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<html>
<head>
<title>
R8LIB - A Double Precision Real Arithmetic Utility Library
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
R8LIB <br> A Double Precision Real Arithmetic Utility Library
</h1>
<hr>
<p>
<b>R8LIB</b>
is a C++ library which
contains a number of utility routines for "R8" or "double precision real"
arithmetic.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>R8LIB</b> is available in
<a href = "../../c_src/r8lib/r8lib.html">a C version</a> and
<a href = "../../cpp_src/r8lib/r8lib.html">a C++ version</a> and
<a href = "../../f77_src/r8lib/r8lib.html">a FORTRAN77 version</a> and
<a href = "../../f_src/r8lib/r8lib.html">a FORTRAN90 version</a> and
<a href = "../../m_src/r8lib/r8lib.html">a MATLAB version</a> and
<a href = "../../py_src/r8lib/r8lib.html">a PYTHON version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../cpp_src/c4lib/c4lib.html">
C4LIB</a>,
a C++ library which
implements certain elementary functions for "C4"
or single precision complex variables;
</p>
<p>
<a href = "../../cpp_src/c8lib/c8lib.html">
C8LIB</a>,
a C++ library which
implements certain elementary functions for "C8"
or double precision complex variables;
</p>
<p>
<a href = "../../cpp_src/i4lib/i4lib.html">
I4LIB</a>,
a C++ library which
contains many utility routines, using "I4" or "single precision integer"
arithmetic.
</p>
<p>
<a href = "../../cpp_src/i8lib/i8lib.html">
I8LIB</a>,
a C++ library which
contains many utility routines, using "I8" or "double precision integer"
arithmetic.
</p>
<p>
<a href = "../../cpp_src/r4lib/r4lib.html">
R4LIB</a>,
a C++ library which
contains many utility routines, using "R4" or
"single precision real" arithmetic.
</p>
<p>
<a href = "../../cpp_src/subpak/subpak.html">
SUBPAK</a>,
a C++ library which
contains many utility routines;
</p>
<h3 align = "center">
References:
</h3>
<p>
<ol>
<li>
Thomas Cormen, Charles Leiserson, Ronald Rivest,<br>
Introduction to Algorithms,<br>
MIT Press, 2001,<br>
ISBN: 0262032937,<br>
LC: QA76.C662.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "r8lib.cpp">r8lib.cpp</a>, the source code;
</li>
<li>
<a href = "r8lib.hpp">r8lib.hpp</a>, the include file.
</li>
<li>
<a href = "r8lib.sh">r8lib.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "r8lib_prb.cpp">r8lib_prb.cpp</a>, the calling program;
</li>
<li>
<a href = "r8lib_prb.sh">r8lib_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "r8lib_prb_output.txt">r8lib_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>GAMMA_VALUES</b> returns some values of the Gamma function.
</li>
<li>
<b>GAMMA_LOG_VALUES</b> returns some values of the Log Gamma function.
</li>
<li>
<b>I4_LOG_10</b> returns the integer part of the logarithm base 10 of an I4.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4_SIGN</b> returns the sign of an I4.
</li>
<li>
<b>I4_UNIFORM_AB</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>I4INT_TO_R8INT</b> maps an I4 interval to an R8 interval.
</li>
<li>
<b>I4VEC_COPY</b> copies an I4VEC.
</li>
<li>
<b>I4VEC_INDICATOR_NEW</b> sets an I4VEC to the indicator vector.
</li>
<li>
<b>I4VEC_PERMUTE</b> permutes an I4VEC in place.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>I4VEC_ZERO</b> zeroes an I4VEC.
</li>
<li>
<b>I4VEC_ZERO_NEW</b> creates and zeroes an I4VEC.
</li>
<li>
<b>LEGENDRE_ZEROS</b> returns the zeros of the Legendre polynomial of degree N.
</li>
<li>
<b>PERM_CHECK</b> checks that a vector represents a permutation.
</li>
<li>
<b>PERM_UNIFORM_NEW</b> selects a random permutation of N objects.
</li>
<li>
<b>R4_NINT</b> returns the nearest integer to an R4.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_ACOS</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>R8_ADD</b> adds two R8's.
</li>
<li>
<b>R8_AINT</b> truncates an R8 argument to an integer.
</li>
<li>
<b>R8_ASIN</b> computes the arc sine function, with argument truncation.
</li>
<li>
<b>R8_ATAN</b> computes the inverse tangent of the ratio Y / X.
</li>
<li>
<b>R8_CAS</b> returns the "casine" of an R8.
</li>
<li>
<b>R8_CEILING</b> rounds an R8 up to the nearest integral R8.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_CHOP</b> chops an R8 to a given number of binary places.
</li>
<li>
<b>R8_CSC</b> returns the cosecant of X.
</li>
<li>
<b>R8_CUBE_ROOT</b> returns the cube root of an R8.
</li>
<li>
<b>R8_DIFF</b> computes (X-Y) to a specified accuracy.
</li>
<li>
<b>R8_DIGIT</b> returns a particular decimal digit of an R8.
</li>
<li>
<b>R8_DIVIDE_I4</b> returns an I4 fraction as an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_EPSILON_COMPUTE</b> computes the R8 roundoff unit.
</li>
<li>
<b>R8_EXP</b> computes the exponential function, avoiding overflow and underflow.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_FLOOR</b> rounds an R8 down to the nearest integral R8.
</li>
<li>
<b>R8_FRACTION</b> uses real arithmetic on an integer ratio.
</li>
<li>
<b>R8_FRACTIONAL</b> returns the fractional part of an R8.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for an R8.
</li>
<li>
<b>R8_GAMMA_LOG</b> evaluates the logarithm of the gamma function.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_HYPOT</b> returns the value of sqrt ( X^2 + Y^2 ).
</li>
<li>
<b>R8_IN_01</b> is TRUE if an R8 is in the range [0,1].
</li>
<li>
<b>R8_INSIGNIFICANT</b> determines if an R8 is insignificant.
</li>
<li>
<b>R8_IS_INT</b> determines if an R8 represents an integer value.
</li>
<li>
<b>R8_LOG_10</b> returns the logarithm base 10 of the absolute value of an R8.
</li>
<li>
<b>R8_LOG_2</b> returns the logarithm base 2 of the absolute value of an R8.
</li>
<li>
<b>R8_LOG_B</b> returns the logarithm base B of an R8.
</li>
<li>
<b>R8_MANT</b> computes the "mantissa" or "fraction part" of an R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_MOD</b> returns the remainder of R8 division.
</li>
<li>
<b>R8_MODP</b> returns the nonnegative remainder of R8 division.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8 value.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8_NORMAL</b> returns a scaled pseudonormal R8.
</li>
<li>
<b>R8_NORMAL_01</b> samples the standard normal probability distribution.
</li>
<li>
<b>R8_PI</b> returns the value of PI as an R8.
</li>
<li>
<b>R8_PI_SQRT</b> returns the square root of PI as an R8.
</li>
<li>
<b>R8_POWER</b> computes an integer power of an R8.
</li>
<li>
<b>R8_POWER_FAST</b> computes the P-th power of R, for real R and integer P.
</li>
<li>
<b>R8_PYTHAG</b> computes sqrt ( A*A + B*B ), avoiding overflow and underflow.
</li>
<li>
<b>R8_REVERSE_BYTES</b> reverses the bytes in an R8.
</li>
<li>
<b>R8_ROUND</b> rounds an R8 to the nearest integral value.
</li>
<li>
<b>R8_ROUND_I4</b> rounds an R8, returning an I4.
</li>
<li>
<b>R8_ROUND2</b> rounds an R8 in base 2.
</li>
<li>
<b>R8_ROUNDB</b> rounds an R8 in a given base.
</li>
<li>
<b>R8_ROUNDX</b> rounds an R8 in base 10.
</li>
<li>
<b>R8_SECH</b> evaluates the hyperbolic secant, while avoiding COSH overflow.
</li>
<li>
<b>R8_SIGN</b> returns the sign of an R8.
</li>
<li>
<b>R8_SIGN_CHAR</b> returns a character indicating the sign of an R8.
</li>
<li>
<b>R8_SIGN_MATCH</b> is TRUE if two R8's are of the same sign.
</li>
<li>
<b>R8_SIGN_MATCH_STRICT</b> is TRUE if two R8's are of the same strict sign.
</li>
<li>
<b>R8_SIGN_OPPOSITE</b> is TRUE if two R8's are not of the same sign.
</li>
<li>
<b>R8_SIGN_OPPOSITE_STRICT</b> is TRUE if two R8's are strictly of opposite sign.
</li>
<li>
<b>R8_SIGN2</b> returns the first argument with the sign of the second.
</li>
<li>
<b>R8_SQRT_I4</b> returns the square root of an I4 as an R8.
</li>
<li>
<b>R8_SUM</b> returns the sum of two R8's.
</li>
<li>
<b>R8_SWAP</b> switches two R8's.
</li>
<li>
<b>R8_SWAP3</b> swaps three R8's.
</li>
<li>
<b>R8_TINY</b> returns a "tiny" R8.
</li>
<li>
<b>R8_TO_DHMS</b> converts an R8 day value into days, hours, minutes, seconds.
</li>
<li>
<b>R8_TO_I4</b> maps real X in [XMIN, XMAX] to integer IX in [IXMIN, IXMAX].
</li>
<li>
<b>R8_TO_R8_DISCRETE</b> maps R to RD in [RMIN, RMAX] with NR possible values.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8_UNIFORM_AB</b> returns a scaled pseudorandom R8.
</li>
<li>
<b>R8_UNSWAP3</b> unswaps three R8's.
</li>
<li>
<b>R8_WALSH_1D</b> evaluates the Walsh function of a real scalar argument.
</li>
<li>
<b>R8_WRAP</b> forces an R8 to lie between given limits by wrapping.
</li>
<li>
<b>R82_DIST_L2</b> returns the L2 distance between a pair of R82's.
</li>
<li>
<b>R82_PRINT</b> prints an R82.
</li>
<li>
<b>R82_UNIFORM_AB</b> returns a random R82 value in a given range.
</li>
<li>
<b>R82POLY2_PRINT</b> prints a second order polynomial in two variables.
</li>
<li>
<b>R82POLY2_TYPE</b> analyzes a second order polynomial in two variables.
</li>
<li>
<b>R82POLY2_TYPE_PRINT</b> prints the meaning of the output from R82POLY2_TYPE.
</li>
<li>
<b>R82VEC_MAX</b> returns the maximum value in an R82VEC.
</li>
<li>
<b>R82VEC_MIN</b> returns the minimum value in an R82VEC.
</li>
<li>
<b>R82VEC_ORDER_TYPE</b> finds if an R82VEC is (non)strictly ascending/descending.
</li>
<li>
<b>R82VEC_PART_QUICK_A</b> reorders an R82VEC as part of a quick sort.
</li>
<li>
<b>R82VEC_PERMUTE</b> permutes an R82VEC in place.
</li>
<li>
<b>R82VEC_PRINT</b> prints an R82VEC.
</li>
<li>
<b>R82VEC_PRINT_PART</b> prints "part" of an R82VEC.
</li>
<li>
<b>R82VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R82VEC.
</li>
<li>
<b>R82VEC_SORT_QUICK_A</b> ascending sorts an R82VEC using quick sort.
</li>
<li>
<b>R83_NORM</b> returns the Euclidean norm of an R83.
</li>
<li>
<b>R83VEC_MAX</b> returns the maximum value in an R83VEC.
</li>
<li>
<b>R83VEC_MIN</b> returns the minimum value in an R83VEC.
</li>
<li>
<b>R83VEC_PART_QUICK_A</b> reorders an R83VEC as part of a quick sort.
</li>
<li>
<b>R83VEC_PRINT_PART</b> prints "part" of an R83VEC.
</li>
<li>
<b>R83VEC_SORT_QUICK_A</b> ascending sorts an R83VEC using quick sort.
</li>
<li>
<b>R8BLOCK_DELETE</b> frees memory associated with an R8BLOCK.
</li>
<li>
<b>R8BLOCK_EXPAND_LINEAR</b> linearly interpolates new data into a 3D block.
</li>
<li>
<b>R8BLOCK_NEW</b> allocates a new R8BLOCK.
</li>
<li>
<b>R8BLOCK_PRINT</b> prints an R8BLOCK block (a 3D matrix).
</li>
<li>
<b>R8BLOCK_ZERO_NEW</b> returns a new zeroed R8BLOCK.
</li>
<li>
<b>R8COL_COMPARE</b> compares two columns in an R8COL.
</li>
<li>
<b>R8COL_DUPLICATES</b> generates an R8COL with some duplicate columns.
</li>
<li>
<b>R8COL_FIND</b> seeks a column value in an R8COL.
</li>
<li>
<b>R8COL_FIRST_INDEX</b> indexes the first occurrence of values in an R8COL.
</li>
<li>
<b>R8COL_INSERT</b> inserts a column into an R8COL.
</li>
<li>
<b>R8COL_MAX</b> returns the column maximums of an R8COL.
</li>
<li>
<b>R8COL_MAX_INDEX</b> returns the indices of column maximums in an R8COL.
</li>
<li>
<b>R8COL_MAX_ONE</b> rescales an R8COL so each column maximum is 1.
</li>
<li>
<b>R8COL_MEAN</b> returns the column means of an R8COL.
</li>
<li>
<b>R8COL_MIN</b> returns the column minimums of an R8COL.
</li>
<li>
<b>R8COL_MIN_INDEX</b> returns the indices of column minimums in an R8COL.
</li>
<li>
<b>R8COL_NORMALIZE_LI</b> normalizes an R8COL with the column infinity norm.
</li>
<li>
<b>R8COL_PART_QUICK_A</b> reorders the columns of an R8COL.
</li>
<li>
<b>R8COL_PERMUTE</b> permutes an R8COL in place.
</li>
<li>
<b>R8COL_SORT_HEAP_A</b> ascending heapsorts an R8COL.
</li>
<li>
<b>R8COL_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R8COL.
</li>
<li>
<b>R8COL_SORT_QUICK_A</b> ascending quick sorts an R8COL.
</li>
<li>
<b>R8COL_SORTED_TOL_UNDEX:</b> index tolerably unique entries of a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_TOL_UNIQUE</b> keeps tolerably unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_TOL_UNIQUE_COUNT</b> counts tolerably unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_UNDEX</b> returns unique sorted indexes for a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_UNIQUE</b> keeps unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SORTED_UNIQUE_COUNT</b> counts unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SORTR_A</b> ascending sorts one column of an R8COL, adjusting all entries.
</li>
<li>
<b>R8COL_SUM</b> sums the columns of an R8COL.
</li>
<li>
<b>R8COL_SWAP</b> swaps columns J1 and J2 of an R8COL.
</li>
<li>
<b>R8COL_TO_R8VEC</b> converts an R8COL to an R8VEC.
</li>
<li>
<b>R8COL_TOL_UNDEX</b> indexes tolerably unique entries of an R8COL.
</li>
<li>
<b>R8COL_TOL_UNIQUE_COUNT</b> counts tolerably unique entries in an R8COL.
</li>
<li>
<b>R8COL_TOL_UNIQUE_INDEX</b> indexes tolerably unique entries in an R8COL.
</li>
<li>
<b>R8COL_UNDEX</b> indexes unique entries in an R8COL.
</li>
<li>
<b>R8COL_UNIFORM_ABVEC_NEW</b> fills an R8COL with scaled pseudorandom numbers.
</li>
<li>
<b>R8COL_UNIQUE_COUNT</b> counts unique entries in an R8COL.
</li>
<li>
<b>R8COL_UNIQUE_INDEX</b> indexes unique entries in an R8COL.
</li>
<li>
<b>R8COL_VARIANCE</b> returns the variances of an R8COL.
</li>
<li>
<b>R8INT_TO_R8INT</b> maps one R8 interval to another.
</li>
<li>
<b>R8INT_TO_I4INT</b> maps an R8 interval to an integer interval.
</li>
<li>
<b>R8MAT_ADD</b> computes C = alpha * A + beta * B for R8MAT's.
</li>
<li>
<b>R8MAT_ADD_NEW</b> computes C = alpha * A + beta * B for R8MAT's.
</li>
<li>
<b>R8MAT_AMAX</b> returns the maximum absolute value entry of an R8MAT.
</li>
<li>
<b>R8MAT_BORDER_ADD</b> adds a "border" to an R8MAT.
</li>
<li>
<b>R8MAT_BORDER_CUT</b> cuts the "border" of an R8MAT.
</li>
<li>
<b>R8MAT_CHOLESKY_FACTOR</b> computes the Cholesky factor of a symmetric R8MAT.
</li>
<li>
<b>R8MAT_CHOLESKY_SOLVE</b> solves a Cholesky factored linear system A * x = b.
</li>
<li>
<b>R8MAT_CHORESKY_FACTOR</b> computes the "Choresky" factor of a symmetric R8MAT.
</li>
<li>
<b>R8MAT_COPY</b> copies one R8MAT to another.
</li>
<li>
<b>R8MAT_COPY_NEW</b> copies one R8MAT to a "new" R8MAT.
</li>
<li>
<b>R8MAT_DELETE</b> frees memory associated with an R8MAT.
</li>
<li>
<b>R8MAT_DET</b> computes the determinant of an R8MAT.
</li>
<li>
<b>R8MAT_DET_2D</b> computes the determinant of a 2 by 2 R8MAT.
</li>
<li>
<b>R8MAT_DET_3D</b> computes the determinant of a 3 by 3 R8MAT.
</li>
<li>
<b>R8MAT_DET_4D</b> computes the determinant of a 4 by 4 R8MAT.
</li>
<li>
<b>R8MAT_DET_5D</b> computes the determinant of a 5 by 5 R8MAT.
</li>
<li>
<b>R8MAT_DIAG_ADD_SCALAR</b> adds a scalar to the diagonal of an R8MAT.
</li>
<li>
<b>R8MAT_DIAG_ADD_VECTOR</b> adds a vector to the diagonal of an R8MAT.
</li>
<li>
<b>R8MAT_DIAG_GET_VECTOR</b> gets the value of the diagonal of an R8MAT.
</li>
<li>
<b>R8MAT_DIAG_SET_SCALAR</b> sets the diagonal of an R8MAT to a scalar value.
</li>
<li>
<b>R8MAT_DIAG_SET_VECTOR</b> sets the diagonal of an R8MAT to a vector.
</li>
<li>
<b>R8MAT_DIF_FRO</b> returns the Frobenius norm of the difference of R8MAT's.
</li>
<li>
<b>R8MAT_EXPAND_LINEAR</b> linearly interpolates new data into an R8MAT.
</li>
<li>
<b>R8MAT_EXPAND_LINEAR2</b> expands an R8MAT by linear interpolation.
</li>
<li>
<b>R8MAT_FLIP_COLS</b> swaps the columns of an R8MAT.
</li>
<li>
<b>R8MAT_FLIP_ROWS</b> swaps the rows of an R8MAT.
</li>
<li>
<b>R8MAT_FS</b> factors and solves a system with one right hand side.
</li>
<li>
<b>R8MAT_FS_NEW</b> factors and solves a system with one right hand side.
</li>
<li>
<b>R8MAT_FSS</b> factors and solves a system with multiple right hand sides.
</li>
<li>
<b>R8MAT_FSS_NEW</b> factors and solves a system with multiple right hand sides.
</li>
<li>
<b>R8MAT_GIVENS_POST</b> computes the Givens postmultiplier rotation matrix.
</li>
<li>
<b>R8MAT_GIVENS_PRE</b> computes the Givens premultiplier rotation matrix.
</li>
<li>
<b>R8MAT_HESS</b> approximates a Hessian matrix via finite differences.
</li>
<li>
<b>R8MAT_HOUSE_AXH</b> computes A*H where H is a compact Householder matrix.
</li>
<li>
<b>R8MAT_HOUSE_AXH_NEW</b> computes A*H where H is a compact Householder matrix.
</li>
<li>
<b>R8MAT_HOUSE_FORM</b> constructs a Householder matrix from its compact form.
</li>
<li>
<b>R8MAT_HOUSE_HXA</b> computes H*A where H is a compact Householder matrix.
</li>
<li>
<b>R8MAT_HOUSE_POST</b> computes a Householder post-multiplier matrix.
</li>
<li>
<b>R8MAT_HOUSE_PRE</b> computes a Householder pre-multiplier matrix.
</li>
<li>
<b>R8MAT_IDENTITY</b> sets the square matrix A to the identity.
</li>
<li>
<b>R8MAT_IDENTITY_NEW</b> returns an identity matrix.
</li>
<li>
<b>R8MAT_IN_01</b> is TRUE if the entries of an R8MAT are in the range [0,1].
</li>
<li>
<b>R8MAT_INDICATOR_NEW</b> sets up an "indicator" R8MAT.
</li>
<li>
<b>R8MAT_INSIGNIFICANT</b> determines if an R8MAT is insignificant.
</li>
<li>
<b>R8MAT_INVERSE_2D</b> inverts a 2 by 2 matrix using Cramer's rule.
</li>
<li>
<b>R8MAT_INVERSE_3D</b> inverts a 3 by 3 matrix using Cramer's rule.
</li>
<li>
<b>R8MAT_INVERSE_4D</b> inverts a 4 by 4 matrix using Cramer's rule.
</li>
<li>
<b>R8MAT_IS_IDENTITY</b> determines if an R8MAT is the identity.
</li>
<li>
<b>R8MAT_IS_SYMMETRIC</b> checks an R8MAT for symmetry.
</li>
<li>
<b>R8MAT_JAC</b> estimates a dense jacobian matrix of the function FX.
</li>
<li>
<b>R8MAT_L_INVERSE</b> inverts a lower triangular R8MAT.
</li>
<li>
<b>R8MAT_L_PRINT</b> prints a lower triangular R8MAT.
</li>
<li>
<b>R8MAT_L_SOLVE</b> solves a lower triangular linear system.
</li>
<li>
<b>R8MAT_L1_INVERSE</b> inverts a unit lower triangular R8MAT.
</li>
<li>
<b>R8MAT_LT_SOLVE</b> solves a transposed lower triangular linear system.
</li>
<li>
<b>R8MAT_LU</b> computes the LU factorization of a rectangular R8MAT.
</li>
<li>
<b>R8MAT_MAX</b> returns the maximum entry of an R8MAT.
</li>
<li>
<b>R8MAT_MAX_INDEX</b> returns the location of the maximum entry of an R8MAT.
</li>
<li>
<b>R8MAT_MAXCOL_MINROW</b> gets the maximum column minimum row of an M by N matrix.
</li>
<li>
<b>R8MAT_MAXROW_MINCOL</b> gets the maximum row minimum column of an M by N matrix.
</li>
<li>
<b>R8MAT_MIN</b> returns the minimum entry of an R8MAT.
</li>
<li>
<b>R8MAT_MIN_INDEX</b> returns the location of the minimum entry of an R8MAT.
</li>
<li>
<b>R8MAT_MINCOL_MAXROW</b> gets the minimum column maximum row of an M by N matrix.
</li>
<li>
<b>R8MAT_MINROW_MAXCOL</b> gets the minimum row maximum column of an M by N matrix.
</li>
<li>
<b>R8MAT_MINVM</b> computes inverse(A) * B for R8MAT's.
</li>
<li>
<b>R8MAT_MINVM_NEW</b> returns inverse(A) * B for R8MAT's.
</li>
<li>
<b>R8MAT_MM</b> multiplies two matrices.
</li>
<li>
<b>R8MAT_MM_NEW</b> multiplies two matrices.
</li>
<li>
<b>R8MAT_MMT_NEW</b> computes C = A * B'.
</li>
<li>
<b>R8MAT_MTM_NEW</b> computes C = A' * B.
</li>
<li>
<b>R8MAT_MTV</b> multiplies a transposed matrix times a vector.
</li>
<li>
<b>R8MAT_MTV_NEW</b> multiplies a transposed matrix times a vector.
</li>
<li>
<b>R8MAT_MV</b> multiplies a matrix times a vector.
</li>
<li>
<b>R8MAT_MV_NEW</b> multiplies a matrix times a vector.
</li>
<li>
<b>R8MAT_MXM</b> multiplies two matrices.
</li>
<li>
<b>R8MAT_MXM_NEW</b> multiplies two matrices.
</li>
<li>
<b>R8MAT_NEW</b> allocates a new R8MAT.
</li>
<li>
<b>R8MAT_NINT</b> rounds the entries of an R8MAT.
</li>
<li>
<b>R8MAT_NORM_EIS</b> returns the EISPACK norm of an R8MAT.
</li>
<li>
<b>R8MAT_NORM_FRO</b> returns the Frobenius norm of an R8MAT.
</li>
<li>
<b>R8MAT_NORM_FRO_AFFINE</b> returns the Frobenius norm of an R8MAT difference.
</li>
<li>
<b>R8MAT_NORM_L1</b> returns the matrix L1 norm of an R8MAT.
</li>
<li>
<b>R8MAT_NORM_L2</b> returns the matrix L2 norm of an R8MAT.
</li>
<li>
<b>R8MAT_NORM_LI</b> returns the matrix L-oo norm of an R8MAT.
</li>
<li>
<b>R8MAT_NORMAL_01_NEW</b> returns a unit pseudonormal R8MAT.
</li>
<li>
<b>R8MAT_NULLSPACE</b> computes the nullspace of a matrix.
</li>
<li>
<b>R8MAT_NULLSPACE_SIZE</b> computes the size of the nullspace of a matrix.
</li>
<li>
<b>R8MAT_ORTH_UNIFORM_NEW</b> returns a random orthogonal matrix.
</li>
<li>
<b>R8MAT_PLOT</b> "plots" an R8MAT.
</li>
<li>
<b>R8MAT_PLOT_SYMBOL</b> returns a symbol for entries of an R8MAT.
</li>
<li>
<b>R8MAT_POLY_CHAR</b> computes the characteristic polynomial of an R8MAT.
</li>
<li>
<b>R8MAT_POWER</b> computes a nonnegative power of an R8MAT.
</li>
<li>
<b>R8MAT_POWER_METHOD</b> applies the power method to a matrix.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_REF</b> computes the row echelon form of a matrix.
</li>
<li>
<b>R8MAT_RMS</b> returns the RMS norm of an R8MAT.
</li>
<li>
<b>R8MAT_RREF</b> computes the reduced row echelon form of a matrix.
</li>
<li>
<b>R8MAT_SCALE</b> multiplies an R8MAT by a scalar.
</li>
<li>
<b>R8MAT_SIGNIFICANT</b> determines if an R8MAT is significant compared to another.
</li>
<li>
<b>R8MAT_SOLVE</b> uses Gauss-Jordan elimination to solve an N by N linear system.
</li>
<li>
<b>R8MAT_SOLVE_2D</b> solves a 2 by 2 linear system using Cramer's rule.
</li>
<li>
<b>R8MAT_SOLVE_3D</b> solves a 3 by 3 linear system using Cramer's rule.
</li>
<li>
<b>R8MAT_SOLVE2</b> computes the solution of an N by N linear system.
</li>
<li>
<b>R8MAT_SUM</b> returns the sum of an R8MAT.
</li>
<li>
<b>R8MAT_SYMM_EIGEN</b> returns a symmetric matrix with given eigensystem.
</li>
<li>
<b>R8MAT_SYMM_JACOBI</b> applies Jacobi eigenvalue iteration to a symmetric matrix.
</li>
<li>
<b>R8MAT_TO_R8PLU</b> factors a general matrix.
</li>
<li>
<b>R8MAT_TRACE</b> computes the trace of an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_IN_PLACE</b> transposes a square R8MAT in place.
</li>
<li>
<b>R8MAT_TRANSPOSE_NEW</b> returns the transpose of an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_U_INVERSE</b> inverts an upper triangular R8MAT.
</li>
<li>
<b>R8MAT_U1_INVERSE</b> inverts a unit upper triangular R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_01_NEW</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_AB</b> returns a scaled pseudorandom R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_AB_NEW</b> returns a new scaled pseudorandom R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_ABVEC</b> returns a scaled pseudorandom R8MAT.