Annotation of rpl/lapack/lapack/dpbequ.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DPBEQU
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DPBEQU + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbequ.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbequ.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbequ.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER UPLO
! 25: * INTEGER INFO, KD, LDAB, N
! 26: * DOUBLE PRECISION AMAX, SCOND
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * DOUBLE PRECISION AB( LDAB, * ), S( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> DPBEQU computes row and column scalings intended to equilibrate a
! 39: *> symmetric positive definite band matrix A and reduce its condition
! 40: *> number (with respect to the two-norm). S contains the scale factors,
! 41: *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
! 42: *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
! 43: *> choice of S puts the condition number of B within a factor N of the
! 44: *> smallest possible condition number over all possible diagonal
! 45: *> scalings.
! 46: *> \endverbatim
! 47: *
! 48: * Arguments:
! 49: * ==========
! 50: *
! 51: *> \param[in] UPLO
! 52: *> \verbatim
! 53: *> UPLO is CHARACTER*1
! 54: *> = 'U': Upper triangular of A is stored;
! 55: *> = 'L': Lower triangular of A is stored.
! 56: *> \endverbatim
! 57: *>
! 58: *> \param[in] N
! 59: *> \verbatim
! 60: *> N is INTEGER
! 61: *> The order of the matrix A. N >= 0.
! 62: *> \endverbatim
! 63: *>
! 64: *> \param[in] KD
! 65: *> \verbatim
! 66: *> KD is INTEGER
! 67: *> The number of superdiagonals of the matrix A if UPLO = 'U',
! 68: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 69: *> \endverbatim
! 70: *>
! 71: *> \param[in] AB
! 72: *> \verbatim
! 73: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
! 74: *> The upper or lower triangle of the symmetric band matrix A,
! 75: *> stored in the first KD+1 rows of the array. The j-th column
! 76: *> of A is stored in the j-th column of the array AB as follows:
! 77: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 78: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[in] LDAB
! 82: *> \verbatim
! 83: *> LDAB is INTEGER
! 84: *> The leading dimension of the array A. LDAB >= KD+1.
! 85: *> \endverbatim
! 86: *>
! 87: *> \param[out] S
! 88: *> \verbatim
! 89: *> S is DOUBLE PRECISION array, dimension (N)
! 90: *> If INFO = 0, S contains the scale factors for A.
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[out] SCOND
! 94: *> \verbatim
! 95: *> SCOND is DOUBLE PRECISION
! 96: *> If INFO = 0, S contains the ratio of the smallest S(i) to
! 97: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 98: *> large nor too small, it is not worth scaling by S.
! 99: *> \endverbatim
! 100: *>
! 101: *> \param[out] AMAX
! 102: *> \verbatim
! 103: *> AMAX is DOUBLE PRECISION
! 104: *> Absolute value of largest matrix element. If AMAX is very
! 105: *> close to overflow or very close to underflow, the matrix
! 106: *> should be scaled.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[out] INFO
! 110: *> \verbatim
! 111: *> INFO is INTEGER
! 112: *> = 0: successful exit
! 113: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 114: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 115: *> \endverbatim
! 116: *
! 117: * Authors:
! 118: * ========
! 119: *
! 120: *> \author Univ. of Tennessee
! 121: *> \author Univ. of California Berkeley
! 122: *> \author Univ. of Colorado Denver
! 123: *> \author NAG Ltd.
! 124: *
! 125: *> \date November 2011
! 126: *
! 127: *> \ingroup doubleOTHERcomputational
! 128: *
! 129: * =====================================================================
1.1 bertrand 130: SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
131: *
1.9 ! bertrand 132: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 133: * -- LAPACK is a software package provided by Univ. of Tennessee, --
134: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 135: * November 2011
1.1 bertrand 136: *
137: * .. Scalar Arguments ..
138: CHARACTER UPLO
139: INTEGER INFO, KD, LDAB, N
140: DOUBLE PRECISION AMAX, SCOND
141: * ..
142: * .. Array Arguments ..
143: DOUBLE PRECISION AB( LDAB, * ), S( * )
144: * ..
145: *
146: * =====================================================================
147: *
148: * .. Parameters ..
149: DOUBLE PRECISION ZERO, ONE
150: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
151: * ..
152: * .. Local Scalars ..
153: LOGICAL UPPER
154: INTEGER I, J
155: DOUBLE PRECISION SMIN
156: * ..
157: * .. External Functions ..
158: LOGICAL LSAME
159: EXTERNAL LSAME
160: * ..
161: * .. External Subroutines ..
162: EXTERNAL XERBLA
163: * ..
164: * .. Intrinsic Functions ..
165: INTRINSIC MAX, MIN, SQRT
166: * ..
167: * .. Executable Statements ..
168: *
169: * Test the input parameters.
170: *
171: INFO = 0
172: UPPER = LSAME( UPLO, 'U' )
173: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
174: INFO = -1
175: ELSE IF( N.LT.0 ) THEN
176: INFO = -2
177: ELSE IF( KD.LT.0 ) THEN
178: INFO = -3
179: ELSE IF( LDAB.LT.KD+1 ) THEN
180: INFO = -5
181: END IF
182: IF( INFO.NE.0 ) THEN
183: CALL XERBLA( 'DPBEQU', -INFO )
184: RETURN
185: END IF
186: *
187: * Quick return if possible
188: *
189: IF( N.EQ.0 ) THEN
190: SCOND = ONE
191: AMAX = ZERO
192: RETURN
193: END IF
194: *
195: IF( UPPER ) THEN
196: J = KD + 1
197: ELSE
198: J = 1
199: END IF
200: *
201: * Initialize SMIN and AMAX.
202: *
203: S( 1 ) = AB( J, 1 )
204: SMIN = S( 1 )
205: AMAX = S( 1 )
206: *
207: * Find the minimum and maximum diagonal elements.
208: *
209: DO 10 I = 2, N
210: S( I ) = AB( J, I )
211: SMIN = MIN( SMIN, S( I ) )
212: AMAX = MAX( AMAX, S( I ) )
213: 10 CONTINUE
214: *
215: IF( SMIN.LE.ZERO ) THEN
216: *
217: * Find the first non-positive diagonal element and return.
218: *
219: DO 20 I = 1, N
220: IF( S( I ).LE.ZERO ) THEN
221: INFO = I
222: RETURN
223: END IF
224: 20 CONTINUE
225: ELSE
226: *
227: * Set the scale factors to the reciprocals
228: * of the diagonal elements.
229: *
230: DO 30 I = 1, N
231: S( I ) = ONE / SQRT( S( I ) )
232: 30 CONTINUE
233: *
234: * Compute SCOND = min(S(I)) / max(S(I))
235: *
236: SCOND = SQRT( SMIN ) / SQRT( AMAX )
237: END IF
238: RETURN
239: *
240: * End of DPBEQU
241: *
242: END
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