GIRTS
Solves a triangular (possibly singular) set of linear systems and/or compute a generalized inverse of an upper triangular matrix.
Required Arguments
R ‑ N by N upper triangular matrix. (Input)
If R contains a zero along the diagonal, the remaining elements of the row must also be zero. Only the upper triangle of R is referenced.
B ‑ N by NB matrix containing the right hand sides of the linear system. (Input, if NB > 0)
If NB = 0, B is not referenced and can be a vector length one.
IRANK ‑ Rank of R. (Output)
X ‑ N by NB matrix containing the solution matrix corresponding to the right hand side B. (Output, if NB > 0)
If B is not needed, then X and B can share the same storage locations. If NB = 0, x is not referenced and can be a vector of length one.
Optional Arguments
N ‑ Order of the upper triangular matrix R. (Input)
Default: N = size (R,2).
LDR ‑ Leading dimension of R exactly as specified in the dimension statement of the calling program. (Input)
Default: LDR = size (R,1).
NB ‑ Number of columns in B. (Input)
NB must be nonnegative. If NB is zero, no linear systems are solved.
Default: NB = size (B,2).
LDB ‑ Leading dimension of B exactly as specified in the dimension statement of the calling program. (Input)
Default: LDB = size (B,1).
IPATH ‑ Path option. (Input)
Default: IPATH = 1.
IPATH | Action |
|---|
1 | Solve R * X = B. |
2 | Solve RT * X = B. |
3 | Solve R * X = B and compute RINV. |
4 | Solve RT * X = B and compute RINV. |
LDX ‑ Leading dimension of X exactly as specified in the dimension statement of the calling program. (Input)
Default: LDX = size (X,1).
RINV ‑ N by N upper triangular matrix that is the inverse of R when R is nonsingular. (Output, if IPATH equals 3 or 4)
(When R is singular, RINV is g3 inverse. See the Algorithm section for an explanation of g3 inverses.) If IPATH = 1 or 2, RINV is not referenced and can be a 1x1 array. If IPATH = 3 or 4 and R is not needed, then R and RINV can share the same storage locations.
LDRINV ‑ Leading dimension of RINV exactly as specified in the dimension statement of the calling program. (Input)
FORTRAN 90 Interface
Generic: CALL GIRTS (R, B, IRANK, X [, …])
Specific: The specific interface names are S_GIRTS and D_GIRTS.
FORTRAN 77 Interface
Single: CALL GIRTS (N, R, LDR, NB, B, LDB, IPATH, IRANK, X, LDX, RINV, LDRINV)
Double: The double precision name is DGIRTS.
Description
Routine GIRTS solves systems of linear algebraic equations with a triangular coefficient matrix. Inversion of the coefficient matrix is an option. The coefficient matrix can contain a zero diagonal element, but if so, the remaining elements in the row must be zero also. (A terminal error message is issued if a nonzero element appears in the row of the coefficient matrix where a zero diagonal element appears.)
If solution of a linear system is requested (i.e., NB > 0) and row i of the coefficient matrix contains elements all equal to zero, the following action is taken:
The i-th row of the solution X is set to zero.
If IPATH is 1 or 3, a warning error is issued when the i-th row of the right-hand side B is not zero.
If IPATH is 2 or 4, a warning error is issued when the i-th row of the reduced
right-hand side (obtained after the first i – 1 variables are eliminated from row i) is not zero within a computed tolerance.
If an inverse of the coefficient matrix is requested and row i contains elements all equal to zero, row i and column i elements of RINV are set to zero. The resulting inverse is a g3 inverse of R. For a matrix G to be g3 inverse of a matrix A, G must satisfy Conditions 1, 2, and 3 for the Moore‑Penrose inverse but generally fail Condition 4. The four conditions for G to be a Moore‑Penrose inverse of A are as follows:
1. AGA = A
2. GAG = G
3. AG is symmetric
4. GA is symmetric
For a detailed description of the algorithm, see Section 2 in Sallas and Lionti (1988).
Comments
1. Informational error
Type | Code | Description |
|---|
3 | 1 | The linear system of equations is inconsistent. |
2. Routine GIRTS assumes that a singular R is represented by zero rows in R. No other forms of singularity in R are allowed.
Example
The following example is taken from Maindonald (1984, pp. 102‑105). A linear system Rx = B is solved, and a g3 inverse of R is computed.
USE GIRTS_INT
USE WRRRN_INT
IMPLICIT NONE
INTEGER LDB, LDR, LDRINV, LDX, N, NB, J
PARAMETER (N=4, NB=1, LDB=N, LDR=N, LDRINV=N, LDX=N)
!
INTEGER IPATH, IRANK
REAL B(LDB,NB), R(LDR,N), RINV(LDRINV,N), X(LDX,NB)
!
DATA (R(1,J),J=1,N)/6.0, 2.0, 5.0, 1.0/, B(1,1)/3.0/
DATA (R(2,J),J=1,N)/0.0, 4.0,-2.0, 2.0/, B(2,1)/4.0/
DATA (R(3,J),J=1,N)/0.0, 0.0, 0.0, 0.0/, B(3,1)/0.0/
DATA (R(4,J),J=1,N)/0.0, 0.0, 0.0, 3.0/, B(4,1)/3.0/
!
IPATH = 3
CALL GIRTS (R, B, IRANK, X, IPATH=IPATH, RINV=RINV)
!
CALL WRRRN ('RINV', RINV)
CALL WRRRN ('X', X)
END
Output
RINV
1 2 3 4
1 0.1667 -0.0833 0.0000 0.0000
2 0.0000 0.2500 0.0000 -0.1667
3 0.0000 0.0000 0.0000 0.0000
4 0.0000 0.0000 0.0000 0.3333
X
1 0.167
2 0.500
3 0.000
4 1.000
