void triangle_o03b ( double w[], double xy[] )
/******************************************************************************/
/*
Purpose:
TRIANGLE_O03B returns a 3 point quadrature rule for the unit triangle.
Discussion:
This rule is precise for monomials through degree 2.
The integration region is:
0 <= X
0 <= Y
X + Y <= 1.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
16 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[3], the weights.
Output, double XY[2*3], the abscissas.
*/
{
int order = 3;
double w_save[3] = {
0.33333333333333333333,
0.33333333333333333333,
0.33333333333333333333 };
double xy_save[2*3] = {
0.0, 0.5,
0.5, 0.0,
0.5, 0.5 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 2*order, xy_save, xy );
return;
}
void pyramid_unit_o01 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O01 returns a 1 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Parameters:
Output, double W[1], the weights.
Output, double XYZ[3*1], the abscissas.
*/
{
int order = 1;
double w_save[1] = { 1.0 };
double xyz_save[3*1] = { 0.00, 0.00, 0.25 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void line_o04 ( double w[], double x[] )
/******************************************************************************/
/*
Purpose:
LINE_O04 returns a 4 point quadrature rule for the unit line.
Discussion:
The integration region is:
- 1.0 <= X <= 1.0
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
17 April 2009
Author:
John Burkardt
Reference:
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Parameters:
Output, double W[4], the weights.
Output, double X[4], the abscissas.
*/
{
int i;
int order = 4;
double w_save[4] = {
0.173927422568727,
0.326072577431273,
0.326072577431273,
0.173927422568727 };
double x_save[4] = {
-0.86113631159405257522,
-0.33998104358485626480,
0.33998104358485626480,
0.86113631159405257522 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( order, x_save, x );
return;
}
void line_o03 ( double w[], double x[] )
/******************************************************************************/
/*
Purpose:
LINE_O03 returns a 3 point quadrature rule for the unit line.
Discussion:
The integration region is:
- 1.0 <= X <= 1.0
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
17 April 2009
Author:
John Burkardt
Reference:
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Parameters:
Output, double W[3], the weights.
Output, double X[3], the abscissas.
*/
{
int i;
int order = 3;
double w_save[3] = {
0.27777777777777777777,
0.44444444444444444444,
0.27777777777777777777 };
double x_save[3] = {
-0.77459666924148337704,
0.00000000000000000000,
0.77459666924148337704 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( order, x_save, x );
return;
}
void line_o02 ( double w[], double x[] )
/******************************************************************************/
/*
Purpose:
LINE_O02 returns a 2 point quadrature rule for the unit line.
Discussion:
The integration region is:
- 1.0 <= X <= 1.0
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
17 April 2009
Author:
John Burkardt
Reference:
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Parameters:
Output, double W[2], the weights.
Output, double X[2], the abscissas.
*/
{
int i;
int order = 2;
double w_save[2] = {
0.5,
0.5 };
double x_save[2] = {
-0.57735026918962576451,
0.57735026918962576451 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( order, x_save, x );
return;
}
void triangle_o12 ( double w[], double xy[] )
/******************************************************************************/
/*
Purpose:
TRIANGLE_O12 returns a 12 point quadrature rule for the unit triangle.
Discussion:
This rule is precise for monomials through degree 6.
The integration region is:
0 <= X
0 <= Y
X + Y <= 1.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
19 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[12], the weights.
Output, double XY[2*12], the abscissas.
*/
{
int order = 12;
double w_save[12] = {
0.050844906370206816921,
0.050844906370206816921,
0.050844906370206816921,
0.11678627572637936603,
0.11678627572637936603,
0.11678627572637936603,
0.082851075618373575194,
0.082851075618373575194,
0.082851075618373575194,
0.082851075618373575194,
0.082851075618373575194,
0.082851075618373575194 };
double xy_save[2*12] = {
0.87382197101699554332, 0.063089014491502228340,
0.063089014491502228340, 0.87382197101699554332,
0.063089014491502228340, 0.063089014491502228340,
0.50142650965817915742, 0.24928674517091042129,
0.24928674517091042129, 0.50142650965817915742,
0.24928674517091042129, 0.24928674517091042129,
0.053145049844816947353, 0.31035245103378440542,
0.31035245103378440542, 0.053145049844816947353,
0.053145049844816947353, 0.63650249912139864723,
0.31035245103378440542, 0.63650249912139864723,
0.63650249912139864723, 0.053145049844816947353,
0.63650249912139864723, 0.31035245103378440542 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 2*order, xy_save, xy );
return;
}
void triangle_o07 ( double w[], double xy[] )
/******************************************************************************/
/*
Purpose:
TRIANGLE_O07 returns a 7 point quadrature rule for the unit triangle.
Discussion:
This rule is precise for monomials through degree 5.
The integration region is:
0 <= X
0 <= Y
X + Y <= 1.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
16 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[7], the weights.
Output, double XY[2*7], the abscissas.
*/
{
int order = 7;
double w_save[7] = {
0.12593918054482715260,
0.12593918054482715260,
0.12593918054482715260,
0.13239415278850618074,
0.13239415278850618074,
0.13239415278850618074,
0.22500000000000000000 };
double xy_save[2*7] = {
0.79742698535308732240, 0.10128650732345633880,
0.10128650732345633880, 0.79742698535308732240,
0.10128650732345633880, 0.10128650732345633880,
0.059715871789769820459, 0.47014206410511508977,
0.47014206410511508977, 0.059715871789769820459,
0.47014206410511508977, 0.47014206410511508977,
0.33333333333333333333, 0.33333333333333333333 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 2*order, xy_save, xy );
return;
}
void triangle_o06 ( double w[], double xy[] )
/******************************************************************************/
/*
Purpose:
TRIANGLE_O06 returns a 6 point quadrature rule for the unit triangle.
Discussion:
This rule is precise for monomials through degree 4.
The integration region is:
0 <= X
0 <= Y
X + Y <= 1.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
16 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[6], the weights.
Output, double XY[2*6], the abscissas.
*/
{
int order = 6;
double w_save[6] = {
0.22338158967801146570,
0.22338158967801146570,
0.22338158967801146570,
0.10995174365532186764,
0.10995174365532186764,
0.10995174365532186764 };
double xy_save[2*6] = {
0.10810301816807022736, 0.44594849091596488632,
0.44594849091596488632, 0.10810301816807022736,
0.44594849091596488632, 0.44594849091596488632,
0.81684757298045851308, 0.091576213509770743460,
0.091576213509770743460, 0.81684757298045851308,
0.091576213509770743460, 0.091576213509770743460 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 2*order, xy_save, xy );
return;
}
void pyramid_unit_o48 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O48 returns a 48 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Parameters:
Output, double W[48], the weights.
Output, double XYZ[3*48], the abscissas.
*/
{
int order = 48;
double w_save[48] = {
2.01241939442682455E-002,
2.01241939442682455E-002,
2.01241939442682455E-002,
2.01241939442682455E-002,
2.60351137043010779E-002,
2.60351137043010779E-002,
2.60351137043010779E-002,
2.60351137043010779E-002,
1.24557795239745531E-002,
1.24557795239745531E-002,
1.24557795239745531E-002,
1.24557795239745531E-002,
1.87873998794808156E-003,
1.87873998794808156E-003,
1.87873998794808156E-003,
1.87873998794808156E-003,
4.32957927807745280E-002,
4.32957927807745280E-002,
4.32957927807745280E-002,
4.32957927807745280E-002,
1.97463249834127288E-002,
1.97463249834127288E-002,
1.97463249834127288E-002,
1.97463249834127288E-002,
5.60127223523590526E-002,
5.60127223523590526E-002,
5.60127223523590526E-002,
5.60127223523590526E-002,
2.55462562927473852E-002,
2.55462562927473852E-002,
2.55462562927473852E-002,
2.55462562927473852E-002,
2.67977366291788643E-002,
2.67977366291788643E-002,
2.67977366291788643E-002,
2.67977366291788643E-002,
1.22218992265373354E-002,
1.22218992265373354E-002,
1.22218992265373354E-002,
1.22218992265373354E-002,
4.04197740453215038E-003,
4.04197740453215038E-003,
4.04197740453215038E-003,
4.04197740453215038E-003,
1.84346316995826843E-003,
1.84346316995826843E-003,
1.84346316995826843E-003,
1.84346316995826843E-003 };
double xyz_save[3*48] = {
0.88091731624450909E+00, 0.00000000000000000E+00, 4.85005494469969989E-02,
-0.88091731624450909E+00, 0.00000000000000000E+00, 4.85005494469969989E-02,
0.00000000000000000E+00, 0.88091731624450909E+00, 4.85005494469969989E-02,
0.00000000000000000E+00, -0.88091731624450909E+00, 4.85005494469969989E-02,
0.70491874112648223E+00, 0.00000000000000000E+00, 0.23860073755186201E+00,
-0.70491874112648223E+00, 0.00000000000000000E+00, 0.23860073755186201E+00,
0.00000000000000000E+00, 0.70491874112648223E+00, 0.23860073755186201E+00,
0.00000000000000000E+00, -0.70491874112648223E+00, 0.23860073755186201E+00,
0.44712732143189760E+00, 0.00000000000000000E+00, 0.51704729510436798E+00,
-0.44712732143189760E+00, 0.00000000000000000E+00, 0.51704729510436798E+00,
0.00000000000000000E+00, 0.44712732143189760E+00, 0.51704729510436798E+00,
0.00000000000000000E+00, -0.44712732143189760E+00, 0.51704729510436798E+00,
0.18900486065123448E+00, 0.00000000000000000E+00, 0.79585141789677305E+00,
-0.18900486065123448E+00, 0.00000000000000000E+00, 0.79585141789677305E+00,
0.00000000000000000E+00, 0.18900486065123448E+00, 0.79585141789677305E+00,
0.00000000000000000E+00, -0.18900486065123448E+00, 0.79585141789677305E+00,
0.36209733410322176E+00, 0.36209733410322176E+00, 4.85005494469969989E-02,
-0.36209733410322176E+00, 0.36209733410322176E+00, 4.85005494469969989E-02,
-0.36209733410322176E+00, -0.36209733410322176E+00, 4.85005494469969989E-02,
0.36209733410322176E+00, -0.36209733410322176E+00, 4.85005494469969989E-02,
0.76688932060387538E+00, 0.76688932060387538E+00, 4.85005494469969989E-02,
-0.76688932060387538E+00, 0.76688932060387538E+00, 4.85005494469969989E-02,
-0.76688932060387538E+00, -0.76688932060387538E+00, 4.85005494469969989E-02,
0.76688932060387538E+00, -0.76688932060387538E+00, 4.85005494469969989E-02,
0.28975386476618070E+00, 0.28975386476618070E+00, 0.23860073755186201E+00,
-0.28975386476618070E+00, 0.28975386476618070E+00, 0.23860073755186201E+00,
-0.28975386476618070E+00, -0.28975386476618070E+00, 0.23860073755186201E+00,
0.28975386476618070E+00, -0.28975386476618070E+00, 0.23860073755186201E+00,
0.61367241226233160E+00, 0.61367241226233160E+00, 0.23860073755186201E+00,
-0.61367241226233160E+00, 0.61367241226233160E+00, 0.23860073755186201E+00,
-0.61367241226233160E+00, -0.61367241226233160E+00, 0.23860073755186201E+00,
0.61367241226233160E+00, -0.61367241226233160E+00, 0.23860073755186201E+00,
0.18378979287798017E+00, 0.18378979287798017E+00, 0.51704729510436798E+00,
-0.18378979287798017E+00, 0.18378979287798017E+00, 0.51704729510436798E+00,
-0.18378979287798017E+00, -0.18378979287798017E+00, 0.51704729510436798E+00,
0.18378979287798017E+00, -0.18378979287798017E+00, 0.51704729510436798E+00,
0.38925011625173161E+00, 0.38925011625173161E+00, 0.51704729510436798E+00,
-0.38925011625173161E+00, 0.38925011625173161E+00, 0.51704729510436798E+00,
-0.38925011625173161E+00, -0.38925011625173161E+00, 0.51704729510436798E+00,
0.38925011625173161E+00, -0.38925011625173161E+00, 0.51704729510436798E+00,
7.76896479525748113E-02, 7.76896479525748113E-02, 0.79585141789677305E+00,
-7.76896479525748113E-02, 7.76896479525748113E-02, 0.79585141789677305E+00,
-7.76896479525748113E-02, -7.76896479525748113E-02, 0.79585141789677305E+00,
7.76896479525748113E-02, -7.76896479525748113E-02, 0.79585141789677305E+00,
0.16453962988669860E+00, 0.16453962988669860E+00, 0.79585141789677305E+00,
-0.16453962988669860E+00, 0.16453962988669860E+00, 0.79585141789677305E+00,
-0.16453962988669860E+00, -0.16453962988669860E+00, 0.79585141789677305E+00,
0.16453962988669860E+00, -0.16453962988669860E+00, 0.79585141789677305E+00 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void pyramid_unit_o13 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O13 returns a 13 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[13], the weights.
Output, double XYZ[3*13], the abscissas.
*/
{
int order = 13;
double w_save[13] = {
0.063061594202898550725,
0.063061594202898550725,
0.063061594202898550725,
0.063061594202898550725,
0.042101946815575556199,
0.042101946815575556199,
0.042101946815575556199,
0.042101946815575556199,
0.13172030707666776585,
0.13172030707666776585,
0.13172030707666776585,
0.13172030707666776585,
0.05246460761943250889 };
double xyz_save[3*13] = {
-0.38510399211870384331, -0.38510399211870384331, 0.428571428571428571429,
0.38510399211870384331, -0.38510399211870384331, 0.428571428571428571429,
0.38510399211870384331, 0.38510399211870384331, 0.428571428571428571429,
-0.38510399211870384331, 0.38510399211870384331, 0.428571428571428571429,
-0.40345831960728204766, 0.00000000000000000000, 0.33928571428571428571,
0.40345831960728204766, 0.00000000000000000000, 0.33928571428571428571,
0.00000000000000000000, -0.40345831960728204766, 0.33928571428571428571,
0.00000000000000000000, 0.40345831960728204766, 0.33928571428571428571,
-0.53157877436961973359, -0.53157877436961973359, 0.08496732026143790850,
0.53157877436961973359, -0.53157877436961973359, 0.08496732026143790850,
0.53157877436961973359, 0.53157877436961973359, 0.08496732026143790850,
-0.53157877436961973359, 0.53157877436961973359, 0.08496732026143790850,
0.00000000000000000000, 0.00000000000000000000, 0.76219701803768503595 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void pyramid_unit_o09 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O09 returns a 9 point quadrature rule for the unit pyramid.
/
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[9], the weights.
Output, double XYZ[3*9], the abscissas.
*/
{
int order = 9;
double w_save[9] = {
0.13073389672275944791,
0.13073389672275944791,
0.13073389672275944791,
0.13073389672275944791,
0.10989110327724055209,
0.10989110327724055209,
0.10989110327724055209,
0.10989110327724055209,
0.03750000000000000000 };
double xyz_save[3*9] = {
-0.52966422253852215131, -0.52966422253852215131, 0.08176876558246862335,
0.52966422253852215131, -0.52966422253852215131, 0.08176876558246862335,
0.52966422253852215131, 0.52966422253852215131, 0.08176876558246862335,
-0.52966422253852215131, 0.52966422253852215131, 0.08176876558246862335,
-0.34819753825720418039, -0.34819753825720418039, 0.400374091560388519511,
0.34819753825720418039, -0.34819753825720418039, 0.400374091560388519511,
0.34819753825720418039, 0.34819753825720418039, 0.400374091560388519511,
-0.34819753825720418039, 0.34819753825720418039, 0.400374091560388519511,
0.00000000000000000000, 0.00000000000000000000, 0.83333333333333333333 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void pyramid_unit_o08b ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O08B returns an 8 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[8], the weights.
Output, double XYZ[3*8], the abscissas.
*/
{
int order = 8;
double w_save[8] = {
0.16438287736328777572,
0.16438287736328777572,
0.16438287736328777572,
0.16438287736328777572,
0.085617122636712224276,
0.085617122636712224276,
0.085617122636712224276,
0.085617122636712224276 };
double xyz_save[3*8] = {
-0.51197009372656270107, -0.51197009372656270107, 0.11024490204163285720,
0.51197009372656270107, -0.51197009372656270107, 0.11024490204163285720,
0.51197009372656270107, 0.51197009372656270107, 0.11024490204163285720,
-0.51197009372656270107, 0.51197009372656270107, 0.11024490204163285720,
-0.28415447557052037456, -0.28415447557052037456, 0.518326526529795714229,
0.28415447557052037456, -0.28415447557052037456, 0.518326526529795714229,
0.28415447557052037456, 0.28415447557052037456, 0.518326526529795714229,
-0.28415447557052037456, 0.28415447557052037456, 0.518326526529795714229 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void pyramid_unit_o08 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O08 returns an 8 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[8], the weights.
Output, double XYZ[3*8], the abscissas.
*/
{
int order = 8;
double w_save[8] = {
0.075589411559869072938,
0.075589411559869072938,
0.075589411559869072938,
0.075589411559869072938,
0.17441058844013092706,
0.17441058844013092706,
0.17441058844013092706,
0.17441058844013092706 };
double xyz_save[3*8] = {
-0.26318405556971359557, -0.26318405556971359557, 0.54415184401122528880,
0.26318405556971359557, -0.26318405556971359557, 0.54415184401122528880,
0.26318405556971359557, 0.26318405556971359557, 0.54415184401122528880,
-0.26318405556971359557, 0.26318405556971359557, 0.54415184401122528880,
-0.50661630334978742377, -0.50661630334978742377, 0.12251482265544137787,
0.50661630334978742377, -0.50661630334978742377, 0.12251482265544137787,
0.50661630334978742377, 0.50661630334978742377, 0.12251482265544137787,
-0.50661630334978742377, 0.50661630334978742377, 0.12251482265544137787 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void pyramid_unit_o06 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O06 returns a 6 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[6], the weights.
Output, double XYZ[3*6], the abscissas.
*/
{
int order = 6;
double w_save[6] = {
0.21000000000000000000,
0.21000000000000000000,
0.21000000000000000000,
0.21000000000000000000,
0.06000000000000000000,
0.10000000000000000000 };
double xyz_save[3*6] = {
-0.48795003647426658968, -0.48795003647426658968, 0.16666666666666666667,
0.48795003647426658968, -0.48795003647426658968, 0.16666666666666666667,
0.48795003647426658968, 0.48795003647426658968, 0.16666666666666666667,
-0.48795003647426658968, 0.48795003647426658968, 0.16666666666666666667,
0.00000000000000000000, 0.00000000000000000000, 0.58333333333333333333,
0.00000000000000000000, 0.00000000000000000000, 0.75000000000000000000 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void pyramid_unit_o18 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O18 returns an 18 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[18], the weights.
Output, double XYZ[3*18], the abscissas.
*/
{
int order = 18;
double w_save[18] = {
0.023330065296255886709,
0.037328104474009418735,
0.023330065296255886709,
0.037328104474009418735,
0.059724967158415069975,
0.037328104474009418735,
0.023330065296255886709,
0.037328104474009418735,
0.023330065296255886709,
0.05383042853090460712,
0.08612868564944737139,
0.05383042853090460712,
0.08612868564944737139,
0.13780589703911579422,
0.08612868564944737139,
0.05383042853090460712,
0.08612868564944737139,
0.05383042853090460712 };
double xyz_save[3*18] = {
-0.35309846330877704481, -0.35309846330877704481, 0.544151844011225288800,
0.00000000000000000000, -0.35309846330877704481, 0.544151844011225288800,
0.35309846330877704481, -0.35309846330877704481, 0.544151844011225288800,
-0.35309846330877704481, 0.00000000000000000000, 0.544151844011225288800,
0.00000000000000000000, 0.00000000000000000000, 0.544151844011225288800,
0.35309846330877704481, 0.00000000000000000000, 0.544151844011225288800,
-0.35309846330877704481, 0.35309846330877704481, 0.544151844011225288800,
0.00000000000000000000, 0.35309846330877704481, 0.544151844011225288800,
0.35309846330877704481, 0.35309846330877704481, 0.544151844011225288800,
-0.67969709567986745790, -0.67969709567986745790, 0.12251482265544137787,
0.00000000000000000000, -0.67969709567986745790, 0.12251482265544137787,
0.67969709567986745790, -0.67969709567986745790, 0.12251482265544137787,
-0.67969709567986745790, 0.00000000000000000000, 0.12251482265544137787,
0.00000000000000000000, 0.00000000000000000000, 0.12251482265544137787,
0.67969709567986745790, 0.00000000000000000000, 0.12251482265544137787,
-0.67969709567986745790, 0.67969709567986745790, 0.12251482265544137787,
0.00000000000000000000, 0.67969709567986745790, 0.12251482265544137787,
0.67969709567986745790, 0.67969709567986745790, 0.12251482265544137787 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void line_o05 ( double w[], double x[] )
/******************************************************************************/
/*
Purpose:
LINE_O05 returns a 5 point quadrature rule for the unit line.
Discussion:
The integration region is:
- 1.0 <= X <= 1.0
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
17 April 2009
Author:
John Burkardt
Reference:
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Parameters:
Output, double W[5], the weights.
Output, double X[5], the abscissas.
*/
{
int i;
int order = 5;
double w_save[5] = {
0.118463442528095,
0.239314335249683,
0.284444444444444,
0.239314335249683,
0.118463442528095 };
double x_save[5] = {
-0.90617984593866399280,
-0.53846931010568309104,
0.00000000000000000000,
0.53846931010568309104,
0.90617984593866399280 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( order, x_save, x );
return;
}
void pyramid_unit_o27 ( double w[], double xyz[] )
/******************************************************************************/
/*
Purpose:
PYRAMID_UNIT_O27 returns a 27 point quadrature rule for the unit pyramid.
Discussion:
The integration region is:
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
02 April 2009
Author:
John Burkardt
Reference:
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Parameters:
Output, double W[27], the weights.
Output, double XYZ[3*27], the abscissas.
*/
{
int order = 27;
double w_save[27] = {
0.036374157653908938268,
0.05819865224625430123,
0.036374157653908938268,
0.05819865224625430123,
0.09311784359400688197,
0.05819865224625430123,
0.036374157653908938268,
0.05819865224625430123,
0.036374157653908938268,
0.033853303069413431019,
0.054165284911061489631,
0.033853303069413431019,
0.054165284911061489631,
0.08666445585769838341,
0.054165284911061489631,
0.033853303069413431019,
0.054165284911061489631,
0.033853303069413431019,
0.006933033103838124540,
0.011092852966140999264,
0.006933033103838124540,
0.011092852966140999264,
0.017748564745825598822,
0.011092852966140999264,
0.006933033103838124540,
0.011092852966140999264,
0.006933033103838124540 };
double xyz_save[3*27] = {
-0.7180557413198889387, -0.7180557413198889387, 0.07299402407314973216,
0.00000000000000000000, -0.7180557413198889387, 0.07299402407314973216,
0.7180557413198889387, -0.7180557413198889387, 0.07299402407314973216,
-0.7180557413198889387, 0.00000000000000000000, 0.07299402407314973216,
0.00000000000000000000, 0.00000000000000000000, 0.07299402407314973216,
0.7180557413198889387, 0.00000000000000000000, 0.07299402407314973216,
-0.7180557413198889387, 0.7180557413198889387, 0.07299402407314973216,
0.00000000000000000000, 0.7180557413198889387, 0.07299402407314973216,
0.7180557413198889387, 0.7180557413198889387, 0.07299402407314973216,
-0.50580870785392503961, -0.50580870785392503961, 0.34700376603835188472,
0.00000000000000000000, -0.50580870785392503961, 0.34700376603835188472,
0.50580870785392503961, -0.50580870785392503961, 0.34700376603835188472,
-0.50580870785392503961, 0.00000000000000000000, 0.34700376603835188472,
0.00000000000000000000, 0.00000000000000000000, 0.34700376603835188472,
0.50580870785392503961, 0.00000000000000000000, 0.34700376603835188472,
-0.50580870785392503961, 0.50580870785392503961, 0.34700376603835188472,
0.00000000000000000000, 0.50580870785392503961, 0.34700376603835188472,
0.50580870785392503961, 0.50580870785392503961, 0.34700376603835188472,
-0.22850430565396735360, -0.22850430565396735360, 0.70500220988849838312,
0.00000000000000000000, -0.22850430565396735360, 0.70500220988849838312,
0.22850430565396735360, -0.22850430565396735360, 0.70500220988849838312,
-0.22850430565396735360, 0.00000000000000000000, 0.70500220988849838312,
0.00000000000000000000, 0.00000000000000000000, 0.70500220988849838312,
0.22850430565396735360, 0.00000000000000000000, 0.70500220988849838312,
-0.22850430565396735360, 0.22850430565396735360, 0.70500220988849838312,
0.00000000000000000000, 0.22850430565396735360, 0.70500220988849838312,
0.22850430565396735360, 0.22850430565396735360, 0.70500220988849838312 };
r8vec_copy ( order, w_save, w );
r8vec_copy ( 3 * order, xyz_save, xyz );
return;
}
void line_cvt_lloyd ( int n, double a, double b, int it_num, char *header,
double x[] )
/******************************************************************************/
/*
Purpose:
LINE_CVT_LLOYD carries out the Lloyd algorithm.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
28 July 2014
Author:
John Burkardt
Parameters:
Input, int N, the number of generators.
Input, double A, B, the left and right endpoints.
Input, int IT_NUM, the number of iterations to take.
Input, char *HEADER, an identifying string.
Input/output, double X[N], the point locations.
*/
{
double e;
double *e_plot;
int i;
int it;
double *x_old;
double *x_plot;
double xm;
double *xm_plot;
e_plot = ( double * ) malloc ( ( it_num + 1 ) * sizeof ( double ) );
e = line_cvt_energy ( n, a, b, x );
e_plot[0] = e;
x_plot = ( double * ) malloc ( ( it_num + 1 ) * n * sizeof ( double ) );
for ( i = 0; i < n; i++ )
{
x_plot[i+0*n] = x[i];
}
xm_plot = ( double * ) malloc ( it_num * sizeof ( double ) );
x_old = ( double * ) malloc ( n * sizeof ( double ) );
for ( it = 1; it <= it_num; it++ )
{
r8vec_copy ( n, x, x_old );
line_cvt_lloyd_step ( n, a, b, x );
for ( i = 0; i < n; i++ )
{
x_plot[i+it*n] = x[i];
}
e = line_cvt_energy ( n, a, b, x );
e_plot[it] = e;
xm = 0.0;
for ( i = 0; i < n; i++ )
{
xm = xm + pow ( x_old[i] - x[i], 2 );
}
xm = xm / ( double ) ( n );
xm_plot[it-1] = xm;
}
energy_plot ( it_num, e_plot, header );
motion_plot ( it_num, xm_plot, header );
evolution_plot ( n, it_num, x_plot, header );
free ( e_plot );
free ( x_old );
free ( x_plot );
free ( xm_plot );
return;
}
double *r8vec_house_column ( int n, double a[], int k )
/******************************************************************************/
/*
Purpose:
R8VEC_HOUSE_COLUMN defines a Householder premultiplier that "packs" a column.
Discussion:
An R8VEC is a vector of R8's.
The routine returns a vector V that defines a Householder
premultiplier matrix H(V) that zeros out the subdiagonal entries of
column K of the matrix A.
H(V) = I - 2 * v * v'
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
21 August 2010
Author:
John Burkardt
Parameters:
Input, int N, the order of the matrix A.
Input, double A[N], column K of the matrix A.
Input, int K, the column of the matrix to be modified.
Output, double R8VEC_HOUSE_COLUMN[N], a vector of unit L2 norm which
defines an orthogonal Householder premultiplier matrix H with the property
that the K-th column of H*A is zero below the diagonal.
*/
{
int i;
double s;
double *v;
v = r8vec_zero_new ( n );
if ( k < 1 || n <= k )
{
return v;
}
s = r8vec_norm_l2 ( n+1-k, a+k-1 );
if ( s == 0.0 )
{
return v;
}
v[k-1] = a[k-1] + r8_abs ( s ) * r8_sign ( a[k-1] );
r8vec_copy ( n-k, a+k, v+k );
s = r8vec_norm_l2 ( n-k+1, v+k-1 );
for ( i = k-1; i < n; i++ )
{
v[i] = v[i] / s;
}
return v;
}
