R8LIB
A Double Precision Real Arithmetic Utility Library

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

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

R8LIB is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a PYTHON version.

Related Programs:

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

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

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

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

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

SUBPAK, a C++ library which contains many utility routines;

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,
   Academic Press, 1978,
   ISBN: 0-12-519260-6,
   LC: QA164.N54.

Source Code:

• r8lib.cpp, the source code;
• r8lib.hpp, the include file.
• r8lib.sh, commands to compile the source code;

Examples and Tests:

• r8lib_prb.cpp, the calling program;
• r8lib_prb.sh, commands to compile, link and run the calling program;
• r8lib_prb_output.txt, the output file.

List of Routines:

• GAMMA_VALUES returns some values of the Gamma function.
• GAMMA_LOG_VALUES returns some values of the Log Gamma function.
• I4_LOG_10 returns the integer part of the logarithm base 10 of an I4.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_POWER returns the value of I^J.
• I4_SIGN returns the sign of an I4.
• 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 I4 interval to an R8 interval.
• I4VEC_COPY copies an I4VEC.
• I4VEC_INDICATOR_NEW sets an I4VEC to the indicator vector.
• I4VEC_PERMUTE permutes an I4VEC in place.
• I4VEC_PRINT prints an I4VEC.
• I4VEC_ZERO zeroes an I4VEC.
• I4VEC_ZERO_NEW creates and zeroes an I4VEC.
• LEGENDRE_ZEROS returns the zeros of the Legendre polynomial of degree N.
• PERM_CHECK checks that a vector represents a permutation.
• PERM_UNIFORM_NEW selects a random permutation of N objects.
• R4_NINT returns the nearest integer to an R4.
• R8_ABS returns the absolute value of an R8.
• R8_ACOS computes the arc cosine function, with argument truncation.
• R8_ADD adds two R8's.
• R8_AINT truncates an R8 argument to an integer.
• R8_ASIN computes the arc sine function, with argument truncation.
• R8_ATAN computes the inverse tangent of the ratio Y / X.
• R8_CAS returns the "casine" of an R8.
• R8_CEILING rounds an R8 up to the nearest integral R8.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_CHOP chops an R8 to a given number of binary places.
• R8_CSC returns the cosecant of X.
• R8_CUBE_ROOT returns the cube root of an R8.
• R8_DIFF computes (X-Y) 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_EPSILON returns the R8 roundoff unit.
• R8_EPSILON_COMPUTE computes the R8 roundoff unit.
• R8_EXP computes the exponential function, avoiding overflow and underflow.
• R8_FACTORIAL computes the factorial of N.
• R8_FACTORIAL2 computes the double factorial function.
• R8_FLOOR rounds an R8 down 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 an R8.
• R8_GAMMA_LOG evaluates the logarithm of the gamma function.
• R8_HUGE returns a "huge" 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_10 returns the logarithm base 10 of the absolute value of an R8.
• R8_LOG_2 returns the logarithm base 2 of the absolute value 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 value.
• R8_NINT returns the nearest integer to an R8.
• R8_NORMAL returns a scaled pseudonormal R8.
• R8_NORMAL_01 samples the standard normal probability distribution.
• 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 an integer power of an R8.
• R8_POWER_FAST computes the P-th power of R, for real R and integer P.
• R8_PYTHAG computes sqrt ( A*A + B*B ), avoiding overflow and underflow.
• R8_REVERSE_BYTES reverses the bytes in an R8.
• R8_ROUND rounds an R8 to the nearest integral value.
• R8_ROUND_I4 rounds an R8, 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_SECH evaluates the hyperbolic secant, while avoiding COSH overflow.
• R8_SIGN returns the 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_SIGN2 returns the first argument with the sign of the second.
• R8_SQRT_I4 returns the square root of an I4 as an R8.
• R8_SUM returns the sum of two R8's.
• R8_SWAP switches two R8's.
• R8_SWAP3 swaps three R8's.
• R8_TINY returns a "tiny" R8.
• R8_TO_DHMS converts an R8 day value into days, hours, minutes, seconds.
• R8_TO_I4 maps real X in [XMIN, XMAX] to integer IX in [IXMIN, IXMAX].
• R8_TO_R8_DISCRETE maps R to RD in [RMIN, RMAX] with NR possible values.
• 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 of a real scalar argument.
• 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_PRINT prints an R82.
• R82_UNIFORM_AB returns a random R82 value in a given range.
• 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.
• R82VEC_MAX returns the maximum value in an R82VEC.
• R82VEC_MIN returns the minimum value in an R82VEC.
• R82VEC_ORDER_TYPE finds if an R82VEC is (non)strictly ascending/descending.
• R82VEC_PART_QUICK_A reorders an R82VEC as part of a quick sort.
• R82VEC_PERMUTE permutes an R82VEC in place.
• R82VEC_PRINT prints an R82VEC.
• R82VEC_PRINT_PART prints "part" of an R82VEC.
• R82VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R82VEC.
• R82VEC_SORT_QUICK_A ascending sorts an R82VEC using quick sort.
• R83_NORM returns the Euclidean norm of an R83.
• R83VEC_MAX returns the maximum value in an R83VEC.
• R83VEC_MIN returns the minimum value in an R83VEC.
• R83VEC_PART_QUICK_A reorders an R83VEC as part of a quick sort.
• R83VEC_PRINT_PART prints "part" of an R83VEC.
• R83VEC_SORT_QUICK_A ascending sorts an R83VEC using quick sort.
• R8BLOCK_DELETE frees memory associated with an R8BLOCK.
• R8BLOCK_EXPAND_LINEAR linearly interpolates new data into a 3D block.
• R8BLOCK_NEW allocates a new R8BLOCK.
• R8BLOCK_PRINT prints an R8BLOCK block (a 3D matrix).
• R8BLOCK_ZERO_NEW returns a new zeroed R8BLOCK.
• R8COL_COMPARE compares two columns in an R8COL.
• R8COL_DUPLICATES generates an R8COL with some duplicate columns.
• R8COL_FIND seeks a column value in an R8COL.
• R8COL_FIRST_INDEX indexes the first occurrence of values in an R8COL.
• R8COL_INSERT inserts a column into an R8COL.
• R8COL_MAX returns the column maximums of an R8COL.
• R8COL_MAX_INDEX returns the indices of column maximums in an R8COL.
• R8COL_MAX_ONE rescales an R8COL so each column maximum is 1.
• R8COL_MEAN returns the column means of an R8COL.
• R8COL_MIN returns the column minimums of an R8COL.
• R8COL_MIN_INDEX returns the indices of column minimums in an R8COL.
• R8COL_NORMALIZE_LI normalizes an R8COL with the column infinity norm.
• R8COL_PART_QUICK_A reorders the columns of an R8COL.
• R8COL_PERMUTE permutes an R8COL in place.
• R8COL_SORT_HEAP_A ascending heapsorts an R8COL.
• R8COL_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8COL.
• R8COL_SORT_QUICK_A ascending quick sorts an R8COL.
• R8COL_SORTED_TOL_UNDEX: index tolerably unique entries of a sorted R8COL.
• R8COL_SORTED_TOL_UNIQUE keeps tolerably unique elements in a sorted R8COL.
• R8COL_SORTED_TOL_UNIQUE_COUNT counts tolerably unique elements in a sorted R8COL.
• R8COL_SORTED_UNDEX returns unique sorted indexes for a sorted R8COL.
• R8COL_SORTED_UNIQUE keeps unique elements in a sorted R8COL.
• R8COL_SORTED_UNIQUE_COUNT counts unique elements in a sorted R8COL.
• R8COL_SORTR_A ascending sorts one column of an R8COL, adjusting all entries.
• R8COL_SUM sums the columns of an R8COL.
• R8COL_SWAP swaps columns J1 and J2 of an R8COL.
• R8COL_TO_R8VEC converts an R8COL to an R8VEC.
• R8COL_TOL_UNDEX indexes tolerably unique entries of an R8COL.
• R8COL_TOL_UNIQUE_COUNT counts tolerably unique entries in an R8COL.
• R8COL_TOL_UNIQUE_INDEX indexes tolerably unique entries in an R8COL.
• R8COL_UNDEX indexes unique entries in an R8COL.
• R8COL_UNIFORM_ABVEC_NEW fills an R8COL with scaled pseudorandom numbers.
• R8COL_UNIQUE_COUNT counts unique entries in an R8COL.
• R8COL_UNIQUE_INDEX indexes unique entries in an R8COL.
• R8COL_VARIANCE returns the variances of an R8COL.
• R8INT_TO_R8INT maps one R8 interval to another.
• R8INT_TO_I4INT maps an R8 interval to an integer interval.
• R8MAT_ADD computes C = alpha * A + beta * B for R8MAT's.
• R8MAT_ADD_NEW computes C = alpha * A + beta * B for R8MAT's.
• R8MAT_AMAX returns the maximum absolute value entry of an R8MAT.
• R8MAT_BORDER_ADD adds a "border" to an R8MAT.
• R8MAT_BORDER_CUT cuts the "border" of an R8MAT.
• R8MAT_CHOLESKY_FACTOR computes the Cholesky factor of a symmetric R8MAT.
• R8MAT_CHOLESKY_SOLVE solves a Cholesky factored linear system A * x = b.
• R8MAT_CHORESKY_FACTOR computes the "Choresky" factor of a symmetric R8MAT.
• R8MAT_COPY copies one R8MAT to another.
• R8MAT_COPY_NEW copies one R8MAT to a "new" R8MAT.
• R8MAT_DELETE frees memory associated with an R8MAT.
• 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_ADD_SCALAR adds a scalar to the diagonal of an R8MAT.
• R8MAT_DIAG_ADD_VECTOR adds a vector to the diagonal of an 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_DIF_FRO returns the Frobenius norm of the difference of R8MAT's.
• R8MAT_EXPAND_LINEAR linearly interpolates new data into an R8MAT.
• R8MAT_EXPAND_LINEAR2 expands an R8MAT by linear interpolation.
• R8MAT_FLIP_COLS swaps the columns of an R8MAT.
• R8MAT_FLIP_ROWS swaps the rows of an R8MAT.
• R8MAT_FS factors and solves a system with one right hand side.
• R8MAT_FS_NEW factors and solves a system with one right hand side.
• R8MAT_FSS factors and solves a system with multiple right hand sides.
• R8MAT_FSS_NEW 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_AXH_NEW 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 sets the square matrix A to the identity.
• R8MAT_IDENTITY_NEW returns an identity matrix.
• R8MAT_IN_01 is TRUE if the entries of an R8MAT are in the range [0,1].
• R8MAT_INDICATOR_NEW sets up an "indicator" R8MAT.
• R8MAT_INSIGNIFICANT determines if an R8MAT is insignificant.
• R8MAT_INVERSE_2D inverts a 2 by 2 matrix using Cramer's rule.
• R8MAT_INVERSE_3D inverts a 3 by 3 matrix using Cramer's rule.
• R8MAT_INVERSE_4D inverts a 4 by 4 matrix using Cramer's rule.
• R8MAT_IS_IDENTITY determines if an R8MAT is the identity.
• R8MAT_IS_SYMMETRIC checks an R8MAT for symmetry.
• R8MAT_JAC estimates a dense jacobian matrix of the function FX.
• 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 matrix.
• R8MAT_MAXROW_MINCOL gets the maximum row minimum column of an M by N matrix.
• R8MAT_MIN returns the minimum entry of an 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 matrix.
• R8MAT_MINROW_MAXCOL gets the minimum row maximum column of an M by N matrix.
• R8MAT_MINVM computes inverse(A) * B for R8MAT's.
• R8MAT_MINVM_NEW returns inverse(A) * B for R8MAT's.
• R8MAT_MM multiplies two matrices.
• R8MAT_MM_NEW multiplies two matrices.
• R8MAT_MMT_NEW computes C = A * B'.
• R8MAT_MTM_NEW computes C = A' * B.
• R8MAT_MTV multiplies a transposed matrix times a vector.
• R8MAT_MTV_NEW multiplies a transposed matrix times a vector.
• R8MAT_MV multiplies a matrix times a vector.
• R8MAT_MV_NEW multiplies a matrix times a vector.
• R8MAT_MXM multiplies two matrices.
• R8MAT_MXM_NEW multiplies two matrices.
• R8MAT_NEW allocates a new R8MAT.
• R8MAT_NINT rounds the entries of 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_NEW 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_NEW returns a random orthogonal matrix.
• R8MAT_PLOT "plots" an R8MAT.
• R8MAT_PLOT_SYMBOL returns a symbol for entries 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 a matrix.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_REF computes the row echelon form of a matrix.
• R8MAT_RMS returns the RMS norm of an R8MAT.
• R8MAT_RREF computes the reduced row echelon form of a matrix.
• R8MAT_SCALE multiplies an R8MAT by a scalar.
• R8MAT_SIGNIFICANT determines if an R8MAT is significant compared to another.
• 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_SUM returns the sum 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_R8PLU factors a general matrix.
• R8MAT_TRACE computes the trace of an R8MAT.
• R8MAT_TRANSPOSE_IN_PLACE transposes a square R8MAT in place.
• R8MAT_TRANSPOSE_NEW returns the transpose of an R8MAT.
• 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_U1_INVERSE inverts a unit upper triangular R8MAT.
• R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
• R8MAT_UNIFORM_01_NEW returns a unit pseudorandom R8MAT.
• R8MAT_UNIFORM_AB returns a scaled pseudorandom R8MAT.
• R8MAT_UNIFORM_AB_NEW returns a new scaled pseudorandom R8MAT.
• R8MAT_UNIFORM_ABVEC returns a scaled pseudorandom R8MAT.
• R8MAT_UNIFORM_ABVEC_NEW returns a new scaled pseudorandom R8MAT.
• R8MAT_VAND2 returns the N by N row Vandermonde matrix A.
• R8MAT_ZERO zeroes an R8MAT.
• R8MAT_ZERO_NEW returns a new zeroed R8MAT.
• R8PLU_DET computes the determinant of a real PLU matrix.
• R8PLU_INVERSE computes the inverse of a real PLU matrix.
• R8PLU_MUL computes A * x using the PLU factors of A.
• R8PLU_SOL 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 evaluates a polynomial.
• R8POLY_VALUE_HORNER evaluates a polynomial using Horner's method.
• R8POLY_VALUE_OLD evaluates an R8POLY.
• 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 the extremal point of a parabola determined by three points.
• 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 function through 3 points in a table.
• R8PP_DELETE frees the memory set aside by R8PP_NEW.
• R8PP_NEW allocates a new R8PP.
• R8R8_COMPARE compares two R8R8's.
• R8R8_PRINT prints an R8R8.
• R8R8R8_COMPARE compares two R8R8R8's.
• R8R8R8VEC_INDEX_INSERT_UNIQUE inserts a unique R8R8R8 value in an indexed sorted list.
• R8R8R8VEC_INDEX_SEARCH searches for an R8R8R8 value in an indexed sorted list.
• R8R8VEC_INDEX_INSERT_UNIQUE inserts a unique R8R8 value in an indexed sorted list.
• R8R8VEC_INDEX_SEARCH searches for an R8R8 value in an indexed sorted list.
• R8ROW_COMPARE compares two rows in an R8ROW.
• R8ROW_MAX returns the row maximums of an R8ROW.
• R8ROW_MEAN returns the row means of an R8ROW.
• R8ROW_MIN returns the row minimums of an R8ROW.
• R8ROW_SORT_HEAP_A ascending heapsorts an R8ROW.
• R8ROW_SUM returns the sums of the rows of an R8ROW.
• R8ROW_SWAP swaps two rows of an R8ROW.
• R8ROW_TO_R8VEC converts an R8ROW into an R8VEC.
• R8ROW_VARIANCE returns the variances of an R8ROW.
• R8SLMAT_PRINT prints a strict lower triangular R8MAT.
• R8VEC_01_TO_AB shifts and rescales data to lie within given bounds.
• R8VEC_AB_TO_01 shifts and rescales data to lie within [0,1].
• R8VEC_AB_TO_CD shifts and rescales data to lie within a given pair of bounds.
• R8VEC_ADD adds one R8VEC 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_BIN computes bins based on a given R8VEC.
• R8VEC_BRACKET searches a sorted array for successive brackets of a value.
• R8VEC_BRACKET2 searches a sorted array for successive brackets of a value.
• R8VEC_BRACKET3 finds the interval containing or nearest a given value.
• R8VEC_BRACKET4 finds the interval containing or nearest a given value.
• R8VEC_BRACKET5 brackets data between successive entries of a sorted R8VEC.
• R8VEC_BRACKET6 brackets data between successive entries of a sorted R8VEC.
• R8VEC_CHEBYSPACE_NEW creates a vector of Chebyshev spaced values in [A,B].
• R8VEC_CHEBY1SPACE_NEW creates Type 1 Chebyshev spaced values in [A,B].
• R8VEC_CHEBY2SPACE_NEW 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_CONVOLUTION returns the convolution of two R8VEC's.
• R8VEC_CONVOLUTION_CIRC returns the discrete circular convolution of two R8VEC's.
• R8VEC_COPY copies an R8VEC.
• R8VEC_COPY_NEW copies an R8VEC to a new 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 R8VEC's 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_NEW computes the cumulutive sums of an R8VEC.
• R8VEC_CUM0_NEW computes the cumulutive sums of an R8VEC.
• R8VEC_DIF computes coefficients for estimating the N-th derivative.
• 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_DIVIDE divides an R8VEC by a nonzero scalar.
• R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
• R8VEC_DOT_PRODUCT_AFFINE computes the affine dot product.
• R8VEC_EQ is true if every pair of entries in two R8VEC's is equal.
• R8VEC_EVEN returns an R8VEC of values evenly spaced between ALO and AHI.
• R8VEC_EVEN_NEW returns an R8VEC of values evenly spaced between ALO and AHI.
• R8VEC_EVEN_SELECT returns the I-th of N evenly spaced values in [ XLO, XHI ].
• R8VEC_EVEN2 linearly interpolates new numbers into an R8VECa.
• 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_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 two R8VEC's.
• R8VEC_HEAP_A reorders an R8VEC into a ascending heap.
• R8VEC_HEAP_D reorders an R8VEC into a descending heap.
• R8VEC_HISTOGRAM histograms an R8VEC.
• R8VEC_HOUSE_COLUMN defines a Householder premultiplier that "packs" a column.
• R8VEC_I4VEC_DOT_PRODUCT computes 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 all occurrences of a value from an indexed sorted list.
• R8VEC_INDEX_DELETE_DUPES deletes duplicates from an indexed sorted list.
• R8VEC_INDEX_DELETE_ONE deletes one copy of a value from an indexed sorted list.
• R8VEC_INDEX_INSERT inserts a value in an indexed sorted list.
• R8VEC_INDEX_INSERT_UNIQUE inserts a unique value in an indexed sorted list.
• 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 sort 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_INDICATOR sets an R8VEC to the indicator vector.
• R8VEC_INDICATOR_NEW 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 an R8VEC is integral.
• R8VEC_IS_NONNEGATIVE is true if all entries in an R8VEC are nonnegative.
• R8VEC_IS_ZERO is true if the entries in an R8VEC are all zero.
• R8VEC_LEGENDRE_NEW creates a vector of Chebyshev spaced values.
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• R8VEC_LINSPACE2_NEW creates a vector of linearly spaced values.
• R8VEC_LT == ( A1 < A2 ) for two R8VEC's.
• R8VEC_MASK_PRINT prints a masked R8VEC.
• R8VEC_MAX returns the value of the maximum element in an R8VEC.
• R8VEC_MAX_ABS_INDEX returns the 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_MEDIAN returns the median of an unsorted R8VEC.
• R8VEC_MIDSPACE_NEW creates a vector of linearly spaced values.
• R8VEC_MIN returns the value of the minimum element in 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: all entries of R8VEC are strictly negative.
• R8VEC_NINT rounds the entries of an R8VEC.
• R8VEC_NORM returns the L2 norm of an R8VEC.
• R8VEC_NORM_AFFINE returns the affine L2 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_NORMAL_NEW returns a scaled pseudonormal R8VEC.
• R8VEC_NORMAL_01_NEW returns a unit pseudonormal R8VEC.
• R8VEC_NORMALIZE normalizes an R8VEC.
• R8VEC_NORMALIZE_L1 normalizes an R8VEC to have unit sum.
• R8VEC_NORMSQ returns the squared L2 norm of an R8VEC.
• R8VEC_NORMSQ_AFFINE returns the squared affine L2 norm of an R8VEC.
• R8VEC_ONES_NEW creates a vector of 1's.
• R8VEC_ORDER_TYPE determines if an 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: all entries of R8VEC are strictly positive.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_PRINT_PART prints "part" of an R8VEC.
• R8VEC_PRINT_SOME prints "some" of 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 R8VEC
• 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_SCALAR_TRIPLE_PRODUCT computes the scalar triple product.
• R8VEC_SCALE multiples 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_A_NEW does an indexed heap ascending sort of an R8VEC
• R8VEC_SORT_HEAP_INDEX_D_NEW does an indexed heap descending sort of an R8VEC.
• R8VEC_SORT_HEAP_INDEX_D_NEW does an indexed heap descending sort of an R8VEC.
• R8VEC_SORT_HEAP_MASK_A: indexed heap ascending sort of a masked 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_QUICK_A ascending sorts an R8VEC using quick sort.
• R8VEC_SORT_SHELL_A ascending sorts an R8VEC using Shell's sort.
• 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 finds the unique elements in a sorted R8VEC.
• R8VEC_SORTED_UNIQUE_COUNT counts unique elements in a sorted R8VEC.
• R8VEC_SORTED_UNIQUE_HIST histograms 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_STUTTER makes a "stuttering" copy of an R8VEC.
• R8VEC_STUTTER_NEW makes a "stuttering" copy of an R8VEC.
• R8VEC_SUM returns the sum of an R8VEC.
• R8VEC_SWAP swaps the entries of two R8VEC's.
• 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_01_NEW returns a new unit pseudorandom R8VEC.
• R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_AB_NEW returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_ABVEC returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_ABVEC_NEW returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_UNIT_NEW generates a 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_ZERO zeroes an R8VEC.
• R8VEC_ZERO_NEW creates and zeroes an R8VEC.
• R8VEC2_COMPARE compares two elements of 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 the unique elements in an 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 a triple of real vectors.
• 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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