sorted unique index for an R8VEC.
• R8VEC_INDEX_SORTED_RANGE: search index sorted vector for elements in a range.
• R8VEC_INDEXED_HEAP_D creates a descending heap from an indexed R8VEC.
• R8VEC_INDEXED_HEAP_D_EXTRACT: extract from heap descending indexed R8VEC.
• R8VEC_INDEXED_HEAP_D_INSERT: insert value into heap descending indexed R8VEC.
• R8VEC_INDEXED_HEAP_D_MAX: maximum value in heap descending indexed R8VEC.
• R8VEC_INDICATOR0 sets an R8VEC to the indicator vector (0,1,2,...).
• R8VEC_INDICATOR1 sets an R8VEC to the indicator vector (1,2,3,...).
• R8VEC_INSERT inserts a value into an R8VEC.
• R8VEC_INSIGNIFICANT determines if an R8VEC is insignificant.
• R8VEC_IS_INT is TRUE if the entries of an R8VEC are integers.
• R8VEC_IS_NONNEGATIVE is TRUE if all the entries of an R8VEC are nonnegative.
• R8VEC_IS_ZERO is TRUE if all the entries of an R8VEC are zero.
• R8VEC_LEGENDRE creates a vector of Legendre-spaced values.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_LINSPACE2 creates a vector of linearly spaced values.
• R8VEC_LT == ( A1 < A2 ) for R8VEC's.
• R8VEC_MAX returns the maximum value in an R8VEC.
• R8VEC_MAX_ABS_INDEX: index of the maximum absolute value in an R8VEC.
• R8VEC_MAX_INDEX returns the index of the maximum value in an R8VEC.
• R8VEC_MEAN returns the mean of an R8VEC.
• R8VEC_MEAN_GEOMETRIC returns the geometric mean of an R8VEC.
• R8VEC_MEDIAN returns the median of an unsorted R8VEC.
• R8VEC_MESH_2D creates a 2D mesh from X and Y vectors.
• R8VEC_MIDSPACE creates a vector of linearly spaced values.
• R8VEC_MIN returns the minimum value of an R8VEC.
• R8VEC_MIN_INDEX returns the index of the minimum value in an R8VEC.
• R8VEC_MIN_POS returns the minimum positive value of an R8VEC.
• R8VEC_MIRROR_NEXT steps through all sign variations of an R8VEC.
• R8VEC_NEGATIVE_STRICT: every element of an R8VEC is strictly negative.
• R8VEC_NINT rounds entries of an R8VEC.
• R8VEC_NORM returns the L2 norm of an R8VEC.
• R8VEC_NORM_AFFINE returns the affine norm of an R8VEC.
• R8VEC_NORM_L0 returns the l0 "norm" of an R8VEC.
• R8VEC_NORM_L1 returns the L1 norm of an R8VEC.
• R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
• R8VEC_NORM_LI returns the L-oo norm of an R8VEC.
• R8VEC_NORM_LP returns the LP norm of an R8VEC.
• R8VEC_NORM_SQUARED returns the square of the L2 norm of an R8VEC.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_NORMALIZE normalizes an R8VEC in the Euclidean norm.
• R8VEC_NORMALIZE_L1 normalizes an R8VEC to have unit sum.
• R8VEC_NORMSQ returns the square of the L2 norm of an R8VEC.
• R8VEC_NORMSQ_AFFINE returns the affine squared norm of an R8VEC.
• R8VEC_ORDER_TYPE determines if R8VEC is (non)strictly ascending/descending.
• R8VEC_PART_QUICK_A reorders an R8VEC as part of a quick sort.
• R8VEC_PERMUTE permutes an R8VEC in place.
• R8VEC_PERMUTE_CYCLIC performs a cyclic permutation of an R8VEC.
• R8VEC_PERMUTE_UNIFORM randomly permutes an R8VEC.
• R8VEC_POLARIZE decomposes an R8VEC into normal and parallel components.
• R8VEC_POSITIVE_STRICT: every element of an R8VEC is strictly positive.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_PRINT_16 prints an R8VEC to 16 decimal places.
• R8VEC_PRINT_PART prints "part" of an R8VEC.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• R8VEC_PRINT2 prints out an R8VEC.
• R8VEC_PRODUCT returns the product of the entries of an R8VEC.
• R8VEC_RANGE finds the range of Y's within a restricted X range.
• R8VEC_RANGE_2 updates a range to include a new array.
• R8VEC_REVERSE reverses the elements of an R8VEC.
• R8VEC_RMS returns the RMS norm of an R8VEC.
• R8VEC_ROTATE "rotates" the entries of an R8VEC in place.
• R8VEC_RUNNING_AVERAGE computes the running average of an R8VEC.
• R8VEC_RUNNING_SIGN3 computes the running threeway sign of an R8VEC.
• R8VEC_RUNNING_SUM computes the running sum of an R8VEC.
• R8VEC_SCALAR_TRIPLE_PRODUCT computes the scalar triple product.
• R8VEC_SCALE multiplies an R8VEC by a scale factor.
• R8VEC_SEARCH_BINARY_A searches an ascending sorted R8VEC.
• R8VEC_SHIFT performs a shift on an R8VEC.
• R8VEC_SHIFT_CIRCULAR performs a circular shift on an R8VEC.
• R8VEC_SORT_BUBBLE_A ascending sorts an R8VEC using bubble sort.
• R8VEC_SORT_BUBBLE_D descending sorts an R8VEC using bubble sort.
• R8VEC_SORT_HEAP_A ascending sorts an R8VEC using heap sort.
• R8VEC_SORT_HEAP_D descending sorts an R8VEC using heap sort.
• R8VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8VEC.
• R8VEC_SORT_HEAP_INDEX_D does an indexed heap descending sort of an R8VEC.
• R8VEC_SORT_INSERT_A ascending sorts an R8VEC using an insertion sort.
• R8VEC_SORT_INSERT_INDEX_A ascending index sorts an R8VEC using insertion.
• R8VEC_SORT_INSERT_INDEX_D descending index sorts an R8VEC using insertion.
• R8VEC_SORT_QUICK_A ascending sorts an R8VEC using quick sort.
• R8VEC_SORT_SHELL_A ascending sorts an R8VEC using Shell's sort.
• R8VEC_SORT2_A ascending sorts an R8VEC and adjusts an associated R8VEC.
• R8VEC_SORTED_MERGE_A merges two ascending sorted R8VEC's.
• R8VEC_SORTED_NEAREST returns the nearest element in a sorted R8VEC.
• R8VEC_SORTED_RANGE searches a sorted vector for elements in a range.
• R8VEC_SORTED_SPLIT "splits" a sorted R8VEC, given a splitting value.
• R8VEC_SORTED_UNDEX returns unique sorted indexes for a sorted R8VEC.
• R8VEC_SORTED_UNIQUE keeps the unique elements in a sorted R8VEC.
• R8VEC_SORTED_UNIQUE_COUNT counts the unique elements in a sorted R8VEC.
• R8VEC_SORTED_UNIQUE_HIST histograms the unique elements of a sorted R8VEC.
• R8VEC_SPLIT "splits" an unsorted R8VEC based on a splitting value.
• R8VEC_STD returns the standard deviation of an R8VEC.
• R8VEC_STEP evaluates a unit step function.
• R8VEC_STUTTER makes a "stuttering" copy of an R8VEC.
• R8VEC_SUM returns the sum of the entries of an R8VEC.
• R8VEC_SWAP swaps the entries of two R8VECs.
• R8VEC_TRANSPOSE_PRINT prints an R8VEC "transposed".
• R8VEC_UNDEX returns unique sorted indexes for an R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_ABVEC returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_UNIT generates a uniformly random unit vector.
• R8VEC_UNIQUE_COUNT counts the unique elements in an unsorted R8VEC.
• R8VEC_UNIQUE_INDEX indexes the unique occurrence of values in an R8VEC.
• R8VEC_VARIANCE returns the variance of an R8VEC.
• R8VEC_VECTOR_TRIPLE_PRODUCT computes the vector triple product.
• R8VEC_WRITE writes an R8VEC to a file.
• R8VEC_ZEROS zeroes out an R8VEC.
• R8VEC2_COMPARE compares two entries in an R8VEC2.
• R8VEC2_PRINT prints an R8VEC2.
• R8VEC2_PRINT_SOME prints "some" of an R8VEC2.
• R8VEC2_SORT_A ascending sorts an R8VEC2.
• R8VEC2_SORT_D descending sorts an R8VEC2.
• R8VEC2_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8VEC2.
• R8VEC2_SORTED_UNIQUE keeps unique elements in a sorted R8VEC2.
• R8VEC2_SORTED_UNIQUE_INDEX indexes unique elements in a sorted R8VEC2.
• R8VEC2_SUM_MAX_INDEX returns the index of the maximum sum of two R8VEC's.
• R8VEC3_PRINT prints an R8VEC3.
• ROOTS_TO_R8POLY converts polynomial roots to polynomial coefficients.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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Last revised on 28 February 2016.