F112: Manipulation of Triangular and Symmetric Matrices

Author(s): W. Hart Library: KERNLIB
Submitter: Submitted: 01.01.1975
Language: Fortran Revised: 12.12.1986

At CERN, matrices are often stored row-wise (TC-convention); furthermore, symmetric matrices are stored packed as the lower left triangular part only, i.e., the Ith diagonal element is found in position I(I+1)/2. The TR-package performs many of the frequently required operations associated with such matrices without resorting to expanding into the unpacked square form. In all the following routines an tex2html_wrap_inline295 symmetric matrix is taken to be stored in the packed form with M(M+1)/2 elements.

Some of these operations produce and require the manipulation of lower triangular matrices which have all elements zero above the leading diagonal. These are also stored in the packed form with all the zeros dropped; therefore, care has to be taken in the interpretation of a packed matrix as to whether it represents a symmetric or lower triangular array. To facilitate this distinction in the Write-up, the following nomenclature has been adopted:

unpacked rectangular matrices (row-wise storage)
packed symmetric matrices
packed lower triangular matrices
On 32-bit machines the calculations are performed internally in double-precision mode.


SUBROUTINE subprograms
User Entry Names:


Choleski Decomposition
CALL TRCHUL(S,W,M) tex2html_wrap_inline299
CALL TRCHLU(S,V,M) tex2html_wrap_inline301

S is an tex2html_wrap_inline303 positive semi-definite symmetric matrix (e.g., error or weight matrix) and the routines calculate the complementary lower triangular Choleski factors. It is allowed to overwrite S by W or V.

Symmetric Multiplication of Lower Triangular Matrices
CALL TRSMUL(W,S,M) tex2html_wrap_inline307 tex2html_wrap_inline305 S
CALL TRSMLU(W,R,M) tex2html_wrap_inline309 tex2html_wrap_inline305 R

W is an tex2html_wrap_inline311 lower triangular matrix and S, R the two symmetric products of the multiplication of W by its transpose. It is allowed to overwrite W by either S or R.

Lower Triangular Matrix Inversion
CALL TRINV(W,V,M) tex2html_wrap_inline315 tex2html_wrap_inline313 V

W is an tex2html_wrap_inline317 lower triangular matrix which is inverted into V (the inverse of a lower triangular matrix is lower triangular). W may have rows and columns of zeros as produced by the Choleski decomposition of a weight matrix with unmeasured variables. It is allowed to overwrite W by V.

Symmetric Matrix Inversion
CALL TRSINV(S,R,M) tex2html_wrap_inline321 tex2html_wrap_inline319 R

S is an tex2html_wrap_inline323 positive semi-definite symmetric matrix which is inverted into R (also stored packed). It is permissible to overwrite S by R.

Triangular - Rectangular Multiplication
CALL TRLA (W,A,B,M,N) tex2html_wrap_inline327 tex2html_wrap_inline325 B
CALL TRLTA(W,A,B,M,N) tex2html_wrap_inline329 tex2html_wrap_inline325 B
CALL TRAL (A,V,B,M,N) AV tex2html_wrap_inline325 B
CALL TRALT(A,V,B,M,N) tex2html_wrap_inline331 tex2html_wrap_inline325 B

A and B are tex2html_wrap_inline333 rectangular matrices, W is an tex2html_wrap_inline335 lower triangular matrix, and V is an tex2html_wrap_inline337 lower triangular matrix. In each call it is allowed to overwrite A by B.

Symmetric - Rectangular Multiplication
CALL TRSA (S,A,C,M,N) SA tex2html_wrap_inline339 C
CALL TRAS (A,R,C,M,N) AR tex2html_wrap_inline339 C
CALL TRSAT(S,B,C,M,N) tex2html_wrap_inline341 tex2html_wrap_inline339 C
CALL TRATS(B,R,C,M,N) tex2html_wrap_inline343 tex2html_wrap_inline339 C

A and C are tex2html_wrap_inline345 rectangular matrices, B is an tex2html_wrap_inline347 matrix, S is an tex2html_wrap_inline349 symmetrix matrix, and R is an tex2html_wrap_inline351 symmetric matrix. It is not allowed to overwrite A or B by the product matrix C.

Symmetric Multiplication of Rectangular Matrices
CALL TRAAT(A,S,M,N) tex2html_wrap_inline355 tex2html_wrap_inline353 S
CALL TRATA(B,R,M,N) tex2html_wrap_inline357 tex2html_wrap_inline353 R

A is an tex2html_wrap_inline359 matrix, B is an tex2html_wrap_inline361 matrix, S is an tex2html_wrap_inline363 symmetric matrix, and R is an tex2html_wrap_inline365 symmetric matrix. No overwriting is allowed.

Transformation of Symmetric Matrix
CALL TRASAT(A,S,R,M,N) tex2html_wrap_inline369 tex2html_wrap_inline367 R
CALL TRATSA(B,S,R,M,N) tex2html_wrap_inline371 tex2html_wrap_inline367 R
CALL TRQSQ (Q,T,R,M) QTQ tex2html_wrap_inline367 R

A is an tex2html_wrap_inline373 matrix, B is an tex2html_wrap_inline375 matrix, S is an tex2html_wrap_inline377 symmetric matrix, and R, Q, T are tex2html_wrap_inline379 symmetric matrices. No overwriting is allowed.

Packing and Unpacking a Symmetric Matrix
CALL TRPCK (A,S,M) A tex2html_wrap_inline381 S
CALL TRUPCK(S,A,M) S tex2html_wrap_inline381 A

A is an tex2html_wrap_inline383 unpacked symmetric matrix (all tex2html_wrap_inline385 elements) and S is the same matrix stored packed. Overwriting is allowed for both TRPCK and TRUPCK.

Michel Goossens Wed Jun 5 05:00:56 METDST 1996