Millepede-II V04-17-03
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Symmetric (band) matrix routines. More...
Go to the source code of this file.
Functions/Subroutines | |
subroutine | dchdec (w, n, aux) |
Decomposition of symmetric matrix. More... | |
subroutine | dchslv (w, n, b, x) |
solution B -> X More... | |
subroutine | dchinv (w, n, v) |
inversion More... | |
real(mps) function | condes (w, n, aux) |
Etimate condition. More... | |
subroutine | dbcdec (w, mp1, n, aux) |
Decomposition of symmetric band matrix. More... | |
subroutine | dbcslv (w, mp1, n, b, x) |
solution B -> X More... | |
subroutine | dbciel (w, mp1, n, v) |
V = inverse band matrix elements. More... | |
subroutine | dbcinb (w, mp1, n, vs) |
VS = band part of inverse symmetric matrix. More... | |
subroutine | dbcinv (w, mp1, n, vs) |
VS = inverse symmetric matrix. More... | |
subroutine | dbcprv (w, mp1, n, b) |
Print corr band matrix and pars. More... | |
subroutine | dbcprb (w, mp1, n) |
Print band matrix. More... | |
subroutine | db2dec (w, n, aux) |
Decomposition (M=1). More... | |
subroutine | db2slv (w, n, b, x) |
solution B -> X More... | |
subroutine | db2iel (w, n, v) |
V = inverse band matrix elements. More... | |
subroutine | db3dec (w, n, aux) |
Decomposition (M=2). More... | |
subroutine | db3slv (w, n, b, x) |
solution B -> X More... | |
subroutine | db3iel (w, n, v) |
V = inverse band matrix elements. More... | |
subroutine | dcfdec (w, n) |
Decomposition of symmetric matrix. More... | |
subroutine | dbfdec (w, mp1, n) |
Decomposition of symmetric band matrix. More... | |
Symmetric (band) matrix routines.
For the original broken lines implementation by V. Blobel (University Hamburg).
************************************************************* * * * Subroutines for symmetric and symmetric band matrices, * * based on the (square root free) Cholesky decomposition. * * * ************************************************************* All floating point arguments are in DOUBLE PRECISION (and all entry names start with a D). The Cholesky decomposition transforms a symmetric matrix W e.g. the matrix from normal equation of least squares, according to W = L D L^ (L^ means L transposed) where D is a diagonal matrix and L is a unit triangular matrix (diagonal elements all ones, all elements above diagonal zero). The above decomposition allows to solve a matrix equation W x = b in two steps, using an auxiliary vector y: solve L y = b for y, and solve D L^ x = y for x. The inverse matrix of W can be calculated from the decomposition. In least-squares normal equations the inverse matrix is equal to the covariance matrix of the fitted parameters. All diagonal elements of the inverse matrix, the parameter variances, are positive, and the matrix is positive-definite (all eigenvalues > 0). The Cholesky algorithm is stable for a positive-definite matrix. The standard form of the Cholesky algorithm includes n square roots for a n-by-n matrix, and is possible only for positive-definite matrices. The version used here is squareroot-free; this algorithm does not necessarily break down in the indefinite case, although it is potentially unstable in this case. All decomposition routines include a check for singularity, and this check needs an auxiliary array AUX of dimension n. Method: The Cholesky algorithm for symmetric matrix decomposition makes use of the symmetry. The operation count (leading term) is n**3/6 (compared to n**3/3 for normal Gaussian elimination). The solution of the two triangular systems involves operations proportional to n**2. The real advantage of the Cholesky algorithm is for band matrices, where all matrix elements outside of a band with total width (2m+1) around the diagonal are zero. The band structure is kept in the decomposition, and allows a fast solution of matrix equations. The operation count (leading term) is proportional to m**2*n and thus (for fixed m) linear in n. Thus for n=100 and m=2 the Cholesky algorithm for the band matrix is 1000 times faster than the standard solution method. The inverse of a band matrix is a full matrix. It is not necessary to calculate the inverse, if only the solution for a matrix equation is needed. However the inverse is often needed, because the elements of the inverse are the variances and covariances of parameters in a least-squares fit. The inverse can be calculated afterwards from the decomposition. Since the inverse matrix is a full matrix, this has of course an operation count proportional to n**3. Usually only the elements of the inverse in and around the diagonal are really needed, and this subset of inverse elements, corresponding to the original band, can be calculated from the decomposition with an operation count, which is linear in n. Thus all variances (the diagonal elements) and covariances between neighbour parameters are calculated in a short time even for large matrices. Matrix storage: the mathematical indexing of matrix elements follows the scheme: ( W11 W12 W13 ... W1n ) ( W21 W22 W23 ... W2n ) W = ( ... ... ... ... ) ( ... ... ... ... ) ( Wn1 Wn2 Wn3 ... Wnn ) and a storage in an array would require n**2 words, although the symmetric matrix has only (n**2+n)/2 different elements, and a band matrix has less than (m+1)*n different elements. Therefore the following storage schemes are used. Symmetric matrix: the elements are in the order W11 W12 W22 W13 W23 W33 W14 ... Wnn with total (n**2+n)/2 array elements. Band matrix: a band matrix of bandwidth m is stored in an array of dimension W(m+1,n), according to W(1,.) W(2,.) W(3,.) -------------------------------- W11 W12 W13 W22 W23 W24 W33 W34 W35 ... Wnn - - The example is for a bandwidth of m=2; three elements at the end are unused. The diagonal elements are in the array elements W(1,.). This package includes subroutines for: (1) Symmetric matrix W: decomposition, solution, inverse (2) Symmetric band matrix: decomposition, solution, complete inverse and band part of the inverse (3) Symmetric band matrix of band width m=1: decomposition, solution, complete, inverse and band part of the inverse (4) Symmetric band matrix of band width m=2: decomposition, solution, complete, inverse and band part of the inverse (5) Symmetric matrix with band structure, bordered by full row/col (not yet included) The subroutines for a fixed band width of m=1 and of m=2 are faster than the general routine, because certain loops are avoided and replaced by the direct code. Historical remark: the square-root algorithm was invented by the french Mathematician Andre-Louis Cholesky (1875 - 1918). Cholesky's method of computing solutions to the normal equations was published 1924, after the death of Cholesky, by Benoit. The method received little attention after its publication in 1924. In 1948 the method was analysed in a paper by Fox, Huskey and Wilkinson, and in the same year Turing published a paper on the stability of the method. The fast method to calculate the band part of the inverse matrix is usually not mentioned in the literature. An exception is: I.S.Duff, A.M.Erisman and J.K.Reid, Direct Methods for Sparse Matrices, Oxford Science Publications, 1986. The following original work is quoted in this book: K.Takahashi, J.Fagan and M.Chin, Formation of a sparse bus impedance matrix and its application to short circuit study. Proceedings 8th PICA Conference, Minneapolis, Minnesota, 1973 A.M.Erisman and W.F.Tinney, On computing certain elements of the inverse of a sparse matrix, CACM 18, 177-179, 1975 symmetric decomposit. solution inv-element inverse ---------------- |-----------|-----------|--------------|-----------| n x n matrix DCHDEC DCHSLV - DCHINV band matrix m,n DBCDEC DBCSLV DBCIEL/DBCINB DBCINV bandwidth m=1 DB2DEC DB2SLV DB2IEL - bandwidth m=2 DB3DEC DB3SLV DB3IEL - The DB2... and DB3... routines are special routines for a fixed bandwidth of 1 and 2, they are faster versions of the general DBG... routines. The complete inverse matrix can be obtained by DBGINV. The routine DBGPRI can be used to print all types of band matrices. The decomposition in routines ...DEC replaces (overwrites) the original matrix (the number of elements is identical). All other routines require W to be the already decomposed matrix. The matrix L is a unit lower triangular matrix, with ones on the diagonal, which have not be stored. Instead the inverse of the diagonal elements of matrix D are stored in those places. In the solution routines ...SLV the array B is the right-hand matrix, the array is the resulting solution. The same array can be used for B and X. W(.) and V(.) are symmetric n-by-n matrices with (N*N+N)/2 elements SUBROUTINE DCHDEC(W,N, AUX) ! decomposition, symmetric matrix SUBROUTINE DCHSLV(W,N,B, X) ! solution B -> X SUBROUTINE DCHINV(W,N, V) ! inversion SUBROUTINE DCFDEC(W,N) ! modified composition, symmetric ! alternative to DCHDEC W(.) and V(.) are band matrices, n rows, band width m (i.e. the total width of the band is (2m+1). With MP1 = m +1, the array has dimension W(MP1,N). The symmetric matrix VS has (N*N+N)/2 elements SUBROUTINE DBCDEC(W,MP1,N, AUX) ! decomposition, bandwidth M SUBROUTINE DBCSLV(W,MP1,N,B, X) ! solution B -> X SUBROUTINE DBCIEL(W,MP1,N, V) ! V = inverse band matrix elements SUBROUTINE DBCINV(W,MP1,N, VS) ! V = inverse symmetric matrix SUBROUTINE DBFDEC(W,MP1,N) ! modified decomposition, bandwidth M ! alternative to DBCDEC SUBROUTINE DBCPRB(W,MP1,N) ! print band matrix SUBROUTINE DBCPRV(W,MP1,N,B) ! print corr band matrix and pars SUBROUTINE DB2DEC(W,N, AUX) ! decomposition (M=1) SUBROUTINE DB2SLV(W,N,B, X) ! solution B -> X SUBROUTINE DB2IEL(W,N, V) ! V = inverse band matrix elements SUBROUTINE DB3DEC(W,N, AUX) ! decomposition (M=2) SUBROUTINE DB3SLV(W,N,B, X) ! solution B -> X SUBROUTINE DB3IEL(W,N, V) ! V = inverse band matrix elements
Definition in file Dbandmatrix.f90.
real(mps) function condes | ( | real(mpd), dimension((n*n+n)/2), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(inout) | aux | ||
) |
Etimate condition.
[in] | W | symmetric matrix |
[in] | N | size |
[in] | AUX | scratch array |
Definition at line 384 of file Dbandmatrix.f90.
References condes(), and mpdef::mps.
Referenced by condes().
subroutine db2dec | ( | real(mpd), dimension(2,n), intent(inout) | w, |
integer(mpi), intent(inout) | n, | ||
real(mpd), dimension(n), intent(out) | aux | ||
) |
Decomposition (M=1).
W is a symmetrix positive definite band matrix of bandwidth 1(+1). W(1,.) are the diagonal elements, W(2,.) is the next diagonals; W(2,N) is never referenced. AUX is an auxiliary array of length N. W is decomposed to L D Lt, where D = diagonal and L unit triangular. A row is set to zero, if the diagonal element is reduced in previous steps by a word length (i.e. global correlation coefficient large). The resulting L and D replace W: the diagonal elements W(1,...) will contain the inverse of the D-elements; the diagonal elements of L are all 1 and not stored. The other elements of L are stored in the corresponding elements of W.
SUBROUTINE DB2SLV(W,N,B, X), solution B -> X
SUBROUTINE DB2IEL(W,N, V), V = inverse band matrix elements
[in,out] | W | symmetric band matrix |
[in] | N | size |
[in] | AUX | scratch array |
Definition at line 743 of file Dbandmatrix.f90.
subroutine db2iel | ( | real(mpd), dimension(2,n), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(2,n), intent(out) | v | ||
) |
V = inverse band matrix elements.
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | N | size |
[out] | V | inverse band matrix elements |
Definition at line 815 of file Dbandmatrix.f90.
subroutine db2slv | ( | real(mpd), dimension(2,n), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(in) | b, | ||
real(mpd), dimension(n), intent(out) | x | ||
) |
solution B -> X
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | N | size |
[in] | B | r.h.s. |
[out] | X | solution |
Definition at line 784 of file Dbandmatrix.f90.
subroutine db3dec | ( | real(mpd), dimension(3,n), intent(inout) | w, |
integer(mpi), intent(inout) | n, | ||
real(mpd), dimension(n), intent(out) | aux | ||
) |
Decomposition (M=2).
W is a symmetrix positive definite band matrix of bandwidth 2(+1). W(1,.) are the diagonal elements, W(2,.) and W(3,.) are the next diagonals; W(3,N-1), W(2,N) and W(3,N) are never referenced. AUX is an auxiliary array of length N. W is decomposed to L D Lt, where D = diagonal and L unit triangular. A row is set to zero, if the diagonal element is reduced in previous steps by a word length (i.e. global correlation coefficient large). The resulting L and D replace W: the diagonal elements W(1,...) will contain the inverse of the D-elements; the diagonal elements of L are all 1 and not stored. The other elements of L are stored in the corresponding elements of W.
SUBROUTINE DB3SLV(W,N,B, X), solution B -> X
SUBROUTINE DB3IEL(W,N, V), V = inverse band matrix elements
[in,out] | W | symmetric band matrix |
[in] | N | size |
[in] | AUX | scratch array |
Definition at line 865 of file Dbandmatrix.f90.
subroutine db3iel | ( | real(mpd), dimension(3,n), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(3,n), intent(out) | v | ||
) |
V = inverse band matrix elements.
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | N | size |
[out] | V | inverse band matrix elements |
Definition at line 953 of file Dbandmatrix.f90.
subroutine db3slv | ( | real(mpd), dimension(3,n), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(in) | b, | ||
real(mpd), dimension(n), intent(out) | x | ||
) |
solution B -> X
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | N | size |
[in] | B | r.h.s. |
[out] | X | solution |
Definition at line 920 of file Dbandmatrix.f90.
subroutine dbcdec | ( | real(mpd), dimension(mp1,n), intent(inout) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(out) | aux | ||
) |
Decomposition of symmetric band matrix.
SUBROUTINE DBCSLV(W,MP1,N,B, X) for solution B -> X
SUBROUTINE DBCIEL(W,MP1,N, V), V = inverse band matrix elements
SUBROUTINE DBCINB(W,MP1,N, VS), VS = band part of inverse symmetric matrix
SUBROUTINE DBCINV(W,MP1,N, VS), VS = inverse symmetric matrix
[in,out] | W | symmetric band matrix, replaced by decomposition |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
[in] | AUX | scratch array |
decomposition, bandwidth M
Definition at line 459 of file Dbandmatrix.f90.
subroutine dbciel | ( | real(mpd), dimension(mp1,n), intent(in) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(mp1,n), intent(out) | v | ||
) |
V = inverse band matrix elements.
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
[out] | V | inverse band matrix elements |
Definition at line 542 of file Dbandmatrix.f90.
subroutine dbcinb | ( | real(mpd), dimension(mp1,n), intent(in) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n, | ||
real(mpd), dimension((n*n+n)/2), intent(out) | vs | ||
) |
VS = band part of inverse symmetric matrix.
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
[out] | VS | band part of inverse symmetric matrix |
Definition at line 576 of file Dbandmatrix.f90.
subroutine dbcinv | ( | real(mpd), dimension(mp1,n), intent(in) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n, | ||
real(mpd), dimension((n*n+n)/2), intent(out) | vs | ||
) |
VS = inverse symmetric matrix.
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
[out] | VS | inverse symmetric matrix |
Definition at line 613 of file Dbandmatrix.f90.
subroutine dbcprb | ( | real(mpd), dimension(mp1,n), intent(inout) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n | ||
) |
Print band matrix.
[in] | W | symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
Definition at line 693 of file Dbandmatrix.f90.
subroutine dbcprv | ( | real(mpd), dimension(mp1,n), intent(inout) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(out) | b | ||
) |
Print corr band matrix and pars.
[in] | W | symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
[in] | B | vector |
Definition at line 647 of file Dbandmatrix.f90.
References mpdef::mpi.
subroutine dbcslv | ( | real(mpd), dimension(mp1,n), intent(in) | w, |
integer(mpi), intent(in) | mp1, | ||
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(in) | b, | ||
real(mpd), dimension(n), intent(out) | x | ||
) |
solution B -> X
[in,out] | W | (decomposition of) )symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
[in] | B | r.h.s. |
[out] | X | solution |
Definition at line 503 of file Dbandmatrix.f90.
subroutine dbfdec | ( | real(mpd), dimension(mp1,n), intent(out) | w, |
integer(mpi), intent(inout) | mp1, | ||
integer(mpi), intent(in) | n | ||
) |
Decomposition of symmetric band matrix.
Band matrix modified Cholesky decomposition, Philip E.Gill, Walter Murray and Margarete H.Wright: Practical Optimization, Academic Press, 1981
[in,out] | W | symmetric band matrix |
[in] | MP1 | band width (M) + 1 |
[in] | N | size |
Definition at line 1092 of file Dbandmatrix.f90.
subroutine dcfdec | ( | real(mpd), dimension((n*n+n)/2), intent(out) | w, |
integer(mpi), intent(in) | n | ||
) |
Decomposition of symmetric matrix.
Modified Cholesky decomposition, Philip E.Gill, Walter Murray and Margarete H.Wright: Practical Optimization, Academic Press, 1981
[in,out] | W | symmetric matrix |
[in] | N | size |
Definition at line 1040 of file Dbandmatrix.f90.
subroutine dchdec | ( | real(mpd), dimension((n*n+n)/2), intent(inout) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(out) | aux | ||
) |
Decomposition of symmetric matrix.
SUBROUTINE DCHSLV(W,N,B, X) for solution B -> X
SUBROUTINE DCHINV(W,N,V) for inversion
[in,out] | W | symmetric matrix, replaced by decomposition |
[in] | N | size |
[in] | AUX | scratch array |
Definition at line 245 of file Dbandmatrix.f90.
subroutine dchinv | ( | real(mpd), dimension((n*n+n)/2), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension((n*n+n)/2), intent(out) | v | ||
) |
inversion
[in,out] | W | (decomposition of) symmetric matrix |
[in] | N | size |
[out] | V | inverse symmetric matrix |
Definition at line 345 of file Dbandmatrix.f90.
subroutine dchslv | ( | real(mpd), dimension((n*n+n)/2), intent(in) | w, |
integer(mpi), intent(in) | n, | ||
real(mpd), dimension(n), intent(in) | b, | ||
real(mpd), dimension(n), intent(out) | x | ||
) |
solution B -> X
[in,out] | W | (decomposition of) symmetric matrix |
[in] | N | size |
[in] | B | r.h.s. |
[out] | X | solution |
Definition at line 295 of file Dbandmatrix.f90.