From d54fe7c1f704a63824c5bfa0ece65245572e9b27 Mon Sep 17 00:00:00 2001 From: Joseph Hunkeler Date: Wed, 4 Mar 2015 21:21:30 -0500 Subject: Initial commit --- src/slalib/sun67.htx/node180.html | 184 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 184 insertions(+) create mode 100644 src/slalib/sun67.htx/node180.html (limited to 'src/slalib/sun67.htx/node180.html') diff --git a/src/slalib/sun67.htx/node180.html b/src/slalib/sun67.htx/node180.html new file mode 100644 index 0000000..2372b60 --- /dev/null +++ b/src/slalib/sun67.htx/node180.html @@ -0,0 +1,184 @@ + + + + +SLA_SVD - Singular Value Decomposition + + + + + + + + + + + + +
+ +next + +up + +previous +
+ Next: SLA_SVDCOV - Covariance Matrix from SVD +
+Up: SUBPROGRAM SPECIFICATIONS +
+ Previous: SLA_SUPGAL - Supergalactic to Galactic +

+

+

SLA_SVD - Singular Value Decomposition +   +

+
+
ACTION: +
Singular value decomposition. +This routine expresses a given matrix A as the product of + three matrices U, W, VT: + 

+ 		 A = U $\cdot$ W $\cdot$ VT
+
+ where: + 

+ 		 A 		 is any m (rows) $\times n$ (columns) matrix,                       where $m \geq n$ 
+		 U 		 is an $m \times n$ column-orthogonal matrix
+		 W 		 is an $n \times n$ diagonal matrix with                       $w_{ii} \geq 0$ 
+		 VT is the transpose of an $n \times n$                             orthogonal matrix
+
+

CALL: +
CALL sla_SVD (M, N, MP, NP, A, W, V, WORK, JSTAT) +

+

+
GIVEN: +
+
+ + + + + + + + + + + + + +
M,NIm, n, the numbers of rows and columns in matrix A
MP,NPIphysical dimensions of array containing matrix A
AD(MP,NP)array containing $m \times n$ matrix A
+

+
RETURNED: +
+
+ + + + + + + + + + + + + + + + + + + + + +
AD(MP,NP)array containing $m \times n$ column-orthogonal +matrix U
WD(N)$n \times n$ diagonal matrix W +(diagonal elements only)
VD(NP,NP)array containing $n \times n$ orthogonal +matrix V (n.b. not VT)
WORKD(N)workspace
JSTATI0 = OK, -1 = array A wrong shape, >0 = index of W +for which convergence failed (see note 3, below)
+

+
NOTES: +
+
1. +
M and N are the logical dimensions of the +matrices and vectors concerned, which can be located in +arrays of larger physical dimensions, given by MP and NP. +
2. +
V contains matrix V, not the transpose of matrix V. +
3. +
If the status JSTAT is greater than zero, this need not + necessarily be treated as a failure. It means that, due to + chance properties of the matrix A, the QR transformation + phase of the routine did not fully converge in a predefined + number of iterations, something that very seldom occurs. + When this condition does arise, it is possible that the + elements of the diagonal matrix W have not been correctly + found. However, in practice the results are likely to + be trustworthy. Applications should report the condition + as a warning, but then proceed normally. +
+

+
REFERENCES: +
The algorithm is an adaptation of the routine SVD in the EISPACK +library (Garbow et al. 1977, EISPACK Guide Extension, + Springer Verlag), which is a FORTRAN 66 implementation of the Algol + routine SVD of Wilkinson & Reinsch 1971 (Handbook for Automatic + Computation, vol 2, ed Bauer et al., Springer Verlag). These + references give full details of the algorithm used here. + A good account of the use of SVD in least squares problems is given + in Numerical Recipes (Press et al. 1987, Cambridge + University Press), which includes another variant of the EISPACK code. +
+
+ +next + +up + +previous +
+ Next: SLA_SVDCOV - Covariance Matrix from SVD +
+Up: SUBPROGRAM SPECIFICATIONS +
+ Previous: SLA_SUPGAL - Supergalactic to Galactic +

+

+

+SLALIB --- Positional Astronomy Library
Starlink User Note 67
P. T. Wallace
12 October 1999
E-mail:ptw@star.rl.ac.uk +
+ + -- cgit