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Purdue University
Purdue e-Pubs Computer Science Technical Reports
Department of Computer Science
1984
A Tensor Product Generalized ADI Method for the Method of Planes Wayne R. Dyksen Report Number: 84-494
Dyksen, Wayne R., "A Tensor Product Generalized ADI Method for the Method of Planes" (1984). Computer Science Technical Reports. Paper 414. http://docs.lib.purdue.edu/cstech/414
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.
A TENSOR PRODUCT GENERALIZED ADI METHOD FOR THE METHOD OF PLANES
Wayne R. Dyksen Department o( Computer Sciences Purdue University West Lafayette, Indiana 479fJ7
CSD-TR 494 October 1984
ABSTRACT
We consider solving separable second order linear elliptic partial differential equations in three independent variables. If the partial differential operator separates into two (acton, one depending on x and y, and one depending on z. then we use the MetluJd of Planel to obtain a discrete problem which we write in tensor product form as (T~C?JBZ1
+/0AZ1 )C =F.
We apply a new iterative method, the Teruor Product Generalized Alternating Direction Implicit method, to solve the discrete problem. We study a specifie implementation which uses Her· mite bicubie collocation in tbe ry direction and symmetric finite differences in the z direction. We demonstrate that this method is a fast and accurate way to solve the large linear systems arising from three dimensional elliptic problems.
1980 Mrrr1l#1IUJlics SUb]«l Crtu~f/kiJrIOfl. Primu)'. 6SIllO, 65M20, 65N35.
A Tensor Product Generalized ADI Method for the Method of Planes
Wayne R. Dyksen
1. [ntrodoct!oD. Let R be a rectangular domain. We consider solving second order linear elliptic problems of the form
LXyu. +Lcu. =/(.r,y.z) inR0[a,..bz l II.
= g,,(x,y)
II.
= gbo(X,y)
onR0az a(x,y)u +13(.r,y)u.. =g(.r,y,z) on aRfi!J(az,bc )
(1.1)
a(;r,Y)13(x.y)", 0 a 2 (:t,y) + 132(x,y» 0
on R 0bp on aR on aR
where L, =-(P(z)uz)z +q(z)u, p>O,q2:0,
(12)
and LZJ is a general elliptic operator in x and y. We express a discrete problem in terms of tensor products of matrices resulting from lower dimensional, Bnd hence much simpler, problems. We apply a new iterative tcchnique to solve this discrete problem. We obtain a fast method for solving a large class of elliptic problenu in three
dim~nsions.
Scction 2 presents a brief introduction to the Tensor Product Genualiud Alterll/lling Direction Implicit (TPGADI) method. In Section 3 we extend the Method of Lines (Jones et ai,
[I8D to the Merhod of Planes to obtain a discrete problem which we write in tensor product form as
2
We present the TPGADI method for solving lIuch discrete problems in Section 4. We apply the Method of Planes 115iog Hermite bicubics in x and y Bnd fini.te differences in z in Sec-
tions S. 6 and 7. We consider a specific implementation in Section 8; we show that the Method of Planes together with the TPGADI method give a powerful tool for solving a large
class of eUiptic problems in three dimensions. The TPGADI method has been used effectively to solve discrete elliptic problems arising from olher discretizations in both two and three dimensions. We have used it in conjunction with the Method of Lines (Dyksen, {8D and the Hermite bicubic coliocatioD equations (Dyksen, [12D. In three dimensions. we have used it to solve elliplic problems on cylindrical
domains with holes (Dykscn. [13D·
2. The Two Dlftctlonal Tenmr Product General.lad Ani Methodlll Let A.t and Bt be matrices of order NI; XNt • and consider the lineElI' IS)'Stem (2.1)
We wi.!!h to solve the two directional problem (2.1) by using methods for one dircctional. simpler problems involving Ah Bit A2 and B2' We usc the term dir~clional rather than di~n· .rional since one direction may encompass more than one dimension. For a glven set of positive tu:ul~ralion paramLur.r Ph k =1.2••••• we define the two dircclional T~lUor Producl G~nuaJi;:ed Alt~rnazlnB Dlr~ctlon Implicit (TPGADI) iteration method by
c lO) (22)
given
[(A I + Pt+lBI).
6.640-02 1.330-01 1.990-01 2.65e-01 3.32e-Ol 3.980-01
g 9 o
9
N
10
~
4.65e-Ol 5.310-01 5.97e-Ql Z
.000
.250
.500
.750
= 1/4
1.000
Figure 8.3 A contour plot of the cross section on the planes z = 1/2 (top) and z = 1/4 (botrom) o[ the computed wlution to a beat conduction problem for the case h =
1/8 and ~
= 1{64
22
9. RefereDces 1.
G. Birkboff. R. S. Varga and D. M. Young. "Alternating direction implicit methods", Advances in Compuur.f, Academic PreM. New York. 3(1962),189-273.
2.
G. Birkboff Bnd R. E. Lynch, NlllMrical Solusiofl of Elliptic Problenu. SlAM, Philadelphia, 1985.
3.
R. F. Boisvert, "Families of high order acc:urate discretizations of some elliptic problems", SIAM J. Sci. and SUJI. Comp., 2(1981), 268-284.
4.
C. de Boor, "Efficient computet manipulation of tcnoor products", ACM Tran.!. Ma/h. So/rwart', 5(1979), 173-182.
S.
C. de Boor and B. Swartz, "Collocation approximation to eigenvalues of an ordinary differential equation: The principle of the thing", MaJh. Comp .• 35(1980), 679-694.
6.
C. de Boor and B. Swartz, "Collocation approximation to eigenvalues of an ordinary differential equation: Numerical illustrations". Mazh. Camp., 36(1981), 1-19.
7.
J. Douglas, Jr. and H. H. Rachford, Jr., "On the numerical solution of beat conduction problems in two or three space variables", Trcuu. kMr. Malh. Soc. 82(1956).421-439.
8.
W. R. Dykscn, ''Tensor Product Generalized Alternating Direction Implicit Methods For Solving Separable Second Order Linear Elliptic Partial Differential Equations", PhD. Thesis, Purdue University, 1982.
9.
W. R. Dyksen and J. R. Rice. "A new ordering scheme for the Hermite bicubic collocalion equations", in Elliptic Problem Sol~er:r IlT, (G. Birkhoff and A. Schoensladl. cds.), Academic Press, 1984, pp. 467-480.
10.
W. R. Dykscn and J. R. Rice, "Scale faclors for the Hermite bicubic collocalion equalions", to appear, Mva/fu:r in Computer Method:r for Partial DifferenllaJ EquatiolLf-V, (R. Vichnevetsky and R. Stepleman, eds.), IMACS, New Brunswick., NJ, 1984.
11.
W. R. Dykscn, R. E. Lynch, J. R. Rice and E. N. Houstis, "The performance of the col· location and Galcrkin methods with Hermite bi-cubics", SIAM J. Numer. Anal., 21(1984), 695-715.
u.
W. R. Dykscn, ''Tensor product generalized ADI methods for elliptic problems", Purdue University, Computer Science Department Report CSD-TR 493, September 1984, submitted for publication.
13.
W. R. DybcD, "A tensor product generalized ADI method for elliptic problems on cylindrical domains", Purdue University, Computer Science Department Report CSD-TR 495, October 1984, submitted for pubHcalion.
14.
W. R. Dyk.sen and J. R. Rice, "Symmetric versus nonsymmetric differencing". SIAM J. Sci. Stat. Comp., 6(1985), 45-48.
lS.
E. N. Houstis, "Collocation me[hods for linear elliptic problems". srr. 18(1978), 301·310.
16.
E. N. Houstis, W. F. Mitchell and J. R. Rice, "Algorithms INTCOL Bnd HERMCOL: Collocation OD rectangular domains with bicubic Hermite polynomials'. Purdue University, Computer Science Department Report CSo.TR 445, June 1983.
23
17.
E. N. Hcustis. W. F. Mitchell and J. R. Rice, "Collocation software tor second order elliptic partial differential equations", Purdue University. Computer Science pepartmcnt Report CSD-TR 446, June 1983.
18.
D. J. Jones. J. C. South and E. B. Klunker, "On the numerical solution of elliptic partial differential equations by the method of lincs", J. Compw. Plrys.• 9(1972). 496-527.
19.
R. E. Lynch. J. R. Rice and D. H. Thomas. "Direct solution of partial difference equations by tensor product methods". NlUMr. MaJh .• 6(1964}. 185-199.
2D.
R. E. Lynch, J. R. Rice and D. H. Thomas, ''Tensor product analysis of alternating direction implicit methods", J. Soc./rulwt. Appl. MaJh., 13(1965). 995-1006.
21.
R. E. Lynch, and J. R. Rice, "Convergence rates of ADI methods with smooth initial error", MaJh. Comp., 22.(1968), 311·355.
22.
R. E. Lynch. and J. R. Rice, "High accuracy approximations to solutions of elliptic partial differenlial equations", Proc. Hm. Acad. of Sci., 75(1978), 2541·2544.
23.
R. S. Martin and J. H. Wilkinson, ""The implicit QL algorithm", N~r. Math., 11(1968), 377·383.
24.
D. W. Peaccman and H. H. Rachford Jr., "The numerical solution of parabolic and elliptic differenlial equations", J. Soc.lndust. Appl. Math., 3(1955), 28-41-
25.
P. Pcrcell and M. F. Wheelcr, "A C 1 finite collocation method for elliptic equations", SIAM J. Numi!r.Anal., 17(1980), 605-622.
26.
J. R. Rice, E. N. Houstis and W. R. Dyksen, "A population of linear, second order, elliplie partial diffcrential equations on rectangular domains, Parts 1 and 2", Malh. Comp., 36(1981),475·484.
27.
J. R. Rice and R. F. Boisvert, Solving Elliptic Probli!11U using EUPACK, Springer-Verlag, New York, 1985.
28.
B. T. Smith. J. M. Boyle, J. J. Dongarra. B. S. Garbow, Y. Ikebe. V. C. Klema and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide, Springer-Verlag, New York, 1976.
29.
R. S. Varga, Matrix lurativi! Analysis, Prentice-Hall, Inc., Englewood 1962.
30.
J. H. Wilkinson. "Calculation of the eigenvalues of a symmetric tridiagonal matriI by the method of biscclion", NlJIMr. Malh., 4(1%2), 362-367.
Clif~,
New Jeney,
24
10. Apptlldls A - A Sample ELLPACK Program
The experimental version of tbe Method of Planes and the TPGADI method is implemented within the ELLPACK system (Rice and Boisvert, [27D. We use an ELLPACK program supplemented with Fortran subprograms. A aample ELLPACK program is given Figure 10.1 for the Poisson problem on the unit cube. Notc that ELLPACK "thinks" that we arc
solving a two dimensional problem. The dlscTelualion module INTERIOR COLLOCATION gencrates the Hermite bieubic collocation equations and computes the unknowns associated
with the boundary conditions (Houslis et al, [16]. [17]). The ELLPACK 'Version of INTERIOR COLLOCATION was modified to compute the Gramian matrix Bry. and to eliminate the equations of the boundary conditions 00. each plane. The formcr is trivial whereas the
latter is rather substantial. The
%
direction operator. L: = -(P(z)u.)~ +q(z)". is specified in
the function subprograms ZPCOE and ZQCOE. The
%
variable is made available to aU sub-
programs through global comnum. The matm T% appro:rimating BILDTZ. The %direction operator, L.
uu
is computed by a
=- (p (% )M.). +q(%)M, is specified in the function sub-
programs ZPCOE and ZQCOE. The discrete problem is solved by TPGADI which implements the TPGADI method (4.1). The routine BLDAXY interfaces the output from INTERIOR COLLOCATION for input to TPGADI. The acceleration parameters Pi are computed to be the eigenvaluC5 of the symmetric positive definile matrix Tz by SETRHO which uses the EISPACK routine IMTQLl (Smith et ai, [28D, (Wilk.inson, [30D. They ELI'e used in increasing order (Lynch and Rice.
[21D. The initial iterate,
U{O),
is alwaY' taken to be zero. Although the source tor these sup-
plementary program could be included in the SUBPROGRAMS segment of the BLLPACK program, we automatically load them from a separate. precompiled library.
2S
...••......••••......••.••...•.•••••..•...•••••••••••..............•...•. • •
SAMPLE Eu..PAlX E'RO:iRAM FOR nIB r.£lHID OF PLANES Willi INIERIOR OJLLOCAT£{N AN) nIB 11'G'DI I1ERATIVE MEIHID
• • • •
......................................................................... GLOBAL.
a::MIJN I TPZZlZ I Z DECLARATICliS. pAR,.V,EIER
(N:iI:lZMX. = 10) .. MDZMX:. 2 ) .. 2· S [ 1NiRY + 3)
pARI\l.ETER (NPLN.« pAJtN,ETER (NBI:M\X PARAMITER (MXlL.MX: p.ARNo£TER (Nn!KLXY (Q,f,ffl J TPRSID I CCP.f,Ool' I TPlNO'l I
JB). NCXJU,O(') • WJRI<M,{( NPI..N.1X. 2) • \\ORKNN( S 11~.N:lX.MI{) , ~(!"o'PLN.OC. S [1r.ND) • lMlKBXY (NJ«LXY) • \\ORK.(NNXLXY) • mD(NPlNo«)
B
C
o E
F
G Ii I 1
m
~.
U'" TR1JE(X,Y)
GRID.
S X POINrS S S Y POINrS
X .. 0.0 CNX"'I.O CNY=O,O CfiY=1.0
FOR'I1
Purdue e-Pubs Computer Science Technical Reports
Department of Computer Science
1984
A Tensor Product Generalized ADI Method for the Method of Planes Wayne R. Dyksen Report Number: 84-494
Dyksen, Wayne R., "A Tensor Product Generalized ADI Method for the Method of Planes" (1984). Computer Science Technical Reports. Paper 414. http://docs.lib.purdue.edu/cstech/414
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.
A TENSOR PRODUCT GENERALIZED ADI METHOD FOR THE METHOD OF PLANES
Wayne R. Dyksen Department o( Computer Sciences Purdue University West Lafayette, Indiana 479fJ7
CSD-TR 494 October 1984
ABSTRACT
We consider solving separable second order linear elliptic partial differential equations in three independent variables. If the partial differential operator separates into two (acton, one depending on x and y, and one depending on z. then we use the MetluJd of Planel to obtain a discrete problem which we write in tensor product form as (T~C?JBZ1
+/0AZ1 )C =F.
We apply a new iterative method, the Teruor Product Generalized Alternating Direction Implicit method, to solve the discrete problem. We study a specifie implementation which uses Her· mite bicubie collocation in tbe ry direction and symmetric finite differences in the z direction. We demonstrate that this method is a fast and accurate way to solve the large linear systems arising from three dimensional elliptic problems.
1980 Mrrr1l#1IUJlics SUb]«l Crtu~f/kiJrIOfl. Primu)'. 6SIllO, 65M20, 65N35.
A Tensor Product Generalized ADI Method for the Method of Planes
Wayne R. Dyksen
1. [ntrodoct!oD. Let R be a rectangular domain. We consider solving second order linear elliptic problems of the form
LXyu. +Lcu. =/(.r,y.z) inR0[a,..bz l II.
= g,,(x,y)
II.
= gbo(X,y)
onR0az a(x,y)u +13(.r,y)u.. =g(.r,y,z) on aRfi!J(az,bc )
(1.1)
a(;r,Y)13(x.y)", 0 a 2 (:t,y) + 132(x,y» 0
on R 0bp on aR on aR
where L, =-(P(z)uz)z +q(z)u, p>O,q2:0,
(12)
and LZJ is a general elliptic operator in x and y. We express a discrete problem in terms of tensor products of matrices resulting from lower dimensional, Bnd hence much simpler, problems. We apply a new iterative tcchnique to solve this discrete problem. We obtain a fast method for solving a large class of elliptic problenu in three
dim~nsions.
Scction 2 presents a brief introduction to the Tensor Product Genualiud Alterll/lling Direction Implicit (TPGADI) method. In Section 3 we extend the Method of Lines (Jones et ai,
[I8D to the Merhod of Planes to obtain a discrete problem which we write in tensor product form as
2
We present the TPGADI method for solving lIuch discrete problems in Section 4. We apply the Method of Planes 115iog Hermite bicubics in x and y Bnd fini.te differences in z in Sec-
tions S. 6 and 7. We consider a specific implementation in Section 8; we show that the Method of Planes together with the TPGADI method give a powerful tool for solving a large
class of eUiptic problems in three dimensions. The TPGADI method has been used effectively to solve discrete elliptic problems arising from olher discretizations in both two and three dimensions. We have used it in conjunction with the Method of Lines (Dyksen, {8D and the Hermite bicubic coliocatioD equations (Dyksen, [12D. In three dimensions. we have used it to solve elliplic problems on cylindrical
domains with holes (Dykscn. [13D·
2. The Two Dlftctlonal Tenmr Product General.lad Ani Methodlll Let A.t and Bt be matrices of order NI; XNt • and consider the lineElI' IS)'Stem (2.1)
We wi.!!h to solve the two directional problem (2.1) by using methods for one dircctional. simpler problems involving Ah Bit A2 and B2' We usc the term dir~clional rather than di~n· .rional since one direction may encompass more than one dimension. For a glven set of positive tu:ul~ralion paramLur.r Ph k =1.2••••• we define the two dircclional T~lUor Producl G~nuaJi;:ed Alt~rnazlnB Dlr~ctlon Implicit (TPGADI) iteration method by
c lO) (22)
given
[(A I + Pt+lBI).
6.640-02 1.330-01 1.990-01 2.65e-01 3.32e-Ol 3.980-01
g 9 o
9
N
10
~
4.65e-Ol 5.310-01 5.97e-Ql Z
.000
.250
.500
.750
= 1/4
1.000
Figure 8.3 A contour plot of the cross section on the planes z = 1/2 (top) and z = 1/4 (botrom) o[ the computed wlution to a beat conduction problem for the case h =
1/8 and ~
= 1{64
22
9. RefereDces 1.
G. Birkboff. R. S. Varga and D. M. Young. "Alternating direction implicit methods", Advances in Compuur.f, Academic PreM. New York. 3(1962),189-273.
2.
G. Birkboff Bnd R. E. Lynch, NlllMrical Solusiofl of Elliptic Problenu. SlAM, Philadelphia, 1985.
3.
R. F. Boisvert, "Families of high order acc:urate discretizations of some elliptic problems", SIAM J. Sci. and SUJI. Comp., 2(1981), 268-284.
4.
C. de Boor, "Efficient computet manipulation of tcnoor products", ACM Tran.!. Ma/h. So/rwart', 5(1979), 173-182.
S.
C. de Boor and B. Swartz, "Collocation approximation to eigenvalues of an ordinary differential equation: The principle of the thing", MaJh. Comp .• 35(1980), 679-694.
6.
C. de Boor and B. Swartz, "Collocation approximation to eigenvalues of an ordinary differential equation: Numerical illustrations". Mazh. Camp., 36(1981), 1-19.
7.
J. Douglas, Jr. and H. H. Rachford, Jr., "On the numerical solution of beat conduction problems in two or three space variables", Trcuu. kMr. Malh. Soc. 82(1956).421-439.
8.
W. R. Dykscn, ''Tensor Product Generalized Alternating Direction Implicit Methods For Solving Separable Second Order Linear Elliptic Partial Differential Equations", PhD. Thesis, Purdue University, 1982.
9.
W. R. Dyksen and J. R. Rice. "A new ordering scheme for the Hermite bicubic collocalion equations", in Elliptic Problem Sol~er:r IlT, (G. Birkhoff and A. Schoensladl. cds.), Academic Press, 1984, pp. 467-480.
10.
W. R. Dykscn and J. R. Rice, "Scale faclors for the Hermite bicubic collocalion equalions", to appear, Mva/fu:r in Computer Method:r for Partial DifferenllaJ EquatiolLf-V, (R. Vichnevetsky and R. Stepleman, eds.), IMACS, New Brunswick., NJ, 1984.
11.
W. R. Dykscn, R. E. Lynch, J. R. Rice and E. N. Houstis, "The performance of the col· location and Galcrkin methods with Hermite bi-cubics", SIAM J. Numer. Anal., 21(1984), 695-715.
u.
W. R. Dykscn, ''Tensor product generalized ADI methods for elliptic problems", Purdue University, Computer Science Department Report CSD-TR 493, September 1984, submitted for publication.
13.
W. R. DybcD, "A tensor product generalized ADI method for elliptic problems on cylindrical domains", Purdue University, Computer Science Department Report CSD-TR 495, October 1984, submitted for pubHcalion.
14.
W. R. Dyk.sen and J. R. Rice, "Symmetric versus nonsymmetric differencing". SIAM J. Sci. Stat. Comp., 6(1985), 45-48.
lS.
E. N. Houstis, "Collocation me[hods for linear elliptic problems". srr. 18(1978), 301·310.
16.
E. N. Houstis, W. F. Mitchell and J. R. Rice, "Algorithms INTCOL Bnd HERMCOL: Collocation OD rectangular domains with bicubic Hermite polynomials'. Purdue University, Computer Science Department Report CSo.TR 445, June 1983.
23
17.
E. N. Hcustis. W. F. Mitchell and J. R. Rice, "Collocation software tor second order elliptic partial differential equations", Purdue University. Computer Science pepartmcnt Report CSD-TR 446, June 1983.
18.
D. J. Jones. J. C. South and E. B. Klunker, "On the numerical solution of elliptic partial differential equations by the method of lincs", J. Compw. Plrys.• 9(1972). 496-527.
19.
R. E. Lynch. J. R. Rice and D. H. Thomas. "Direct solution of partial difference equations by tensor product methods". NlUMr. MaJh .• 6(1964}. 185-199.
2D.
R. E. Lynch, J. R. Rice and D. H. Thomas, ''Tensor product analysis of alternating direction implicit methods", J. Soc./rulwt. Appl. MaJh., 13(1965). 995-1006.
21.
R. E. Lynch, and J. R. Rice, "Convergence rates of ADI methods with smooth initial error", MaJh. Comp., 22.(1968), 311·355.
22.
R. E. Lynch. and J. R. Rice, "High accuracy approximations to solutions of elliptic partial differenlial equations", Proc. Hm. Acad. of Sci., 75(1978), 2541·2544.
23.
R. S. Martin and J. H. Wilkinson, ""The implicit QL algorithm", N~r. Math., 11(1968), 377·383.
24.
D. W. Peaccman and H. H. Rachford Jr., "The numerical solution of parabolic and elliptic differenlial equations", J. Soc.lndust. Appl. Math., 3(1955), 28-41-
25.
P. Pcrcell and M. F. Wheelcr, "A C 1 finite collocation method for elliptic equations", SIAM J. Numi!r.Anal., 17(1980), 605-622.
26.
J. R. Rice, E. N. Houstis and W. R. Dyksen, "A population of linear, second order, elliplie partial diffcrential equations on rectangular domains, Parts 1 and 2", Malh. Comp., 36(1981),475·484.
27.
J. R. Rice and R. F. Boisvert, Solving Elliptic Probli!11U using EUPACK, Springer-Verlag, New York, 1985.
28.
B. T. Smith. J. M. Boyle, J. J. Dongarra. B. S. Garbow, Y. Ikebe. V. C. Klema and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide, Springer-Verlag, New York, 1976.
29.
R. S. Varga, Matrix lurativi! Analysis, Prentice-Hall, Inc., Englewood 1962.
30.
J. H. Wilkinson. "Calculation of the eigenvalues of a symmetric tridiagonal matriI by the method of biscclion", NlJIMr. Malh., 4(1%2), 362-367.
Clif~,
New Jeney,
24
10. Apptlldls A - A Sample ELLPACK Program
The experimental version of tbe Method of Planes and the TPGADI method is implemented within the ELLPACK system (Rice and Boisvert, [27D. We use an ELLPACK program supplemented with Fortran subprograms. A aample ELLPACK program is given Figure 10.1 for the Poisson problem on the unit cube. Notc that ELLPACK "thinks" that we arc
solving a two dimensional problem. The dlscTelualion module INTERIOR COLLOCATION gencrates the Hermite bieubic collocation equations and computes the unknowns associated
with the boundary conditions (Houslis et al, [16]. [17]). The ELLPACK 'Version of INTERIOR COLLOCATION was modified to compute the Gramian matrix Bry. and to eliminate the equations of the boundary conditions 00. each plane. The formcr is trivial whereas the
latter is rather substantial. The
%
direction operator. L: = -(P(z)u.)~ +q(z)". is specified in
the function subprograms ZPCOE and ZQCOE. The
%
variable is made available to aU sub-
programs through global comnum. The matm T% appro:rimating BILDTZ. The %direction operator, L.
uu
is computed by a
=- (p (% )M.). +q(%)M, is specified in the function sub-
programs ZPCOE and ZQCOE. The discrete problem is solved by TPGADI which implements the TPGADI method (4.1). The routine BLDAXY interfaces the output from INTERIOR COLLOCATION for input to TPGADI. The acceleration parameters Pi are computed to be the eigenvaluC5 of the symmetric positive definile matrix Tz by SETRHO which uses the EISPACK routine IMTQLl (Smith et ai, [28D, (Wilk.inson, [30D. They ELI'e used in increasing order (Lynch and Rice.
[21D. The initial iterate,
U{O),
is alwaY' taken to be zero. Although the source tor these sup-
plementary program could be included in the SUBPROGRAMS segment of the BLLPACK program, we automatically load them from a separate. precompiled library.
2S
...••......••••......••.••...•.•••••..•...•••••••••••..............•...•. • •
SAMPLE Eu..PAlX E'RO:iRAM FOR nIB r.£lHID OF PLANES Willi INIERIOR OJLLOCAT£{N AN) nIB 11'G'DI I1ERATIVE MEIHID
• • • •
......................................................................... GLOBAL.
a::MIJN I TPZZlZ I Z DECLARATICliS. pAR,.V,EIER
(N:iI:lZMX. = 10) .. MDZMX:. 2 ) .. 2· S [ 1NiRY + 3)
pARI\l.ETER (NPLN.« pAJtN,ETER (NBI:M\X PARAMITER (MXlL.MX: p.ARNo£TER (Nn!KLXY (Q,f,ffl J TPRSID I CCP.f,Ool' I TPlNO'l I
JB). NCXJU,O(') • WJRI<M,{( NPI..N.1X. 2) • \\ORKNN( S 11~.N:lX.MI{) , ~(!"o'PLN.OC. S [1r.ND) • lMlKBXY (NJ«LXY) • \\ORK.(NNXLXY) • mD(NPlNo«)
B
C
o E
F
G Ii I 1
m
~.
U'" TR1JE(X,Y)
GRID.
S X POINrS S S Y POINrS
X .. 0.0 CNX"'I.O CNY=O,O CfiY=1.0
FOR'I1
