> with(linalg); -1
 

MATH 372 -- Numerical Linear Algebra 

Hilbert Matrix Example 

February 12, 2007 

 

The `*`(n*x, n) Hilbert matrix  

 

   H = (1/(i+j-1))[1 <= i and i <= n, 1 <= j and j <= n] 

 

is a classical example showing what can happen when linear  

equations are solved using floating-point arithmetic. 

 

The exact solution of the system  hilbert(7) x = (1, 1, 1, 1, 1, 1, 1)^tis: 

 

> linsolve(hilbert(7), [1, 1, 1, 1, 1, 1, 1]); 1
 

table( [( 1 ) = 7, ( 2 ) = -336, ( 3 ) = 3780, ( 5 ) = 34650, ( 4 ) = -16800, ( 7 ) = 12012, ( 6 ) = -33264 ] ) 

> H7Aug := concat(hilbert(7), matrix([[1], [1], [1], [1], [1], [1], [1]])); -1
 

 

Unlike MATLAB, Maple allows you to specify "custom" floating-point number 

systems by means of the Digits system variable.  This gives the number of  

decimal digits in the normalized floating-point numbers.  We will start 

with 5-digit arithmetic, increase to 10-digit arithmetic. 

 

> for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
for i from 5 to 10 do Digits := i; H7AugFlPt := map(proc (x) options operator, arrow; evalf(x) end proc, H7Aug); backsub(gausselim(H7AugFlPt)) end do; 1
 

5 

table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =...
table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =...
table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =...
table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =...
table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =...
table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =...
table( [( 6, 3 ) = .12500, ( 5, 5 ) = .11111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16667, ( 3, 5 ) = .14286, ( 5, 4 ) = .12500, ( 7, 3 ) = .11111, ( 1, 5 ) = .20000, ( 3, 4 ) = .16667, ( 2, 3 ) =... 

table( [( 1 ) = .68400, ( 2 ) = 7.5434, ( 3 ) = -238.33, ( 5 ) = -1088.5, ( 4 ) = 979.50, ( 7 ) = 553.48, ( 6 ) = -181.58 ] ) 

6 

table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ...
table( [( 6, 3 ) = .125000, ( 5, 5 ) = .111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166667, ( 3, 5 ) = .142857, ( 5, 4 ) = .125000, ( 7, 3 ) = .111111, ( 1, 5 ) = .200000, ( 3, 4 ) = .166667, ( ... 

table( [( 1 ) = -5.28500, ( 2 ) = 156.463, ( 3 ) = -982.554, ( 5 ) = 484.606, ( 4 ) = 1779.49, ( 7 ) = 2270.47, ( 6 ) = -3663.44 ] ) 

7 

table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166...
table( [( 6, 3 ) = .1250000, ( 5, 5 ) = .1111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666667, ( 3, 5 ) = .1428571, ( 5, 4 ) = .1250000, ( 7, 3 ) = .1111111, ( 1, 5 ) = .2000000, ( 3, 4 ) = .166... 

table( [( 1 ) = -7.312100, ( 2 ) = 221.6359, ( 3 ) = -1510.621, ( 5 ) = -2365.405, ( 4 ) = 3550.110, ( 7 ) = 1623.186, ( 6 ) = -1471.774 ] )
table( [( 1 ) = -7.312100, ( 2 ) = 221.6359, ( 3 ) = -1510.621, ( 5 ) = -2365.405, ( 4 ) = 3550.110, ( 7 ) = 1623.186, ( 6 ) = -1471.774 ] ) 

8 

table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )...
table( [( 6, 3 ) = .12500000, ( 5, 5 ) = .11111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .16666667, ( 3, 5 ) = .14285714, ( 5, 4 ) = .12500000, ( 7, 3 ) = .11111111, ( 1, 5 ) = .20000000, ( 3, 4 )... 

table( [( 1 ) = 28.598300, ( 2 ) = -1163.5335, ( 3 ) = 11547.864, ( 5 ) = 88280.435, ( 4 ) = -46453.217, ( 7 ) = 26943.039, ( 6 ) = -79121.445 ] )
table( [( 1 ) = 28.598300, ( 2 ) = -1163.5335, ( 3 ) = 11547.864, ( 5 ) = 88280.435, ( 4 ) = -46453.217, ( 7 ) = 26943.039, ( 6 ) = -79121.445 ] ) 

9 

table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (...
table( [( 6, 3 ) = .125000000, ( 5, 5 ) = .111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .166666667, ( 3, 5 ) = .142857143, ( 5, 4 ) = .125000000, ( 7, 3 ) = .111111111, ( 1, 5 ) = .200000000, (... 

table( [( 1 ) = 7.34442000, ( 2 ) = -349.168633, ( 3 ) = 3903.21351, ( 5 ) = 35495.1365, ( 4 ) = -17268.7792, ( 7 ) = 12245.9966, ( 6 ) = -33984.5511 ] )
table( [( 1 ) = 7.34442000, ( 2 ) = -349.168633, ( 3 ) = 3903.21351, ( 5 ) = 35495.1365, ( 4 ) = -17268.7792, ( 7 ) = 12245.9966, ( 6 ) = -33984.5511 ] ) 

10 

table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000...
table( [( 6, 3 ) = .1250000000, ( 5, 5 ) = .1111111111, ( 2, 8 ) = 1., ( 6, 8 ) = 1., ( 1, 6 ) = .1666666667, ( 3, 5 ) = .1428571429, ( 5, 4 ) = .1250000000, ( 7, 3 ) = .1111111111, ( 1, 5 ) = .200000... 

table( [( 1 ) = 7.020389000, ( 2 ) = -336.8489881, ( 3 ) = 3788.402527, ( 5 ) = 34711.87333, ( 4 ) = -16833.28496, ( 7 ) = 12029.89536, ( 6 ) = -33318.03986 ] )
table( [( 1 ) = 7.020389000, ( 2 ) = -336.8489881, ( 3 ) = 3788.402527, ( 5 ) = 34711.87333, ( 4 ) = -16833.28496, ( 7 ) = 12029.89536, ( 6 ) = -33318.03986 ] ) 

>
 

"Moral" -- it is always possible to get better results by increasing the number of  

digits used in the floating point computations (if you have that capability).   

 

However, by experimenting some, you will see that the larger  n  is for these matrices,  

the more digits you need for reliable computed solutions(!) 

>