8 June 2023 6:54:59.294 AM quad_rule_test(): FORTRAN77 version Test the quad_rule library. BASHFORTH_SET_TEST BASHFORTH_SET sets up an Adams-Bashforth rule; Index X W 1 0.000000000000000 1.0000000000000000 1 0.000000000000000 1.5000000000000000 2 -1.000000000000000 -0.5000000000000000 1 0.000000000000000 1.9166666666666667 2 -1.000000000000000 -1.3333333333333333 3 -2.000000000000000 0.4166666666666667 1 0.000000000000000 2.2916666666666665 2 -1.000000000000000 -2.4583333333333335 3 -2.000000000000000 1.5416666666666667 4 -3.000000000000000 -0.3750000000000000 1 0.000000000000000 2.6402777777777779 2 -1.000000000000000 -3.8527777777777779 3 -2.000000000000000 3.6333333333333333 4 -3.000000000000000 -1.7694444444444444 5 -4.000000000000000 0.3486111111111111 1 0.000000000000000 2.9701388888888891 2 -1.000000000000000 -5.5020833333333332 3 -2.000000000000000 6.9319444444444445 4 -3.000000000000000 -5.0680555555555555 5 -4.000000000000000 1.9979166666666666 6 -5.000000000000000 -0.3298611111111111 1 0.000000000000000 3.2857308201058202 2 -1.000000000000000 -7.3956349206349206 3 -2.000000000000000 11.6658234126984119 4 -3.000000000000000 -11.3798941798941797 5 -4.000000000000000 6.7317956349206352 6 -5.000000000000000 -2.2234126984126985 7 -6.000000000000000 0.3155919312169312 1 0.000000000000000 3.5899553571428573 2 -1.000000000000000 -9.5252066798941790 3 -2.000000000000000 18.0545386904761891 4 -3.000000000000000 -22.0277529761904773 5 -4.000000000000000 17.3796544312169310 6 -5.000000000000000 -8.6121279761904770 7 -6.000000000000000 2.4451636904761904 8 -7.000000000000000 -0.3050512566137566 1 0.000000000000000 3.8848233575837741 2 -1.000000000000000 -11.8841506834215167 3 -2.000000000000000 26.3108427028218692 4 -3.000000000000000 -38.5403610008818376 5 -4.000000000000000 38.0204144620811277 6 -5.000000000000000 -25.1247360008818355 7 -6.000000000000000 10.7014677028218692 8 -7.000000000000000 -2.6631685405643739 9 -8.000000000000000 0.2948680004409171 1 0.000000000000000 4.1717988040123455 2 -1.000000000000000 -14.4669297012786604 3 -2.000000000000000 36.6419587742504405 4 -3.000000000000000 -62.6462985008818336 5 -4.000000000000000 74.1793207120811218 6 -5.000000000000000 -61.2836422508818330 7 -6.000000000000000 34.8074052028218688 8 -7.000000000000000 -12.9942846119929456 9 -8.000000000000000 2.8776470182980600 10 -9.000000000000000 -0.2869754464285714 TEST04 CHEBYSHEV_SET sets up a Chebyshev rule; SUM_SUB carries it out. The integration interval is [ 0.0000, 1.0000] Number of subintervals is 1 Quadrature order will vary. Integrand will vary. Order 1 X X^2 X^3 X^4 1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 2 1.00000000 0.50000000 0.33333333 0.25000000 0.19444444 3 1.00000000 0.50000000 0.33333333 0.25000000 0.19791667 4 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 5 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 6 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 7 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 9 1.00000000 0.50000000 0.33333333 0.25000000 0.20000000 Order X^5 X^6 X^7 SIN(X) EXP(X) 1 0.03125000 0.01562500 0.00781250 0.47942554 1.64872127 2 0.15277778 0.12037037 0.09490741 0.45958781 1.71789638 3 0.16145833 0.13411458 0.11263021 0.45965669 1.71813657 4 0.16666667 0.14259259 0.12407407 0.45969787 1.71828122 5 0.16666667 0.14272280 0.12452980 0.45969778 1.71828152 6 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 7 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 9 0.16666667 0.14285714 0.12500000 0.45969769 1.71828183 Order SQRT(|X|) 1 0.70710678 2 0.67388734 3 0.67122325 4 0.66890560 5 0.66839073 6 0.66777528 7 0.66758934 9 0.66724897 CHEBYSHEV1_COMPUTE_TEST CHEBYSHEV1_COMPUTE computes a Chebyshev Type 1 quadrature rule over [-1,1] of given order. Index X W 1 0.6123233995736766E-16 3.1415926535897931 1 -0.7071067811865475 1.5707963267948966 2 0.7071067811865476 1.5707963267948966 1 -0.8660254037844387 1.0471975511965976 2 0.6123233995736766E-16 1.0471975511965976 3 0.8660254037844387 1.0471975511965976 1 -0.9238795325112867 0.7853981633974483 2 -0.3826834323650897 0.7853981633974483 3 0.3826834323650898 0.7853981633974483 4 0.9238795325112867 0.7853981633974483 1 -0.9510565162951535 0.6283185307179586 2 -0.5877852522924730 0.6283185307179586 3 0.6123233995736766E-16 0.6283185307179586 4 0.5877852522924731 0.6283185307179586 5 0.9510565162951535 0.6283185307179586 1 -0.9659258262890682 0.5235987755982988 2 -0.7071067811865475 0.5235987755982988 3 -0.2588190451025206 0.5235987755982988 4 0.2588190451025207 0.5235987755982988 5 0.7071067811865476 0.5235987755982988 6 0.9659258262890683 0.5235987755982988 1 -0.9749279121818237 0.4487989505128276 2 -0.7818314824680295 0.4487989505128276 3 -0.4338837391175581 0.4487989505128276 4 0.6123233995736766E-16 0.4487989505128276 5 0.4338837391175582 0.4487989505128276 6 0.7818314824680298 0.4487989505128276 7 0.9749279121818236 0.4487989505128276 1 -0.9807852804032304 0.3926990816987241 2 -0.8314696123025453 0.3926990816987241 3 -0.5555702330196020 0.3926990816987241 4 -0.1950903220161282 0.3926990816987241 5 0.1950903220161283 0.3926990816987241 6 0.5555702330196023 0.3926990816987241 7 0.8314696123025452 0.3926990816987241 8 0.9807852804032304 0.3926990816987241 1 -0.9848077530122080 0.3490658503988659 2 -0.8660254037844385 0.3490658503988659 3 -0.6427876096865394 0.3490658503988659 4 -0.3420201433256685 0.3490658503988659 5 0.6123233995736766E-16 0.3490658503988659 6 0.3420201433256688 0.3490658503988659 7 0.6427876096865394 0.3490658503988659 8 0.8660254037844387 0.3490658503988659 9 0.9848077530122080 0.3490658503988659 1 -0.9876883405951377 0.3141592653589793 2 -0.8910065241883678 0.3141592653589793 3 -0.7071067811865475 0.3141592653589793 4 -0.4539904997395467 0.3141592653589793 5 -0.1564344650402306 0.3141592653589793 6 0.1564344650402309 0.3141592653589793 7 0.4539904997395468 0.3141592653589793 8 0.7071067811865476 0.3141592653589793 9 0.8910065241883679 0.3141592653589793 10 0.9876883405951378 0.3141592653589793 TEST065 Approximate the integral of f(x,y) over the semicircle -1 <= x <= 1, y = sqrt ( 1 - x^2 ) using N Chebyshev points. If p(x,y) involves any term of odd degree in y, the estimate will only be approximate. Polynomial N Integral Estimate Error 1 10 1.57080 1.57080 0.222045E-15 x 10 0.00000 -0.104083E-16 0.104083E-16 y 10 0.666667 0.666723 0.565402E-04 x^2 10 0.392699 0.392699 0.555112E-16 x y 10 0.00000 -0.221177E-16 0.221177E-16 y^2 10 0.392699 0.392699 0.00000 x^3 10 0.00000 0.242861E-16 0.242861E-16 x^2y 10 0.133333 0.133392 0.588566E-04 x y^2 10 0.00000 0.477049E-17 0.477049E-17 y^3 10 0.266667 0.266666 0.115821E-05 x^4 10 0.196350 0.196350 0.546392E-08 x^2y^2 10 0.654498E-01 0.654498E-01 0.138778E-16 y^4 10 0.196350 0.196350 0.555112E-16 x^4y 10 0.571429E-01 0.572043E-01 0.613939E-04 x^2y^3 10 0.380952E-01 0.380940E-01 0.126862E-05 y^5 10 0.152381 0.152381 0.736100E-07 x^6 10 0.122718 0.122718 0.277556E-16 x^4y^2 10 0.245437E-01 0.245437E-01 0.00000 x^2y^4 10 0.245437E-01 0.245437E-01 0.346945E-17 y^6 10 0.122718 0.122718 0.277556E-16 CHEBYSHEV3_COMPUTE_TEST CHEBYSHEV3_COMPUTE computes a Chebyshev Type 3 quadrature rule over [-1,1] of given order. Index X W 1 0.000000000000000 3.1415926535897931 1 -1.000000000000000 1.5707963267948966 2 1.000000000000000 1.5707963267948966 1 -1.000000000000000 0.7853981633974483 2 0.6123233995736766E-16 1.5707963267948966 3 1.000000000000000 0.7853981633974483 1 -1.000000000000000 0.5235987755982988 2 -0.4999999999999998 1.0471975511965976 3 0.5000000000000001 1.0471975511965976 4 1.000000000000000 0.5235987755982988 1 -1.000000000000000 0.3926990816987241 2 -0.7071067811865475 0.7853981633974483 3 0.6123233995736766E-16 0.7853981633974483 4 0.7071067811865476 0.7853981633974483 5 1.000000000000000 0.3926990816987241 1 -1.000000000000000 0.3141592653589793 2 -0.8090169943749473 0.6283185307179586 3 -0.3090169943749473 0.6283185307179586 4 0.3090169943749475 0.6283185307179586 5 0.8090169943749475 0.6283185307179586 6 1.000000000000000 0.3141592653589793 1 -1.000000000000000 0.2617993877991494 2 -0.8660254037844387 0.5235987755982988 3 -0.4999999999999998 0.5235987755982988 4 0.6123233995736766E-16 0.5235987755982988 5 0.5000000000000001 0.5235987755982988 6 0.8660254037844387 0.5235987755982988 7 1.000000000000000 0.2617993877991494 1 -1.000000000000000 0.2243994752564138 2 -0.9009688679024190 0.4487989505128276 3 -0.6234898018587335 0.4487989505128276 4 -0.2225209339563143 0.4487989505128276 5 0.2225209339563144 0.4487989505128276 6 0.6234898018587336 0.4487989505128276 7 0.9009688679024191 0.4487989505128276 8 1.000000000000000 0.2243994752564138 1 -1.000000000000000 0.1963495408493621 2 -0.9238795325112867 0.3926990816987241 3 -0.7071067811865475 0.3926990816987241 4 -0.3826834323650897 0.3926990816987241 5 0.6123233995736766E-16 0.3926990816987241 6 0.3826834323650898 0.3926990816987241 7 0.7071067811865476 0.3926990816987241 8 0.9238795325112867 0.3926990816987241 9 1.000000000000000 0.1963495408493621 1 -1.000000000000000 0.1745329251994329 2 -0.9396926207859083 0.3490658503988659 3 -0.7660444431189779 0.3490658503988659 4 -0.4999999999999998 0.3490658503988659 5 -0.1736481776669303 0.3490658503988659 6 0.1736481776669304 0.3490658503988659 7 0.5000000000000001 0.3490658503988659 8 0.7660444431189780 0.3490658503988659 9 0.9396926207859084 0.3490658503988659 10 1.000000000000000 0.1745329251994329 CLENSHAW_CURTIS_COMPUTE_TEST CLENSHAW_CURTIS_COMPUTE computes a Clenshaw-Curtis quadrature rule over [-1,1] of given order. Index X W 1 0.000000000000000 2.0000000000000000 1 -1.000000000000000 1.0000000000000000 2 1.000000000000000 1.0000000000000000 1 -1.000000000000000 0.3333333333333334 2 0.6123233995736766E-16 1.3333333333333333 3 1.000000000000000 0.3333333333333334 1 -1.000000000000000 0.1111111111111111 2 -0.4999999999999998 0.8888888888888892 3 0.5000000000000001 0.8888888888888888 4 1.000000000000000 0.1111111111111111 1 -1.000000000000000 0.0666666666666667 2 -0.7071067811865475 0.5333333333333334 3 0.6123233995736766E-16 0.7999999999999999 4 0.7071067811865476 0.5333333333333333 5 1.000000000000000 0.0666666666666667 1 -1.000000000000000 0.0400000000000000 2 -0.8090169943749473 0.3607430412000113 3 -0.3090169943749473 0.5992569587999887 4 0.3090169943749475 0.5992569587999889 5 0.8090169943749475 0.3607430412000112 6 1.000000000000000 0.0400000000000000 1 -1.000000000000000 0.0285714285714286 2 -0.8660254037844387 0.2539682539682539 3 -0.4999999999999998 0.4571428571428573 4 0.6123233995736766E-16 0.5206349206349206 5 0.5000000000000001 0.4571428571428571 6 0.8660254037844387 0.2539682539682539 7 1.000000000000000 0.0285714285714286 1 -1.000000000000000 0.0204081632653061 2 -0.9009688679024190 0.1901410072182084 3 -0.6234898018587335 0.3522424237181591 4 -0.2225209339563143 0.4372084057983264 5 0.2225209339563144 0.4372084057983264 6 0.6234898018587336 0.3522424237181591 7 0.9009688679024191 0.1901410072182084 8 1.000000000000000 0.0204081632653061 1 -1.000000000000000 0.0158730158730159 2 -0.9238795325112867 0.1462186492160182 3 -0.7071067811865475 0.2793650793650794 4 -0.3826834323650897 0.3617178587204898 5 0.6123233995736766E-16 0.3936507936507936 6 0.3826834323650898 0.3617178587204897 7 0.7071067811865476 0.2793650793650794 8 0.9238795325112867 0.1462186492160181 9 1.000000000000000 0.0158730158730159 1 -1.000000000000000 0.0123456790123457 2 -0.9396926207859083 0.1165674565720372 3 -0.7660444431189779 0.2252843233381044 4 -0.4999999999999998 0.3019400352733687 5 -0.1736481776669303 0.3438625058041442 6 0.1736481776669304 0.3438625058041442 7 0.5000000000000001 0.3019400352733685 8 0.7660444431189780 0.2252843233381044 9 0.9396926207859084 0.1165674565720371 10 1.000000000000000 0.0123456790123457 CLENSHAW_CURTIS_SET_TEST CLENSHAW_CURTIS_SET sets up a Clenshaw-Curtis rule; Estimate the integral of sqrt(abs(x)) over [-1,+1]. N Estimate Error 1 0.000000000000000 0.133333E+01 2 2.000000000000000 0.666667E+00 3 0.6666666666666666 0.666667E+00 4 1.479300944331640 0.145968E+00 5 1.030289509603962 0.303044E+00 6 1.395188802266437 0.618555E-01 7 1.176328668057338 0.157005E+00 8 1.370529534325666 0.371962E-01 9 1.230194560888044 0.103139E+00 10 1.358632996768151 0.252997E-01 FEJER1_COMPUTE_TEST FEJER1_COMPUTE computes a Fejer type 1 rule. Order W X 1 2.00000 0.00000 2 1.00000 -0.707107 1.00000 0.707107 3 0.444444 -0.866025 1.11111 0.612323E-16 0.444444 0.866025 4 0.264298 -0.923880 0.735702 -0.382683 0.735702 0.382683 0.264298 0.923880 5 0.167781 -0.951057 0.525552 -0.587785 0.613333 0.612323E-16 0.525552 0.587785 0.167781 0.951057 6 0.118661 -0.965926 0.377778 -0.707107 0.503561 -0.258819 0.503561 0.258819 0.377778 0.707107 0.118661 0.965926 7 0.867162E-01 -0.974928 0.287831 -0.781831 0.398242 -0.433884 0.454422 0.612323E-16 0.398242 0.433884 0.287831 0.781831 0.867162E-01 0.974928 8 0.669829E-01 -0.980785 0.222988 -0.831470 0.324153 -0.555570 0.385877 -0.195090 0.385877 0.195090 0.324153 0.555570 0.222988 0.831470 0.669829E-01 0.980785 9 0.527366E-01 -0.984808 0.179189 -0.866025 0.264037 -0.642788 0.330845 -0.342020 0.346384 0.612323E-16 0.330845 0.342020 0.264037 0.642788 0.179189 0.866025 0.527366E-01 0.984808 10 0.429391E-01 -0.987688 0.145875 -0.891007 0.220317 -0.707107 0.280879 -0.453990 0.309989 -0.156434 0.309989 0.156434 0.280879 0.453990 0.220317 0.707107 0.145875 0.891007 0.429391E-01 0.987688 FEJER1_SET_TEST FEJER1_SET looks up a Fejer type 1 rule. Order W X 1 2.00000 0.00000 2 1.00000 -0.707107 1.00000 0.707107 3 0.444444 -0.866025 1.11111 0.00000 0.444444 0.866025 4 0.264298 -0.923880 0.735702 -0.382683 0.735702 0.382683 0.264298 0.923880 5 0.167781 -0.951057 0.525552 -0.587785 0.613333 0.00000 0.525552 0.587785 0.167781 0.951057 6 0.118661 -0.965926 0.377778 -0.707107 0.503561 -0.258819 0.503561 0.258819 0.377778 0.707107 0.118661 0.965926 7 0.867162E-01 -0.974928 0.287831 -0.781831 0.398242 -0.433884 0.454422 0.00000 0.398242 0.433884 0.287831 0.781831 0.867162E-01 0.974928 8 0.669829E-01 -0.980785 0.222988 -0.831470 0.324153 -0.555570 0.385877 -0.195090 0.385877 0.195090 0.324153 0.555570 0.222988 0.831470 0.669829E-01 0.980785 9 0.527366E-01 -0.984808 0.179189 -0.866025 0.264037 -0.642788 0.330845 -0.342020 0.346384 0.00000 0.330845 0.342020 0.264037 0.642788 0.179189 0.866025 0.527366E-01 0.984808 10 0.429391E-01 -0.987688 0.145875 -0.891007 0.220317 -0.707107 0.280879 -0.453990 0.309989 -0.156434 0.309989 0.156434 0.280879 0.453990 0.220317 0.707107 0.145875 0.891007 0.429391E-01 0.987688 FEJER2_COMPUTE_TEST FEJER2_COMPUTE computes a Fejer type 2 rule. Order W X 1 2.00000 0.00000 2 1.00000 -0.500000 1.00000 0.500000 3 0.666667 -0.707107 0.666667 0.612323E-16 0.666667 0.707107 4 0.425464 -0.809017 0.574536 -0.309017 0.574536 0.309017 0.425464 0.809017 5 0.311111 -0.866025 0.400000 -0.500000 0.577778 0.612323E-16 0.400000 0.500000 0.311111 0.866025 6 0.226915 -0.900969 0.326794 -0.623490 0.446291 -0.222521 0.446291 0.222521 0.326794 0.623490 0.226915 0.900969 7 0.177965 -0.923880 0.247619 -0.707107 0.393464 -0.382683 0.361905 0.612323E-16 0.393464 0.382683 0.247619 0.707107 0.177965 0.923880 8 0.139770 -0.939693 0.206370 -0.766044 0.314286 -0.500000 0.339575 -0.173648 0.339575 0.173648 0.314286 0.500000 0.206370 0.766044 0.139770 0.939693 9 0.114781 -0.951057 0.165433 -0.809017 0.273790 -0.587785 0.279011 -0.309017 0.333968 0.612323E-16 0.279011 0.309017 0.273790 0.587785 0.165433 0.809017 0.114781 0.951057 10 0.944195E-01 -0.959493 0.141135 -0.841254 0.226387 -0.654861 0.253051 -0.415415 0.285007 -0.142315 0.285007 0.142315 0.253051 0.415415 0.226387 0.654861 0.141135 0.841254 0.944195E-01 0.959493 FEJER2_SET_TEST FEJER2_SET looks up a Fejer type 2 rule. Order W X 1 2.00000 0.00000 2 1.00000 -0.500000 1.00000 0.500000 3 0.666667 -0.707107 0.666667 0.00000 0.666667 0.707107 4 0.425464 -0.809017 0.574536 -0.309017 0.574536 0.309017 0.425464 0.809017 5 0.311111 -0.866025 0.400000 -0.500000 0.577778 0.00000 0.400000 0.500000 0.311111 0.866025 6 0.226915 -0.900969 0.326794 -0.623490 0.446291 -0.222521 0.446291 0.222521 0.326794 0.623490 0.226915 0.900969 7 0.177965 -0.923880 0.247619 -0.707107 0.393464 -0.382683 0.361905 0.00000 0.393464 0.382683 0.247619 0.707107 0.177965 0.923880 8 0.139770 -0.939693 0.206370 -0.766044 0.314286 -0.500000 0.339575 -0.173648 0.339575 0.173648 0.314286 0.500000 0.206370 0.766044 0.139770 0.939693 9 0.114781 -0.951057 0.165433 -0.809017 0.273790 -0.587785 0.279011 -0.309017 0.333968 0.00000 0.279011 0.309017 0.273790 0.587785 0.165433 0.809017 0.114781 0.951057 10 0.944195E-01 -0.959493 0.141135 -0.841254 0.226387 -0.654861 0.253051 -0.415415 0.285007 -0.142315 0.285007 0.142315 0.253051 0.415415 0.226387 0.654861 0.141135 0.841254 0.944195E-01 0.959493 GEGENBAUER_COMPUTE_TEST GEGENBAUER_COMPUTE computes a Gauss-Gegenbauer rule; The printed output can be inserted into a FORTRAN77 program. c c Abscissas X and weights W for c Gegenbauer rule of order = 5 c with ALPHA = 0.250000 c x( 1) = -0.8855262926812553D+00 x( 2) = -0.5181455045810590D+00 x( 3) = 0.0000000000000000D+00 x( 4) = 0.5181455045810590D+00 x( 5) = 0.8855262926812553D+00 w( 1) = 0.1710472328817660D+00 w( 2) = 0.4305503877999760D+00 w( 3) = 0.5448431281645963D+00 w( 4) = 0.4305503877999757D+00 w( 5) = 0.1710472328817660D+00 GEGENBAUER_COMPUTE_TEST GEGENBAUER_COMPUTE computes a Gauss-Gegenbauer rule; The printed output can be inserted into a FORTRAN77 program. c c Abscissas X and weights W for c Gegenbauer rule of order = 5 c with ALPHA = -0.500000 c x( 1) = -0.9510565162951535D+00 x( 2) = -0.5877852522924731D+00 x( 3) = 0.0000000000000000D+00 x( 4) = 0.5877852522924731D+00 x( 5) = 0.9510565162951535D+00 w( 1) = 0.6283185307179600D+00 w( 2) = 0.6283185307179586D+00 w( 3) = 0.6283185307179587D+00 w( 4) = 0.6283185307179586D+00 w( 5) = 0.6283185307179600D+00 TEST087 HERMITE_EK_COMPUTE computes a Gauss-Hermite rule; Compute the data for ORDER = 31 x( 1) = -0.69956801237185395336837245849892D+01 x( 2) = -0.62750787049428602415446221129969D+01 x( 3) = -0.56739614446185875351602589944378D+01 x( 4) = -0.51335955771123744639794495014939D+01 x( 5) = -0.46315595063128585096023925871123D+01 x( 6) = -0.41562717558181461185995431151241D+01 x( 7) = -0.37007434032314692196052874351153D+01 x( 8) = -0.32603207323135410256043087429134D+01 x( 9) = -0.28316804533902013574220291047823D+01 x( 10) = -0.24123177054804201269178065558663D+01 x( 11) = -0.20002585489356383696701868757373D+01 x( 12) = -0.15938858604721393152914288293687D+01 x( 13) = -0.11918269983500462405601183490944D+01 x( 14) = -0.79287697691530878429944095842075D+00 x( 15) = -0.39594273647142280703192795954237D+00 x( 16) = 0.00000000000000000000000000000000D+00 x( 17) = 0.39594273647142314009883534708933D+00 x( 18) = 0.79287697691530778509871879577986D+00 x( 19) = 0.11918269983500460185155134240631D+01 x( 20) = 0.15938858604721388712022189793061D+01 x( 21) = 0.20002585489356397019378164259251D+01 x( 22) = 0.24123177054804174623825474554906D+01 x( 23) = 0.28316804533902062424033374554710D+01 x( 24) = 0.32603207323135428019611481431639D+01 x( 25) = 0.37007434032314701077837071352405D+01 x( 26) = 0.41562717558181425658858643146232D+01 x( 27) = 0.46315595063128558450671334867366D+01 x( 28) = 0.51335955771123815694068071024958D+01 x( 29) = 0.56739614446185893115170983946882D+01 x( 30) = 0.62750787049428549124741039122455D+01 x( 31) = 0.69956801237185413100405639852397D+01 w( 1) = 0.46189683944640472613158859365632D-21 w( 2) = 0.51106090079271744720308741564632D-17 w( 3) = 0.58995564987538567419077633901066D-14 w( 4) = 0.18603735214521792303325971626682D-11 w( 5) = 0.23524920032086516150127716937718D-09 w( 6) = 0.14611988344910378378436693436009D-07 w( 7) = 0.50437125589398611003142270520216D-06 w( 8) = 0.10498602757675530266329687323346D-04 w( 9) = 0.13952090395047135303399632455523D-03 w( 10) = 0.12336833073068882282025127139491D-02 w( 11) = 0.74827999140351904652779246873706D-02 w( 12) = 0.31847230731300399386718424921128D-01 w( 13) = 0.96717948160870648166564933490008D-01 w( 14) = 0.21213278866876472683600240998203D+00 w( 15) = 0.33877265789410798690894921492145D+00 w( 16) = 0.39577855609861034569263438243070D+00 w( 17) = 0.33877265789410748730858813360101D+00 w( 18) = 0.21213278866876492112503171938442D+00 w( 19) = 0.96717948160870509388686855345441D-01 w( 20) = 0.31847230731300447958975752271726D-01 w( 21) = 0.74827999140351705159579509540890D-02 w( 22) = 0.12336833073068838913938227719314D-02 w( 23) = 0.13952090395046929304986860209681D-03 w( 24) = 0.10498602757675648850942302925393D-04 w( 25) = 0.50437125589398176898756802691293D-06 w( 26) = 0.14611988344910669546012312101750D-07 w( 27) = 0.23524920032086381733278177070721D-09 w( 28) = 0.18603735214521178380215092426454D-11 w( 29) = 0.58995564987538685748213417052837D-14 w( 30) = 0.51106090079273316279143361549088D-17 w( 31) = 0.46189683944640181090559855438330D-21 TEST087 HERMITE_EK_COMPUTE computes a Gauss-Hermite rule; Compute the data for ORDER = 63 x( 1) = -0.10435499877854182315672915137839D+02 x( 2) = -0.98028759912975136359136740793474D+01 x( 3) = -0.92792019543050443530773918610066D+01 x( 4) = -0.88118581437284539958909590495750D+01 x( 5) = -0.83807683451863397294800961390138D+01 x( 6) = -0.79755950801420372187067187041976D+01 x( 7) = -0.75901395198641026240693463478237D+01 x( 8) = -0.72203167078889656238516181474552D+01 x( 9) = -0.68632544331795353187430919206236D+01 x( 10) = -0.65168348106821154530621242884081D+01 x( 11) = -0.61794379922705946484029482235201D+01 x( 12) = -0.58497884000810662641356429958250D+01 x( 13) = -0.55268572526403039191222887893673D+01 x( 14) = -0.52097979830408336354707898863126D+01 x( 15) = -0.48979018644975740315317125350703D+01 x( 16) = -0.45905665744435193431627340032719D+01 x( 17) = -0.42872733352824408115111509687267D+01 x( 18) = -0.39875699104197170896668467321433D+01 x( 19) = -0.36910577000963473714989504514961D+01 x( 20) = -0.33973817713303859910922710696468D+01 x( 21) = -0.31062230279282547762420563230990D+01 x( 22) = -0.28172919672837988258606856106780D+01 x( 23) = -0.25303236304711975712677940464346D+01 x( 24) = -0.22450734604812057071399067353923D+01 x( 25) = -0.19613138583081475285752048876020D+01 x( 26) = -0.16788312791720132466366521839518D+01 x( 27) = -0.13974237486049629897166823866428D+01 x( 28) = -0.11168987050996463938901115398039D+01 x( 29) = -0.83707109558947601080092226766283D+00 x( 30) = -0.55776166427908191458584497013362D+00 x( 31) = -0.27879538567115252911676748226455D+00 x( 32) = 0.00000000000000000000000000000000D+00 x( 33) = 0.27879538567115136338259162585018D+00 x( 34) = 0.55776166427908180356354250761797D+00 x( 35) = 0.83707109558947667693473704275675D+00 x( 36) = 0.11168987050996459498009016897413D+01 x( 37) = 0.13974237486049614354044479114236D+01 x( 38) = 0.16788312791720136907258620340144D+01 x( 39) = 0.19613138583081464183521802624455D+01 x( 40) = 0.22450734604812061512291165854549D+01 x( 41) = 0.25303236304712011239814728469355D+01 x( 42) = 0.28172919672837983817714757606154D+01 x( 43) = 0.31062230279282538880636366229737D+01 x( 44) = 0.33973817713303930965196286706487D+01 x( 45) = 0.36910577000963469274097406014334D+01 x( 46) = 0.39875699104197162014884270320181D+01 x( 47) = 0.42872733352824399233327312686015D+01 x( 48) = 0.45905665744435202313411537033971D+01 x( 49) = 0.48979018644975775842453913355712D+01 x( 50) = 0.52097979830408345236492095864378D+01 x( 51) = 0.55268572526403003664086099888664D+01 x( 52) = 0.58497884000810644877788035955746D+01 x( 53) = 0.61794379922705946484029482235201D+01 x( 54) = 0.65168348106821136767052848881576D+01 x( 55) = 0.68632544331795335423862525203731D+01 x( 56) = 0.72203167078889620711379393469542D+01 x( 57) = 0.75901395198641106176751236489508D+01 x( 58) = 0.79755950801420398832419778045733D+01 x( 59) = 0.83807683451863379531232567387633D+01 x( 60) = 0.88118581437284557722477984498255D+01 x( 61) = 0.92792019543050496821479100617580D+01 x( 62) = 0.98028759912975207413410316803493D+01 x( 63) = 0.10435499877854171657531878736336D+02 w( 1) = 0.37099206434902337820724039305906D-47 w( 2) = 0.10400778615223436231367446126448D-41 w( 3) = 0.19796804708320132123823036759110D-37 w( 4) = 0.84687478191906799900482849134302D-34 w( 5) = 0.13071305930819457523986823021419D-30 w( 6) = 0.93437837175664140840050244140485D-28 w( 7) = 0.36027426635286631182527024160908D-25 w( 8) = 0.82963863116210472037129864070371D-23 w( 9) = 0.12266629909143528817474732457275D-20 w( 10) = 0.12288435628835350117472102998385D-18 w( 11) = 0.86925536958461996552078832928693D-17 w( 12) = 0.44857058689315770719814254585641D-15 w( 13) = 0.17335817955789411249041638005010D-13 w( 14) = 0.51265062385198185599513829775971D-12 w( 15) = 0.11808921844569651759792898724500D-10 w( 16) = 0.21508698297875722692438014741162D-09 w( 17) = 0.31371929535383214311784921439008D-08 w( 18) = 0.37041625984896954203546496752625D-07 w( 19) = 0.35734732949990455765592158379751D-06 w( 20) = 0.28393114498469816954473975145756D-05 w( 21) = 0.18709113003788956975693980044895D-04 w( 22) = 0.10284880800685642558787297184963D-03 w( 23) = 0.47411702610320431575466337825731D-03 w( 24) = 0.18409222622442040443596633636503D-02 w( 25) = 0.60436044551375545444416026441559D-02 w( 26) = 0.16829299199652036911345476255519D-01 w( 27) = 0.39858264027817079389048871007617D-01 w( 28) = 0.80467087994200908740438649147109D-01 w( 29) = 0.13871950817658507126850508939242D+00 w( 30) = 0.20448695346897388658291561114311D+00 w( 31) = 0.25799889943138243353359939646907D+00 w( 32) = 0.27876694884925218298477034295502D+00 w( 33) = 0.25799889943138337722317032785213D+00 w( 34) = 0.20448695346897496905036462067073D+00 w( 35) = 0.13871950817658493249062701124785D+00 w( 36) = 0.80467087994200506284592222527863D-01 w( 37) = 0.39858264027817259800290372595555D-01 w( 38) = 0.16829299199651922419596061786251D-01 w( 39) = 0.60436044551375823000172182730694D-02 w( 40) = 0.18409222622442023096361873868432D-02 w( 41) = 0.47411702610320523732650999093607D-03 w( 42) = 0.10284880800685772663047995445496D-03 w( 43) = 0.18709113003788807897895263288035D-04 w( 44) = 0.28393114498469168974269325605997D-05 w( 45) = 0.35734732949990704581520414330476D-06 w( 46) = 0.37041625984896921116321994631518D-07 w( 47) = 0.31371929535383723027861641551028D-08 w( 48) = 0.21508698297874781774491235672182D-09 w( 49) = 0.11808921844569267250055032278042D-10 w( 50) = 0.51265062385197337416268536144078D-12 w( 51) = 0.17335817955789035751250752803390D-13 w( 52) = 0.44857058689317821758167829216335D-15 w( 53) = 0.86925536958458098469871393163326D-17 w( 54) = 0.12288435628834984190782669423573D-18 w( 55) = 0.12266629909143194036683618269792D-20 w( 56) = 0.82963863116215085852456841548840D-23 w( 57) = 0.36027426635283709665805498065490D-25 w( 58) = 0.93437837175658255386500079908787D-28 w( 59) = 0.13071305930819028376332123546191D-30 w( 60) = 0.84687478191904725835067817590480D-34 w( 61) = 0.19796804708317172245716352628385D-37 w( 62) = 0.10400778615223136730088153225420D-41 w( 63) = 0.37099206434904999618274530692691D-47 TEST089 HERMITE_PROBABILIST_SET sets a probabilist Hermite rule; The integration interval is ( -oo, +oo ). Weight function is exp ( - x * x / 2 ) / sqrt ( 2 * pi ). Order 1 X X^2 X^3 X^4 1 1.00000000 0.00000000 0.00000000 0.00000000 0.00000000 2 1.00000000 0.00000000 1.00000000 0.00000000 1.00000000 3 1.00000000 0.00000000 1.00000000 0.00000000 3.00000000 4 1.00000000 0.00000000 1.00000000 0.00000000 3.00000000 5 1.00000000 0.00000000 1.00000000 -0.00000000 3.00000000 6 1.00000000 0.00000000 1.00000000 -0.00000000 3.00000000 7 1.00000000 0.00000000 1.00000000 -0.00000000 3.00000000 8 1.00000000 0.00000000 1.00000000 -0.00000000 3.00000000 9 1.00000000 -0.00000000 1.00000000 -0.00000000 3.00000000 10 1.00000000 0.00000000 1.00000000 0.00000000 3.00000000 Order X^5 X^6 X^7 SIN(X) EXP(X) 1 0.00000000 0.00000000 0.00000000 0.00000000 1.00000000 2 0.00000000 1.00000000 0.00000000 0.00000000 1.54308063 3 0.00000000 9.00000000 0.00000000 0.00000000 1.63819248 4 0.00000000 15.00000000 0.00000000 -0.00000000 1.64796896 5 0.00000000 15.00000000 0.00000000 0.00000000 1.64867943 6 0.00000000 15.00000000 0.00000000 -0.00000000 1.64871937 7 0.00000000 15.00000000 0.00000000 0.00000000 1.64872120 8 0.00000000 15.00000000 0.00000000 0.00000000 1.64872127 9 0.00000000 15.00000000 -0.00000000 0.00000000 1.64872127 10 0.00000000 15.00000000 -0.00000000 -0.00000000 1.64872127 Order SQRT(|X|) 1 0.00000000 2 1.00000000 3 0.43869134 4 0.92252624 5 0.55518788 6 0.89503104 7 0.61275111 8 0.88042681 9 0.64779296 10 0.87121620 TEST095 HERMITE_GK16_SET sets up a nested rule for the Hermite integration problem. The integration interval is ( -oo, +oo ). The weight function is exp ( - x * x ) HERMITE_INTEGRAL determines the exact value of the integal when f(x) = x^m. M N Estimate Exact Error 0 1 1.77245 1.77245 0.00000 2 1 0.00000 0.886227 0.886227 0 3 1.77245 1.77245 0.222045E-15 2 3 0.886227 0.886227 0.222045E-15 4 3 1.32934 1.32934 0.444089E-15 6 3 1.99401 3.32335 1.32934 0 7 1.77245 1.77245 0.222045E-15 2 7 0.886227 0.886227 0.444089E-15 4 7 1.32934 1.32934 0.888178E-15 6 7 3.32335 3.32335 0.133227E-14 8 7 16.9910 11.6317 5.35929 0 9 1.77245 1.77245 0.222045E-15 2 9 0.886227 0.886227 0.111022E-15 4 9 1.32934 1.32934 0.222045E-15 6 9 3.32335 3.32335 0.00000 8 9 11.6317 11.6317 0.177636E-14 10 9 52.3428 52.3428 0.142109E-13 12 9 287.885 287.885 0.568434E-13 14 9 1871.25 1871.25 0.909495E-12 16 9 13798.9 14034.4 235.543 0 17 1.77245 1.77245 0.222045E-15 2 17 0.886227 0.886227 0.444089E-15 4 17 1.32934 1.32934 0.666134E-15 6 17 3.32335 3.32335 0.177636E-14 8 17 11.6317 11.6317 0.532907E-14 10 17 52.3428 52.3428 0.213163E-13 12 17 287.885 287.885 0.113687E-12 14 17 1871.25 1871.25 0.227374E-12 16 17 14034.4 14034.4 0.00000 18 17 122706. 119292. 3413.58 0 19 1.77245 1.77245 0.888178E-15 2 19 0.886227 0.886227 0.00000 4 19 1.32934 1.32934 0.888178E-15 6 19 3.32335 3.32335 0.222045E-14 8 19 11.6317 11.6317 0.106581E-13 10 19 52.3428 52.3428 0.426326E-13 12 19 287.885 287.885 0.113687E-12 14 19 1871.25 1871.25 0.136424E-11 16 19 14034.4 14034.4 0.200089E-10 18 19 119292. 119292. 0.247383E-09 20 19 0.113328E+07 0.113328E+07 0.186265E-08 0 31 1.77245 1.77245 0.222045E-15 2 31 0.886227 0.886227 0.333067E-15 4 31 1.32934 1.32934 0.222045E-15 6 31 3.32335 3.32335 0.444089E-15 8 31 11.6317 11.6317 0.355271E-14 10 31 52.3428 52.3428 0.213163E-13 12 31 287.885 287.885 0.341061E-12 14 31 1871.25 1871.25 0.409273E-11 16 31 14034.4 14034.4 0.418368E-10 18 31 119292. 119292. 0.378350E-09 20 31 0.113328E+07 0.113328E+07 0.349246E-08 0 33 1.77245 1.77245 0.888178E-15 2 33 0.886227 0.886227 0.111022E-15 4 33 1.32934 1.32934 0.888178E-15 6 33 3.32335 3.32335 0.177636E-14 8 33 11.6317 11.6317 0.355271E-14 10 33 52.3428 52.3428 0.284217E-13 12 33 287.885 287.885 0.284217E-12 14 33 1871.25 1871.25 0.272848E-11 16 33 14034.4 14034.4 0.236469E-10 18 33 119292. 119292. 0.247383E-09 20 33 0.113328E+07 0.113328E+07 0.209548E-08 0 35 1.77245 1.77245 0.888178E-15 2 35 0.886227 0.886227 0.888178E-15 4 35 1.32934 1.32934 0.133227E-14 6 35 3.32335 3.32335 0.177636E-14 8 35 11.6317 11.6317 0.177636E-14 10 35 52.3428 52.3428 0.213163E-13 12 35 287.885 287.885 0.284217E-12 14 35 1871.25 1871.25 0.318323E-11 16 35 14034.4 14034.4 0.272848E-10 18 35 119292. 119292. 0.261934E-09 20 35 0.113328E+07 0.113328E+07 0.209548E-08 LOBATTO_COMPUTE_TEST LOBATTO_COMPUTE computes a Lobatto rule; I X W 1 -1.00000000 0.16666667 2 -0.44721360 0.83333333 3 0.44721360 0.83333333 4 1.00000000 0.16666667 1 -1.00000000 0.04761905 2 -0.83022390 0.27682605 3 -0.46884879 0.43174538 4 0.00000000 0.48761905 5 0.46884879 0.43174538 6 0.83022390 0.27682605 7 1.00000000 0.04761905 1 -1.00000000 0.02222222 2 -0.91953391 0.13330599 3 -0.73877387 0.22488934 4 -0.47792495 0.29204268 5 -0.16527896 0.32753976 6 0.16527896 0.32753976 7 0.47792495 0.29204268 8 0.73877387 0.22488934 9 0.91953391 0.13330599 10 1.00000000 0.02222222 LOBATTO_SET_TEST LOBATTO_SET sets a Lobatto rule; I X W 1 -1.00000000 0.16666667 2 -0.44721360 0.83333333 3 0.44721360 0.83333333 4 1.00000000 0.16666667 1 -1.00000000 0.04761905 2 -0.83022390 0.27682605 3 -0.46884879 0.43174538 4 0.00000000 0.48761905 5 0.46884879 0.43174538 6 0.83022390 0.27682605 7 1.00000000 0.04761905 1 -1.00000000 0.02222222 2 -0.91953391 0.13330599 3 -0.73877387 0.22488934 4 -0.47792495 0.29204268 5 -0.16527896 0.32753976 6 0.16527896 0.32753976 7 0.47792495 0.29204268 8 0.73877387 0.22488934 9 0.91953391 0.13330599 10 1.00000000 0.02222222 MOULTON_SET_TEST MOULTON_SET sets up an Adams-Moulton rule; Index X W 1 1.000000000000000 1.0000000000000000 1 1.000000000000000 0.5000000000000000 2 0.000000000000000 0.5000000000000000 1 1.000000000000000 0.4166666666666667 2 0.000000000000000 0.6666666666666666 3 -1.000000000000000 -0.0833333333333333 1 1.000000000000000 0.3750000000000000 2 0.000000000000000 0.7916666666666666 3 -1.000000000000000 -0.2083333333333333 4 -2.000000000000000 0.0416666666666667 1 1.000000000000000 0.3486111111111111 2 0.000000000000000 0.8972222222222223 3 -1.000000000000000 -0.3666666666666666 4 -2.000000000000000 0.1472222222222222 5 -3.000000000000000 -0.0263888888888889 1 1.000000000000000 0.3298611111111111 2 0.000000000000000 0.9909722222222223 3 -1.000000000000000 -0.5541666666666667 4 -2.000000000000000 0.3347222222222222 5 -3.000000000000000 -0.1201388888888889 6 -4.000000000000000 0.0187500000000000 1 1.000000000000000 0.3155919312169312 2 0.000000000000000 1.0765873015873015 3 -1.000000000000000 -0.7682043650793651 4 -2.000000000000000 0.6201058201058202 5 -3.000000000000000 -0.3341765873015873 6 -4.000000000000000 0.1043650793650794 7 -5.000000000000000 -0.0142691798941799 1 1.000000000000000 0.3042245370370371 2 0.000000000000000 1.1561590608465608 3 -1.000000000000000 -1.0069196428571427 4 -2.000000000000000 1.0179646164021163 5 -3.000000000000000 -0.7320353835978836 6 -4.000000000000000 0.3430803571428571 7 -5.000000000000000 -0.0938409391534391 8 -6.000000000000000 0.0113673941798942 1 1.000000000000000 0.2948680004409171 2 0.000000000000000 1.2310113536155203 3 -1.000000000000000 -1.2689026675485009 4 -2.000000000000000 1.5419306657848324 5 -3.000000000000000 -1.3869929453262786 6 -4.000000000000000 0.8670464065255732 7 -5.000000000000000 -0.3558239638447972 8 -6.000000000000000 0.0862196869488536 9 -7.000000000000000 -0.0093565365961199 1 1.000000000000000 0.2869754464285714 2 0.000000000000000 1.3020443397266315 3 -1.000000000000000 -1.5530346119929452 4 -2.000000000000000 2.2049052028218696 5 -3.000000000000000 -2.3814547508818342 6 -4.000000000000000 1.8615082120811288 7 -5.000000000000000 -1.0187985008818343 8 -6.000000000000000 0.3703516313932981 9 -7.000000000000000 -0.0803895227072310 10 -8.000000000000000 0.0078925540123457 NCC_SET_TEST NCC_SET sets up a Newton-Cotes Closed rule; Index X W 1 0.000000000000000 2.0000000000000000 1 -1.000000000000000 1.0000000000000000 2 1.000000000000000 1.0000000000000000 1 -1.000000000000000 0.3333333333333333 2 0.000000000000000 1.3333333333333333 3 1.000000000000000 0.3333333333333333 1 -1.000000000000000 0.2500000000000000 2 -0.3333333333333333 0.7500000000000000 3 0.3333333333333333 0.7500000000000000 4 1.000000000000000 0.2500000000000000 1 -1.000000000000000 0.1555555555555556 2 -0.5000000000000000 0.7111111111111111 3 0.000000000000000 0.2666666666666667 4 0.5000000000000000 0.7111111111111111 5 1.000000000000000 0.1555555555555556 1 -1.000000000000000 0.1319444444444444 2 -0.6000000000000000 0.5208333333333334 3 -0.2000000000000000 0.3472222222222222 4 0.2000000000000000 0.3472222222222222 5 0.6000000000000000 0.5208333333333334 6 1.000000000000000 0.1319444444444444 1 -1.000000000000000 0.0976190476190476 2 -0.6666666666666666 0.5142857142857142 3 -0.3333333333333333 0.0642857142857143 4 0.000000000000000 0.6476190476190476 5 0.3333333333333333 0.0642857142857143 6 0.6666666666666666 0.5142857142857142 7 1.000000000000000 0.0976190476190476 1 -1.000000000000000 0.0869212962962963 2 -0.7142857142857143 0.4140046296296296 3 -0.4285714285714285 0.1531250000000000 4 -0.1428571428571428 0.3459490740740740 5 0.1428571428571428 0.3459490740740740 6 0.4285714285714285 0.1531250000000000 7 0.7142857142857143 0.4140046296296296 8 1.000000000000000 0.0869212962962963 1 -1.000000000000000 0.0697707231040564 2 -0.7500000000000000 0.4153791887125221 3 -0.5000000000000000 -0.0654673721340388 4 -0.2500000000000000 0.7404585537918871 5 0.000000000000000 -0.3202821869488536 6 0.2500000000000000 0.7404585537918871 7 0.5000000000000000 -0.0654673721340388 8 0.7500000000000000 0.4153791887125221 9 1.000000000000000 0.0697707231040564 1 -1.000000000000000 0.0637723214285714 2 -0.7777777777777778 0.3513616071428571 3 -0.5555555555555556 0.0241071428571429 4 -0.3333333333333333 0.4317857142857143 5 -0.1111111111111111 0.1289732142857143 6 0.1111111111111111 0.1289732142857143 7 0.3333333333333333 0.4317857142857143 8 0.5555555555555556 0.0241071428571429 9 0.7777777777777778 0.3513616071428571 10 1.000000000000000 0.0637723214285714 quad_rule_test(): Normal end of execution. 8 June 2023 6:54:59.297 AM