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SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C ) |
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* |
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* -- LAPACK auxiliary routine (version 3.2) -- |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* November 2006 |
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* |
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* .. Scalar Arguments .. |
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INTEGER J, JOB |
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DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR |
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* .. |
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* .. Array Arguments .. |
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DOUBLE PRECISION W( J ), X( J ) |
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* .. |
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* |
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* Purpose |
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* ======= |
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* |
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* DLAIC1 applies one step of incremental condition estimation in |
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* its simplest version: |
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* |
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* Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j |
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* lower triangular matrix L, such that |
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* twonorm(L*x) = sest |
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* Then DLAIC1 computes sestpr, s, c such that |
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* the vector |
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* [ s*x ] |
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* xhat = [ c ] |
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* is an approximate singular vector of |
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* [ L 0 ] |
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* Lhat = [ w' gamma ] |
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* in the sense that |
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* twonorm(Lhat*xhat) = sestpr. |
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* |
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* Depending on JOB, an estimate for the largest or smallest singular |
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* value is computed. |
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* |
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* Note that [s c]' and sestpr**2 is an eigenpair of the system |
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* |
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* diag(sest*sest, 0) + [alpha gamma] * [ alpha ] |
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* [ gamma ] |
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* |
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* where alpha = x'*w. |
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* |
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* Arguments |
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* ========= |
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* |
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* JOB (input) INTEGER |
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* = 1: an estimate for the largest singular value is computed. |
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* = 2: an estimate for the smallest singular value is computed. |
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* |
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* J (input) INTEGER |
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* Length of X and W |
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* |
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* X (input) DOUBLE PRECISION array, dimension (J) |
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* The j-vector x. |
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* |
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* SEST (input) DOUBLE PRECISION |
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* Estimated singular value of j by j matrix L |
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* |
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* W (input) DOUBLE PRECISION array, dimension (J) |
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* The j-vector w. |
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* |
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* GAMMA (input) DOUBLE PRECISION |
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* The diagonal element gamma. |
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* |
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* SESTPR (output) DOUBLE PRECISION |
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* Estimated singular value of (j+1) by (j+1) matrix Lhat. |
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* |
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* S (output) DOUBLE PRECISION |
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* Sine needed in forming xhat. |
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* |
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* C (output) DOUBLE PRECISION |
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* Cosine needed in forming xhat. |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION ZERO, ONE, TWO |
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) |
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DOUBLE PRECISION HALF, FOUR |
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PARAMETER ( HALF = 0.5D0, FOUR = 4.0D0 ) |
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* .. |
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* .. Local Scalars .. |
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DOUBLE PRECISION ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS, |
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$ NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2 |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC ABS, MAX, SIGN, SQRT |
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* .. |
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* .. External Functions .. |
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DOUBLE PRECISION DDOT, DLAMCH |
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EXTERNAL DDOT, DLAMCH |
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* .. |
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* .. Executable Statements .. |
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* |
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EPS = DLAMCH( 'Epsilon' ) |
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ALPHA = DDOT( J, X, 1, W, 1 ) |
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* |
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ABSALP = ABS( ALPHA ) |
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ABSGAM = ABS( GAMMA ) |
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ABSEST = ABS( SEST ) |
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* |
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IF( JOB.EQ.1 ) THEN |
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* |
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* Estimating largest singular value |
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* |
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* special cases |
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* |
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IF( SEST.EQ.ZERO ) THEN |
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S1 = MAX( ABSGAM, ABSALP ) |
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IF( S1.EQ.ZERO ) THEN |
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S = ZERO |
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C = ONE |
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SESTPR = ZERO |
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ELSE |
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S = ALPHA / S1 |
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C = GAMMA / S1 |
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TMP = SQRT( S*S+C*C ) |
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S = S / TMP |
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C = C / TMP |
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SESTPR = S1*TMP |
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END IF |
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RETURN |
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ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN |
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S = ONE |
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C = ZERO |
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TMP = MAX( ABSEST, ABSALP ) |
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S1 = ABSEST / TMP |
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S2 = ABSALP / TMP |
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SESTPR = TMP*SQRT( S1*S1+S2*S2 ) |
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RETURN |
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ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN |
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S1 = ABSGAM |
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S2 = ABSEST |
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IF( S1.LE.S2 ) THEN |
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S = ONE |
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C = ZERO |
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SESTPR = S2 |
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ELSE |
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S = ZERO |
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C = ONE |
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SESTPR = S1 |
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END IF |
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RETURN |
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ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN |
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S1 = ABSGAM |
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S2 = ABSALP |
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IF( S1.LE.S2 ) THEN |
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TMP = S1 / S2 |
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S = SQRT( ONE+TMP*TMP ) |
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SESTPR = S2*S |
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C = ( GAMMA / S2 ) / S |
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S = SIGN( ONE, ALPHA ) / S |
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ELSE |
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TMP = S2 / S1 |
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C = SQRT( ONE+TMP*TMP ) |
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SESTPR = S1*C |
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S = ( ALPHA / S1 ) / C |
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C = SIGN( ONE, GAMMA ) / C |
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END IF |
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RETURN |
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ELSE |
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* |
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* normal case |
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* |
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ZETA1 = ALPHA / ABSEST |
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ZETA2 = GAMMA / ABSEST |
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* |
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B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF |
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C = ZETA1*ZETA1 |
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IF( B.GT.ZERO ) THEN |
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T = C / ( B+SQRT( B*B+C ) ) |
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ELSE |
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T = SQRT( B*B+C ) - B |
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END IF |
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* |
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SINE = -ZETA1 / T |
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COSINE = -ZETA2 / ( ONE+T ) |
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TMP = SQRT( SINE*SINE+COSINE*COSINE ) |
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S = SINE / TMP |
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C = COSINE / TMP |
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SESTPR = SQRT( T+ONE )*ABSEST |
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RETURN |
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END IF |
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* |
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ELSE IF( JOB.EQ.2 ) THEN |
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* |
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* Estimating smallest singular value |
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* |
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* special cases |
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* |
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IF( SEST.EQ.ZERO ) THEN |
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SESTPR = ZERO |
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IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN |
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SINE = ONE |
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COSINE = ZERO |
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ELSE |
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SINE = -GAMMA |
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COSINE = ALPHA |
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END IF |
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S1 = MAX( ABS( SINE ), ABS( COSINE ) ) |
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S = SINE / S1 |
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C = COSINE / S1 |
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TMP = SQRT( S*S+C*C ) |
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S = S / TMP |
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C = C / TMP |
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RETURN |
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ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN |
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S = ZERO |
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C = ONE |
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SESTPR = ABSGAM |
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RETURN |
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ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN |
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S1 = ABSGAM |
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S2 = ABSEST |
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IF( S1.LE.S2 ) THEN |
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S = ZERO |
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C = ONE |
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SESTPR = S1 |
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ELSE |
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S = ONE |
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C = ZERO |
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SESTPR = S2 |
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END IF |
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RETURN |
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ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN |
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S1 = ABSGAM |
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S2 = ABSALP |
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IF( S1.LE.S2 ) THEN |
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TMP = S1 / S2 |
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C = SQRT( ONE+TMP*TMP ) |
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SESTPR = ABSEST*( TMP / C ) |
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S = -( GAMMA / S2 ) / C |
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C = SIGN( ONE, ALPHA ) / C |
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ELSE |
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TMP = S2 / S1 |
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S = SQRT( ONE+TMP*TMP ) |
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SESTPR = ABSEST / S |
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C = ( ALPHA / S1 ) / S |
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S = -SIGN( ONE, GAMMA ) / S |
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END IF |
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RETURN |
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ELSE |
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* |
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* normal case |
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* |
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ZETA1 = ALPHA / ABSEST |
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ZETA2 = GAMMA / ABSEST |
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* |
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NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ), |
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$ ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 ) |
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* |
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* See if root is closer to zero or to ONE |
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* |
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TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 ) |
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IF( TEST.GE.ZERO ) THEN |
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* |
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* root is close to zero, compute directly |
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* |
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B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF |
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C = ZETA2*ZETA2 |
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T = C / ( B+SQRT( ABS( B*B-C ) ) ) |
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SINE = ZETA1 / ( ONE-T ) |
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COSINE = -ZETA2 / T |
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SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST |
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ELSE |
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* |
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* root is closer to ONE, shift by that amount |
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* |
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B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF |
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C = ZETA1*ZETA1 |
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IF( B.GE.ZERO ) THEN |
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T = -C / ( B+SQRT( B*B+C ) ) |
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ELSE |
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T = B - SQRT( B*B+C ) |
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END IF |
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SINE = -ZETA1 / T |
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COSINE = -ZETA2 / ( ONE+T ) |
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SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST |
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END IF |
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TMP = SQRT( SINE*SINE+COSINE*COSINE ) |
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S = SINE / TMP |
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C = COSINE / TMP |
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RETURN |
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* |
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END IF |
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END IF |
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RETURN |
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* |
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* End of DLAIC1 |
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* |
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END |