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// $Id$ |
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/* |
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****************************************************************************** |
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* * |
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* COPYRIGHT ACcESS 2004 - All Rights Reserved * |
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* * |
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* This software is the property of ACcESS. No part of this code * |
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* may be copied in any form or by any means without the expressed written * |
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* consent of ACcESS. Copying, use or modification of this software * |
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* by any unauthorised person is illegal unless that person has a software * |
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* license agreement with ACcESS. * |
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* * |
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****************************************************************************** |
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*/ |
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|
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#if !defined escript_LocalOps_H |
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#define escript_LocalOps_H |
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#include <math.h> |
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namespace escript { |
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|
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|
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/** |
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\brief |
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solves a 1x1 eigenvalue A*V=ev*V problem |
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|
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\param A00 Input - A_00 |
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\param ev0 Output - eigenvalue |
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*/ |
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inline |
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void eigenvalues1(const double A00,double* ev0) { |
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|
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*ev0=A00; |
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|
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} |
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/** |
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\brief |
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solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A |
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|
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\param A00 Input - A_00 |
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\param A01 Input - A_01 |
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\param A11 Input - A_11 |
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\param ev0 Output - smallest eigenvalue |
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\param ev1 Output - largest eigenvalue |
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*/ |
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inline |
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void eigenvalues2(const double A00,const double A01,const double A11, |
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double* ev0, double* ev1) { |
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const register double trA=(A00+A11)/2.; |
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const register double A_00=A00-trA; |
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const register double A_11=A11-trA; |
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const register double s=sqrt(A01*A01-A_00*A_11); |
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*ev0=trA-s; |
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*ev1=trA+s; |
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} |
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/** |
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\brief |
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solves a 3x3 eigenvalue A*V=ev*V problem for symmetric A |
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|
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\param A00 Input - A_00 |
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\param A01 Input - A_01 |
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\param A02 Input - A_02 |
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\param A11 Input - A_11 |
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\param A12 Input - A_12 |
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\param A22 Input - A_22 |
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\param ev0 Output - smallest eigenvalue |
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\param ev1 Output - eigenvalue |
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\param ev2 Output - largest eigenvalue |
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*/ |
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inline |
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void eigenvalues3(const double A00, const double A01, const double A02, |
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const double A11, const double A12, |
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const double A22, |
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double* ev0, double* ev1,double* ev2) { |
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|
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const register double trA=(A00+A11+A22)/3.; |
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const register double A_00=A00-trA; |
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const register double A_11=A11-trA; |
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const register double A_22=A22-trA; |
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const register double A01_2=A01*A01; |
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const register double A02_2=A02*A02; |
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const register double A12_2=A12*A12; |
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const register double p=A02_2+A12_2+A01_2+(A_00*A_00+A_11*A_11+A_22*A_22)/2.; |
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const register double q=(A02_2*A_11+A12_2*A_00+A01_2*A_22)-(A_00*A_11*A_22+2*A01*A12*A02); |
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const register double sq_p=sqrt(p/3.); |
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register double z=-q/(2*pow(sq_p,3)); |
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if (z<-1.) { |
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z=-1.; |
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} else if (z>1.) { |
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z=1.; |
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} |
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const register double alpha_3=acos(z)/3.; |
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*ev2=trA+2.*sq_p*cos(alpha_3); |
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*ev1=trA-2.*sq_p*cos(alpha_3+M_PI/3.); |
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*ev0=trA-2.*sq_p*cos(alpha_3-M_PI/3.); |
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} |
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/** |
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\brief |
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solves a 1x1 eigenvalue A*V=ev*V problem for symmetric A |
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|
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\param A00 Input - A_00 |
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\param ev0 Output - eigenvalue |
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\param V00 Output - eigenvector |
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\param tol Input - tolerance to identify to eigenvalues |
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*/ |
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inline |
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void eigenvalues_and_eigenvectors1(const double A00,double* ev0,double* V00,const double tol) |
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{ |
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eigenvalues1(A00,ev0); |
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*V00=1.; |
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return; |
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} |
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/** |
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\brief |
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returns a non-zero vector in the kernel of [[A00,A01],[A01,A11]] assuming that the kernel dimension is at least 1. |
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|
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\param A00 Input - matrix component |
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\param A10 Input - matrix component |
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\param A01 Input - matrix component |
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\param A11 Input - matrix component |
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\param V0 Output - vector component |
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\param V1 Output - vector component |
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*/ |
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inline |
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void vectorInKernel2(const double A00,const double A10,const double A01,const double A11, |
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double* V0, double*V1) |
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{ |
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register double absA00=fabs(A00); |
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register double absA01=fabs(A01); |
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register double absA10=fabs(A10); |
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register double absA11=fabs(A11); |
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register double m=absA11>absA01 ? absA11 : absA01; |
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if (absA00>m || absA10>m) { |
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*V0=-A10; |
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*V1=A00; |
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} else { |
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if (m<=0) { |
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*V0=1.; |
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*V1=0.; |
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} else { |
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*V0=A11; |
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*V1=-A01; |
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} |
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} |
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} |
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/** |
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\brief |
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returns a non-zero vector in the kernel of [[A00,A01,A02],[A10,A11,A12],[A20,A21,A22]] |
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assuming that the kernel dimension is at least 1 and A00 is non zero. |
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|
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\param A00 Input - matrix component |
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\param A10 Input - matrix component |
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\param A20 Input - matrix component |
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\param A01 Input - matrix component |
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\param A11 Input - matrix component |
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\param A21 Input - matrix component |
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\param A02 Input - matrix component |
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\param A12 Input - matrix component |
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\param A22 Input - matrix component |
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\param V0 Output - vector component |
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\param V1 Output - vector component |
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\param V2 Output - vector component |
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*/ |
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inline |
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void vectorInKernel3__nonZeroA00(const double A00,const double A10,const double A20, |
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const double A01,const double A11,const double A21, |
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const double A02,const double A12,const double A22, |
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double* V0,double* V1,double* V2) |
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{ |
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double TEMP0,TEMP1; |
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register const double I00=1./A00; |
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register const double IA10=I00*A10; |
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register const double IA20=I00*A20; |
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vectorInKernel2(A11-IA10*A01,A21-IA20*A01,A12-IA10*A02,A22-IA20*A02,&TEMP0,&TEMP1); |
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*V0=-(A10*TEMP0+A20*TEMP1); |
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*V1=A00*TEMP0; |
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*V2=A00*TEMP1; |
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} |
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|
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/** |
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\brief |
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solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A. Eigenvectors are |
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ordered by increasing value and eigen vectors are normalizeVector3d such that |
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length is zero and first non-zero component is positive. |
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|
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\param A00 Input - A_00 |
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\param A01 Input - A_01 |
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\param A11 Input - A_11 |
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\param ev0 Output - smallest eigenvalue |
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\param ev1 Output - eigenvalue |
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\param V00 Output - eigenvector componenent coresponding to ev0 |
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\param V10 Output - eigenvector componenent coresponding to ev0 |
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\param V01 Output - eigenvector componenent coresponding to ev1 |
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\param V11 Output - eigenvector componenent coresponding to ev1 |
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\param tol Input - tolerance to identify to eigenvalues |
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*/ |
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inline |
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void eigenvalues_and_eigenvectors2(const double A00,const double A01,const double A11, |
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double* ev0, double* ev1, |
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double* V00, double* V10, double* V01, double* V11, |
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const double tol) |
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{ |
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double TEMP0,TEMP1; |
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eigenvalues2(A00,A01,A11,ev0,ev1); |
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const register double absev0=fabs(*ev0); |
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const register double absev1=fabs(*ev1); |
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register double max_ev=absev0>absev1 ? absev0 : absev1; |
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if (fabs((*ev0)-(*ev1))<tol*max_ev) { |
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*V00=1.; |
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*V10=0.; |
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*V01=0.; |
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*V11=1.; |
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} else { |
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vectorInKernel2(A00-(*ev0),A01,A01,A11-(*ev0),&TEMP0,&TEMP1); |
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const register double scale=1./sqrt(TEMP0*TEMP0+TEMP1*TEMP1); |
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if (TEMP0<0.) { |
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*V00=-TEMP0*scale; |
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*V10=-TEMP1*scale; |
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if (TEMP1<0.) { |
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*V01= *V10; |
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*V11=-(*V00); |
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} else { |
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*V01=-(*V10); |
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*V11= (*V00); |
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} |
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} else if (TEMP0>0.) { |
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*V00=TEMP0*scale; |
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*V10=TEMP1*scale; |
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if (TEMP1<0.) { |
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*V01=-(*V10); |
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*V11= (*V00); |
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} else { |
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*V01= (*V10); |
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*V11=-(*V00); |
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} |
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} else { |
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*V00=0.; |
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*V10=1; |
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*V11=0.; |
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*V01=1.; |
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} |
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} |
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} |
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/** |
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\brief |
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nomalizes a 3-d vector such that length is one and first non-zero component is positive. |
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|
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\param V0 - vector componenent |
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\param V1 - vector componenent |
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\param V2 - vector componenent |
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*/ |
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inline |
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void normalizeVector3(double* V0,double* V1,double* V2) |
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{ |
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register double s; |
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if (*V0>0) { |
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s=1./sqrt((*V0)*(*V0)+(*V1)*(*V1)+(*V2)*(*V2)); |
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*V0*=s; |
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*V1*=s; |
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*V2*=s; |
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} else if (*V0<0) { |
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s=-1./sqrt((*V0)*(*V0)+(*V1)*(*V1)+(*V2)*(*V2)); |
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*V0*=s; |
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*V1*=s; |
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*V2*=s; |
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} else { |
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if (*V1>0) { |
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s=1./sqrt((*V1)*(*V1)+(*V2)*(*V2)); |
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*V1*=s; |
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*V2*=s; |
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} else if (*V1<0) { |
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s=-1./sqrt((*V1)*(*V1)+(*V2)*(*V2)); |
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*V1*=s; |
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*V2*=s; |
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} else { |
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*V2=1.; |
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} |
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} |
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} |
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/** |
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\brief |
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solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A. Eigenvectors are |
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ordered by increasing value and eigen vectors are normalizeVector3d such that |
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length is zero and first non-zero component is positive. |
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|
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\param A00 Input - A_00 |
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\param A01 Input - A_01 |
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\param A11 Input - A_11 |
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\param ev0 Output - smallest eigenvalue |
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\param ev1 Output - eigenvalue |
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\param V00 Output - eigenvector componenent coresponding to ev0 |
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\param V10 Output - eigenvector componenent coresponding to ev0 |
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\param V01 Output - eigenvector componenent coresponding to ev1 |
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\param V11 Output - eigenvector componenent coresponding to ev1 |
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\param tol Input - tolerance to identify to eigenvalues |
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*/ |
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inline |
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void eigenvalues_and_eigenvectors3(const double A00, const double A01, const double A02, |
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const double A11, const double A12, const double A22, |
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double* ev0, double* ev1, double* ev2, |
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double* V00, double* V10, double* V20, |
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double* V01, double* V11, double* V21, |
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double* V02, double* V12, double* V22, |
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const double tol) |
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{ |
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register const double absA01=fabs(A01); |
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register const double absA02=fabs(A02); |
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register const double m=absA01>absA02 ? absA01 : absA02; |
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if (m<=0) { |
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double TEMP_V00,TEMP_V10,TEMP_V01,TEMP_V11,TEMP_EV0,TEMP_EV1; |
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eigenvalues_and_eigenvectors2(A11,A12,A22, |
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&TEMP_EV0,&TEMP_EV1, |
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&TEMP_V00,&TEMP_V10,&TEMP_V01,&TEMP_V11,tol); |
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if (A00<=TEMP_EV0) { |
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*V00=1.; |
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*V10=0.; |
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*V20=0.; |
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*V01=0.; |
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*V11=TEMP_V00; |
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*V21=TEMP_V10; |
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*V02=0.; |
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*V12=TEMP_V01; |
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*V22=TEMP_V11; |
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*ev0=A00; |
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*ev1=TEMP_EV0; |
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*ev2=TEMP_EV1; |
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} else if (A00>TEMP_EV1) { |
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*V00=TEMP_V00; |
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*V10=TEMP_V10; |
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*V20=0.; |
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*V01=TEMP_V01; |
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*V11=TEMP_V11; |
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*V21=0.; |
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*V02=0.; |
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*V12=0.; |
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*V22=1.; |
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*ev0=TEMP_EV0; |
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*ev1=TEMP_EV1; |
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*ev0=A00; |
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} else { |
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*V00=TEMP_V00; |
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*V10=0; |
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*V20=TEMP_V10; |
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*V01=0.; |
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*V11=1.; |
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*V21=0.; |
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*V02=TEMP_V01; |
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*V12=0.; |
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*V22=TEMP_V11; |
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*ev0=TEMP_EV0; |
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*ev1=A00; |
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*ev2=TEMP_EV1; |
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} |
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} else { |
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eigenvalues3(A00,A01,A02,A11,A12,A22,ev0,ev1,ev2); |
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const register double absev0=fabs(*ev0); |
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const register double absev1=fabs(*ev1); |
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const register double absev2=fabs(*ev2); |
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register double max_ev=absev0>absev1 ? absev0 : absev1; |
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max_ev=max_ev>absev2 ? max_ev : absev2; |
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const register double d_01=fabs((*ev0)-(*ev1)); |
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const register double d_12=fabs((*ev1)-(*ev2)); |
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const register double max_d=d_01>d_12 ? d_01 : d_12; |
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if (max_d<=tol*max_ev) { |
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*V00=1.; |
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*V10=0; |
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*V20=0; |
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*V01=0; |
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*V11=1.; |
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*V21=0; |
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*V02=0; |
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*V12=0; |
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*V22=1.; |
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} else { |
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const register double S00=A00-(*ev0); |
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const register double absS00=fabs(S00); |
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if (fabs(S00)>m) { |
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vectorInKernel3__nonZeroA00(S00,A01,A02,A01,A11-(*ev0),A12,A02,A12,A22-(*ev0),V00,V10,V20); |
| 378 |
} else if (absA02<m) { |
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vectorInKernel3__nonZeroA00(A01,A11-(*ev0),A12,S00,A01,A02,A02,A12,A22-(*ev0),V00,V10,V20); |
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} else { |
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vectorInKernel3__nonZeroA00(A02,A12,A22-(*ev0),S00,A01,A02,A01,A11-(*ev0),A12,V00,V10,V20); |
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} |
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normalizeVector3(V00,V10,V20);; |
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const register double T00=A00-(*ev2); |
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const register double absT00=fabs(T00); |
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if (fabs(T00)>m) { |
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vectorInKernel3__nonZeroA00(T00,A01,A02,A01,A11-(*ev2),A12,A02,A12,A22-(*ev2),V02,V12,V22); |
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} else if (absA02<m) { |
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vectorInKernel3__nonZeroA00(A01,A11-(*ev2),A12,T00,A01,A02,A02,A12,A22-(*ev2),V02,V12,V22); |
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} else { |
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vectorInKernel3__nonZeroA00(A02,A12,A22-(*ev2),T00,A01,A02,A01,A11-(*ev2),A12,V02,V12,V22); |
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} |
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const register double dot=(*V02)*(*V00)+(*V12)*(*V10)+(*V22)*(*V20); |
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*V02-=dot*(*V00); |
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*V12-=dot*(*V10); |
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*V22-=dot*(*V20); |
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normalizeVector3(V02,V12,V22); |
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*V01=(*V10)*(*V22)-(*V12)*(*V20); |
| 399 |
*V11=(*V20)*(*V02)-(*V00)*(*V22); |
| 400 |
*V21=(*V00)*(*V12)-(*V02)*(*V10); |
| 401 |
normalizeVector3(V01,V11,V21); |
| 402 |
} |
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} |
| 404 |
} |
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} // end of namespace |
| 406 |
#endif |
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