*DECK CTRCO SUBROUTINE CTRCO (T, LDT, N, RCOND, Z, JOB) C***BEGIN PROLOGUE CTRCO C***PURPOSE Estimate the condition number of a triangular matrix. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2C3 C***TYPE COMPLEX (STRCO-S, DTRCO-D, CTRCO-C) C***KEYWORDS CONDITION NUMBER, LINEAR ALGEBRA, LINPACK, C TRIANGULAR MATRIX C***AUTHOR Moler, C. B., (U. of New Mexico) C***DESCRIPTION C C CTRCO estimates the condition of a complex triangular matrix. C C On Entry C C T COMPLEX(LDT,N) C T contains the triangular matrix. The zero C elements of the matrix are not referenced, and C the corresponding elements of the array can be C used to store other information. C C LDT INTEGER C LDT is the leading dimension of the array T. C C N INTEGER C N is the order of the system. C C JOB INTEGER C = 0 T is lower triangular. C = nonzero T is upper triangular. C C On Return C C RCOND REAL C an estimate of the reciprocal condition of T . C For the system T*X = B , relative perturbations C in T and B of size EPSILON may cause C relative perturbations in X of size EPSILON/RCOND . C If RCOND is so small that the logical expression C 1.0 + RCOND .EQ. 1.0 C is true, then T may be singular to working C precision. In particular, RCOND is zero if C exact singularity is detected or the estimate C underflows. C C Z COMPLEX(N) C a work vector whose contents are usually unimportant. C If T is close to a singular matrix, then Z is C an approximate null vector in the sense that C NORM(A*Z) = RCOND*NORM(A)*NORM(Z) . C C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. C Stewart, LINPACK Users' Guide, SIAM, 1979. C***ROUTINES CALLED CAXPY, CSSCAL, SCASUM C***REVISION HISTORY (YYMMDD) C 780814 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900326 Removed duplicate information from DESCRIPTION section. C (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE CTRCO INTEGER LDT,N,JOB COMPLEX T(LDT,*),Z(*) REAL RCOND C COMPLEX W,WK,WKM,EK REAL TNORM,YNORM,S,SM,SCASUM INTEGER I1,J,J1,J2,K,KK,L LOGICAL LOWER COMPLEX ZDUM,ZDUM1,ZDUM2,CSIGN1 REAL CABS1 CABS1(ZDUM) = ABS(REAL(ZDUM)) + ABS(AIMAG(ZDUM)) CSIGN1(ZDUM1,ZDUM2) = CABS1(ZDUM1)*(ZDUM2/CABS1(ZDUM2)) C C***FIRST EXECUTABLE STATEMENT CTRCO LOWER = JOB .EQ. 0 C C COMPUTE 1-NORM OF T C TNORM = 0.0E0 DO 10 J = 1, N L = J IF (LOWER) L = N + 1 - J I1 = 1 IF (LOWER) I1 = J TNORM = MAX(TNORM,SCASUM(L,T(I1,J),1)) 10 CONTINUE C C RCOND = 1/(NORM(T)*(ESTIMATE OF NORM(INVERSE(T)))) . C ESTIMATE = NORM(Z)/NORM(Y) WHERE T*Z = Y AND CTRANS(T)*Y = E . C CTRANS(T) IS THE CONJUGATE TRANSPOSE OF T . C THE COMPONENTS OF E ARE CHOSEN TO CAUSE MAXIMUM LOCAL C GROWTH IN THE ELEMENTS OF Y . C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW. C C SOLVE CTRANS(T)*Y = E C EK = (1.0E0,0.0E0) DO 20 J = 1, N Z(J) = (0.0E0,0.0E0) 20 CONTINUE DO 100 KK = 1, N K = KK IF (LOWER) K = N + 1 - KK IF (CABS1(Z(K)) .NE. 0.0E0) EK = CSIGN1(EK,-Z(K)) IF (CABS1(EK-Z(K)) .LE. CABS1(T(K,K))) GO TO 30 S = CABS1(T(K,K))/CABS1(EK-Z(K)) CALL CSSCAL(N,S,Z,1) EK = CMPLX(S,0.0E0)*EK 30 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = CABS1(WK) SM = CABS1(WKM) IF (CABS1(T(K,K)) .EQ. 0.0E0) GO TO 40 WK = WK/CONJG(T(K,K)) WKM = WKM/CONJG(T(K,K)) GO TO 50 40 CONTINUE WK = (1.0E0,0.0E0) WKM = (1.0E0,0.0E0) 50 CONTINUE IF (KK .EQ. N) GO TO 90 J1 = K + 1 IF (LOWER) J1 = 1 J2 = N IF (LOWER) J2 = K - 1 DO 60 J = J1, J2 SM = SM + CABS1(Z(J)+WKM*CONJG(T(K,J))) Z(J) = Z(J) + WK*CONJG(T(K,J)) S = S + CABS1(Z(J)) 60 CONTINUE IF (S .GE. SM) GO TO 80 W = WKM - WK WK = WKM DO 70 J = J1, J2 Z(J) = Z(J) + W*CONJG(T(K,J)) 70 CONTINUE 80 CONTINUE 90 CONTINUE Z(K) = WK 100 CONTINUE S = 1.0E0/SCASUM(N,Z,1) CALL CSSCAL(N,S,Z,1) C YNORM = 1.0E0 C C SOLVE T*Z = Y C DO 130 KK = 1, N K = N + 1 - KK IF (LOWER) K = KK IF (CABS1(Z(K)) .LE. CABS1(T(K,K))) GO TO 110 S = CABS1(T(K,K))/CABS1(Z(K)) CALL CSSCAL(N,S,Z,1) YNORM = S*YNORM 110 CONTINUE IF (CABS1(T(K,K)) .NE. 0.0E0) Z(K) = Z(K)/T(K,K) IF (CABS1(T(K,K)) .EQ. 0.0E0) Z(K) = (1.0E0,0.0E0) I1 = 1 IF (LOWER) I1 = K + 1 IF (KK .GE. N) GO TO 120 W = -Z(K) CALL CAXPY(N-KK,W,T(I1,K),1,Z(I1),1) 120 CONTINUE 130 CONTINUE C MAKE ZNORM = 1.0 S = 1.0E0/SCASUM(N,Z,1) CALL CSSCAL(N,S,Z,1) YNORM = S*YNORM C IF (TNORM .NE. 0.0E0) RCOND = YNORM/TNORM IF (TNORM .EQ. 0.0E0) RCOND = 0.0E0 RETURN END