SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C ) * * -- LAPACK auxiliary routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER J, JOB DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR * .. * .. Array Arguments .. DOUBLE PRECISION W( J ), X( J ) * .. * * Purpose * ======= * * DLAIC1 applies one step of incremental condition estimation in * its simplest version: * * Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j * lower triangular matrix L, such that * twonorm(L*x) = sest * Then DLAIC1 computes sestpr, s, c such that * the vector * [ s*x ] * xhat = [ c ] * is an approximate singular vector of * [ L 0 ] * Lhat = [ w' gamma ] * in the sense that * twonorm(Lhat*xhat) = sestpr. * * Depending on JOB, an estimate for the largest or smallest singular * value is computed. * * Note that [s c]' and sestpr**2 is an eigenpair of the system * * diag(sest*sest, 0) + [alpha gamma] * [ alpha ] * [ gamma ] * * where alpha = x'*w. * * Arguments * ========= * * JOB (input) INTEGER * = 1: an estimate for the largest singular value is computed. * = 2: an estimate for the smallest singular value is computed. * * J (input) INTEGER * Length of X and W * * X (input) DOUBLE PRECISION array, dimension (J) * The j-vector x. * * SEST (input) DOUBLE PRECISION * Estimated singular value of j by j matrix L * * W (input) DOUBLE PRECISION array, dimension (J) * The j-vector w. * * GAMMA (input) DOUBLE PRECISION * The diagonal element gamma. * * SESTPR (output) DOUBLE PRECISION * Estimated singular value of (j+1) by (j+1) matrix Lhat. * * S (output) DOUBLE PRECISION * Sine needed in forming xhat. * * C (output) DOUBLE PRECISION * Cosine needed in forming xhat. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) DOUBLE PRECISION HALF, FOUR PARAMETER ( HALF = 0.5D0, FOUR = 4.0D0 ) * .. * .. Local Scalars .. DOUBLE PRECISION ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS, \$ NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, SIGN, SQRT * .. * .. External Functions .. DOUBLE PRECISION DDOT, DLAMCH EXTERNAL DDOT, DLAMCH * .. * .. Executable Statements .. * EPS = DLAMCH( 'Epsilon' ) ALPHA = DDOT( J, X, 1, W, 1 ) * ABSALP = ABS( ALPHA ) ABSGAM = ABS( GAMMA ) ABSEST = ABS( SEST ) * IF( JOB.EQ.1 ) THEN * * Estimating largest singular value * * special cases * IF( SEST.EQ.ZERO ) THEN S1 = MAX( ABSGAM, ABSALP ) IF( S1.EQ.ZERO ) THEN S = ZERO C = ONE SESTPR = ZERO ELSE S = ALPHA / S1 C = GAMMA / S1 TMP = SQRT( S*S+C*C ) S = S / TMP C = C / TMP SESTPR = S1*TMP END IF RETURN ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN S = ONE C = ZERO TMP = MAX( ABSEST, ABSALP ) S1 = ABSEST / TMP S2 = ABSALP / TMP SESTPR = TMP*SQRT( S1*S1+S2*S2 ) RETURN ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN S1 = ABSGAM S2 = ABSEST IF( S1.LE.S2 ) THEN S = ONE C = ZERO SESTPR = S2 ELSE S = ZERO C = ONE SESTPR = S1 END IF RETURN ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN S1 = ABSGAM S2 = ABSALP IF( S1.LE.S2 ) THEN TMP = S1 / S2 S = SQRT( ONE+TMP*TMP ) SESTPR = S2*S C = ( GAMMA / S2 ) / S S = SIGN( ONE, ALPHA ) / S ELSE TMP = S2 / S1 C = SQRT( ONE+TMP*TMP ) SESTPR = S1*C S = ( ALPHA / S1 ) / C C = SIGN( ONE, GAMMA ) / C END IF RETURN ELSE * * normal case * ZETA1 = ALPHA / ABSEST ZETA2 = GAMMA / ABSEST * B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF C = ZETA1*ZETA1 IF( B.GT.ZERO ) THEN T = C / ( B+SQRT( B*B+C ) ) ELSE T = SQRT( B*B+C ) - B END IF * SINE = -ZETA1 / T COSINE = -ZETA2 / ( ONE+T ) TMP = SQRT( SINE*SINE+COSINE*COSINE ) S = SINE / TMP C = COSINE / TMP SESTPR = SQRT( T+ONE )*ABSEST RETURN END IF * ELSE IF( JOB.EQ.2 ) THEN * * Estimating smallest singular value * * special cases * IF( SEST.EQ.ZERO ) THEN SESTPR = ZERO IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN SINE = ONE COSINE = ZERO ELSE SINE = -GAMMA COSINE = ALPHA END IF S1 = MAX( ABS( SINE ), ABS( COSINE ) ) S = SINE / S1 C = COSINE / S1 TMP = SQRT( S*S+C*C ) S = S / TMP C = C / TMP RETURN ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN S = ZERO C = ONE SESTPR = ABSGAM RETURN ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN S1 = ABSGAM S2 = ABSEST IF( S1.LE.S2 ) THEN S = ZERO C = ONE SESTPR = S1 ELSE S = ONE C = ZERO SESTPR = S2 END IF RETURN ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN S1 = ABSGAM S2 = ABSALP IF( S1.LE.S2 ) THEN TMP = S1 / S2 C = SQRT( ONE+TMP*TMP ) SESTPR = ABSEST*( TMP / C ) S = -( GAMMA / S2 ) / C C = SIGN( ONE, ALPHA ) / C ELSE TMP = S2 / S1 S = SQRT( ONE+TMP*TMP ) SESTPR = ABSEST / S C = ( ALPHA / S1 ) / S S = -SIGN( ONE, GAMMA ) / S END IF RETURN ELSE * * normal case * ZETA1 = ALPHA / ABSEST ZETA2 = GAMMA / ABSEST * NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ), \$ ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 ) * * See if root is closer to zero or to ONE * TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 ) IF( TEST.GE.ZERO ) THEN * * root is close to zero, compute directly * B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF C = ZETA2*ZETA2 T = C / ( B+SQRT( ABS( B*B-C ) ) ) SINE = ZETA1 / ( ONE-T ) COSINE = -ZETA2 / T SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST ELSE * * root is closer to ONE, shift by that amount * B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF C = ZETA1*ZETA1 IF( B.GE.ZERO ) THEN T = -C / ( B+SQRT( B*B+C ) ) ELSE T = B - SQRT( B*B+C ) END IF SINE = -ZETA1 / T COSINE = -ZETA2 / ( ONE+T ) SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST END IF TMP = SQRT( SINE*SINE+COSINE*COSINE ) S = SINE / TMP C = COSINE / TMP RETURN * END IF END IF RETURN * * End of DLAIC1 * END