SVBKSB Procedure

Uses “back substitution” to solve the set of simultaneous linear equations Ax = b, given the u, w, and v arrays created by the SVD procedure from the matrix a.

Usage

SVBKSB, u, w, v, b, x 

Input Parameters

u—The m-by-n column matrix of the decomposition of a, as returned by SVD.

w—The vector of singular values, as returned by SVD.

v—The n-by-n orthogonal matrix of the decomposition of A, as returned by SVD.

b—The vector containing the right-hand side of the equation.

Output Parameters

x—A variable containing the result.

Keywords

None.

Discussion

Since no input quantities are destroyed, SVBKSB may be called numerous times with different b vectors.

Example

Here’s a typical use of SVD and SVBKSB to solve the system Ax = b:

; Decompose square matrix A.

s = 0

A = RANDOMU(s, 3, 3) * 10.0

SVD, A, w, u, v

; Get subscripts of singular values, using a given threshold. 

small = WHERE(w LT MAX(w) * 1.0e-6, count)

; Zero singular values less than the threshold.

; See the Numerical Recipes book for details.

b = [0.375, 0.612, 0.457]

IF count NE 0 THEN w(small) = 0.0

; The vector x now contains the solution.

SVBKSB, u, w, v, b, x 

; The residual, |Ax - b| should be near 0.

PRINT, TOTAL(ABS(A # x - b)), Format='(F20.16)'

See Also

SVD

SVBKSB is based on the routine of the same name in Numerical Recipes in C: The Art of Scientific Computing, by Flannery, Press, Teukolsky, and Vetterling, Cambridge University Press, Cambridge, MA, 1988. It is used by permission.