root / src / lapack / double / dlasq1.f @ 1
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| 1 | 1 | equemene | SUBROUTINE DLASQ1( N, D, E, WORK, INFO ) |
|---|---|---|---|
| 2 | 1 | equemene | * |
| 3 | 1 | equemene | * -- LAPACK routine (version 3.2) -- |
| 4 | 1 | equemene | * |
| 5 | 1 | equemene | * -- Contributed by Osni Marques of the Lawrence Berkeley National -- |
| 6 | 1 | equemene | * -- Laboratory and Beresford Parlett of the Univ. of California at -- |
| 7 | 1 | equemene | * -- Berkeley -- |
| 8 | 1 | equemene | * -- November 2008 -- |
| 9 | 1 | equemene | * |
| 10 | 1 | equemene | * -- LAPACK is a software package provided by Univ. of Tennessee, -- |
| 11 | 1 | equemene | * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
| 12 | 1 | equemene | * |
| 13 | 1 | equemene | * .. Scalar Arguments .. |
| 14 | 1 | equemene | INTEGER INFO, N |
| 15 | 1 | equemene | * .. |
| 16 | 1 | equemene | * .. Array Arguments .. |
| 17 | 1 | equemene | DOUBLE PRECISION D( * ), E( * ), WORK( * ) |
| 18 | 1 | equemene | * .. |
| 19 | 1 | equemene | * |
| 20 | 1 | equemene | * Purpose |
| 21 | 1 | equemene | * ======= |
| 22 | 1 | equemene | * |
| 23 | 1 | equemene | * DLASQ1 computes the singular values of a real N-by-N bidiagonal |
| 24 | 1 | equemene | * matrix with diagonal D and off-diagonal E. The singular values |
| 25 | 1 | equemene | * are computed to high relative accuracy, in the absence of |
| 26 | 1 | equemene | * denormalization, underflow and overflow. The algorithm was first |
| 27 | 1 | equemene | * presented in |
| 28 | 1 | equemene | * |
| 29 | 1 | equemene | * "Accurate singular values and differential qd algorithms" by K. V. |
| 30 | 1 | equemene | * Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, |
| 31 | 1 | equemene | * 1994, |
| 32 | 1 | equemene | * |
| 33 | 1 | equemene | * and the present implementation is described in "An implementation of |
| 34 | 1 | equemene | * the dqds Algorithm (Positive Case)", LAPACK Working Note. |
| 35 | 1 | equemene | * |
| 36 | 1 | equemene | * Arguments |
| 37 | 1 | equemene | * ========= |
| 38 | 1 | equemene | * |
| 39 | 1 | equemene | * N (input) INTEGER |
| 40 | 1 | equemene | * The number of rows and columns in the matrix. N >= 0. |
| 41 | 1 | equemene | * |
| 42 | 1 | equemene | * D (input/output) DOUBLE PRECISION array, dimension (N) |
| 43 | 1 | equemene | * On entry, D contains the diagonal elements of the |
| 44 | 1 | equemene | * bidiagonal matrix whose SVD is desired. On normal exit, |
| 45 | 1 | equemene | * D contains the singular values in decreasing order. |
| 46 | 1 | equemene | * |
| 47 | 1 | equemene | * E (input/output) DOUBLE PRECISION array, dimension (N) |
| 48 | 1 | equemene | * On entry, elements E(1:N-1) contain the off-diagonal elements |
| 49 | 1 | equemene | * of the bidiagonal matrix whose SVD is desired. |
| 50 | 1 | equemene | * On exit, E is overwritten. |
| 51 | 1 | equemene | * |
| 52 | 1 | equemene | * WORK (workspace) DOUBLE PRECISION array, dimension (4*N) |
| 53 | 1 | equemene | * |
| 54 | 1 | equemene | * INFO (output) INTEGER |
| 55 | 1 | equemene | * = 0: successful exit |
| 56 | 1 | equemene | * < 0: if INFO = -i, the i-th argument had an illegal value |
| 57 | 1 | equemene | * > 0: the algorithm failed |
| 58 | 1 | equemene | * = 1, a split was marked by a positive value in E |
| 59 | 1 | equemene | * = 2, current block of Z not diagonalized after 30*N |
| 60 | 1 | equemene | * iterations (in inner while loop) |
| 61 | 1 | equemene | * = 3, termination criterion of outer while loop not met |
| 62 | 1 | equemene | * (program created more than N unreduced blocks) |
| 63 | 1 | equemene | * |
| 64 | 1 | equemene | * ===================================================================== |
| 65 | 1 | equemene | * |
| 66 | 1 | equemene | * .. Parameters .. |
| 67 | 1 | equemene | DOUBLE PRECISION ZERO |
| 68 | 1 | equemene | PARAMETER ( ZERO = 0.0D0 ) |
| 69 | 1 | equemene | * .. |
| 70 | 1 | equemene | * .. Local Scalars .. |
| 71 | 1 | equemene | INTEGER I, IINFO |
| 72 | 1 | equemene | DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX |
| 73 | 1 | equemene | * .. |
| 74 | 1 | equemene | * .. External Subroutines .. |
| 75 | 1 | equemene | EXTERNAL DCOPY, DLAS2, DLASCL, DLASQ2, DLASRT, XERBLA |
| 76 | 1 | equemene | * .. |
| 77 | 1 | equemene | * .. External Functions .. |
| 78 | 1 | equemene | DOUBLE PRECISION DLAMCH |
| 79 | 1 | equemene | EXTERNAL DLAMCH |
| 80 | 1 | equemene | * .. |
| 81 | 1 | equemene | * .. Intrinsic Functions .. |
| 82 | 1 | equemene | INTRINSIC ABS, MAX, SQRT |
| 83 | 1 | equemene | * .. |
| 84 | 1 | equemene | * .. Executable Statements .. |
| 85 | 1 | equemene | * |
| 86 | 1 | equemene | INFO = 0 |
| 87 | 1 | equemene | IF( N.LT.0 ) THEN |
| 88 | 1 | equemene | INFO = -2 |
| 89 | 1 | equemene | CALL XERBLA( 'DLASQ1', -INFO ) |
| 90 | 1 | equemene | RETURN |
| 91 | 1 | equemene | ELSE IF( N.EQ.0 ) THEN |
| 92 | 1 | equemene | RETURN |
| 93 | 1 | equemene | ELSE IF( N.EQ.1 ) THEN |
| 94 | 1 | equemene | D( 1 ) = ABS( D( 1 ) ) |
| 95 | 1 | equemene | RETURN |
| 96 | 1 | equemene | ELSE IF( N.EQ.2 ) THEN |
| 97 | 1 | equemene | CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX ) |
| 98 | 1 | equemene | D( 1 ) = SIGMX |
| 99 | 1 | equemene | D( 2 ) = SIGMN |
| 100 | 1 | equemene | RETURN |
| 101 | 1 | equemene | END IF |
| 102 | 1 | equemene | * |
| 103 | 1 | equemene | * Estimate the largest singular value. |
| 104 | 1 | equemene | * |
| 105 | 1 | equemene | SIGMX = ZERO |
| 106 | 1 | equemene | DO 10 I = 1, N - 1 |
| 107 | 1 | equemene | D( I ) = ABS( D( I ) ) |
| 108 | 1 | equemene | SIGMX = MAX( SIGMX, ABS( E( I ) ) ) |
| 109 | 1 | equemene | 10 CONTINUE |
| 110 | 1 | equemene | D( N ) = ABS( D( N ) ) |
| 111 | 1 | equemene | * |
| 112 | 1 | equemene | * Early return if SIGMX is zero (matrix is already diagonal). |
| 113 | 1 | equemene | * |
| 114 | 1 | equemene | IF( SIGMX.EQ.ZERO ) THEN |
| 115 | 1 | equemene | CALL DLASRT( 'D', N, D, IINFO ) |
| 116 | 1 | equemene | RETURN |
| 117 | 1 | equemene | END IF |
| 118 | 1 | equemene | * |
| 119 | 1 | equemene | DO 20 I = 1, N |
| 120 | 1 | equemene | SIGMX = MAX( SIGMX, D( I ) ) |
| 121 | 1 | equemene | 20 CONTINUE |
| 122 | 1 | equemene | * |
| 123 | 1 | equemene | * Copy D and E into WORK (in the Z format) and scale (squaring the |
| 124 | 1 | equemene | * input data makes scaling by a power of the radix pointless). |
| 125 | 1 | equemene | * |
| 126 | 1 | equemene | EPS = DLAMCH( 'Precision' ) |
| 127 | 1 | equemene | SAFMIN = DLAMCH( 'Safe minimum' ) |
| 128 | 1 | equemene | SCALE = SQRT( EPS / SAFMIN ) |
| 129 | 1 | equemene | CALL DCOPY( N, D, 1, WORK( 1 ), 2 ) |
| 130 | 1 | equemene | CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 ) |
| 131 | 1 | equemene | CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1, |
| 132 | 1 | equemene | $ IINFO ) |
| 133 | 1 | equemene | * |
| 134 | 1 | equemene | * Compute the q's and e's. |
| 135 | 1 | equemene | * |
| 136 | 1 | equemene | DO 30 I = 1, 2*N - 1 |
| 137 | 1 | equemene | WORK( I ) = WORK( I )**2 |
| 138 | 1 | equemene | 30 CONTINUE |
| 139 | 1 | equemene | WORK( 2*N ) = ZERO |
| 140 | 1 | equemene | * |
| 141 | 1 | equemene | CALL DLASQ2( N, WORK, INFO ) |
| 142 | 1 | equemene | * |
| 143 | 1 | equemene | IF( INFO.EQ.0 ) THEN |
| 144 | 1 | equemene | DO 40 I = 1, N |
| 145 | 1 | equemene | D( I ) = SQRT( WORK( I ) ) |
| 146 | 1 | equemene | 40 CONTINUE |
| 147 | 1 | equemene | CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO ) |
| 148 | 1 | equemene | END IF |
| 149 | 1 | equemene | * |
| 150 | 1 | equemene | RETURN |
| 151 | 1 | equemene | * |
| 152 | 1 | equemene | * End of DLASQ1 |
| 153 | 1 | equemene | * |
| 154 | 1 | equemene | END |