Annotation of rpl/lapack/lapack/zgebal.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 7: *
! 8: * .. Scalar Arguments ..
! 9: CHARACTER JOB
! 10: INTEGER IHI, ILO, INFO, LDA, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION SCALE( * )
! 14: COMPLEX*16 A( LDA, * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * ZGEBAL balances a general complex matrix A. This involves, first,
! 21: * permuting A by a similarity transformation to isolate eigenvalues
! 22: * in the first 1 to ILO-1 and last IHI+1 to N elements on the
! 23: * diagonal; and second, applying a diagonal similarity transformation
! 24: * to rows and columns ILO to IHI to make the rows and columns as
! 25: * close in norm as possible. Both steps are optional.
! 26: *
! 27: * Balancing may reduce the 1-norm of the matrix, and improve the
! 28: * accuracy of the computed eigenvalues and/or eigenvectors.
! 29: *
! 30: * Arguments
! 31: * =========
! 32: *
! 33: * JOB (input) CHARACTER*1
! 34: * Specifies the operations to be performed on A:
! 35: * = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
! 36: * for i = 1,...,N;
! 37: * = 'P': permute only;
! 38: * = 'S': scale only;
! 39: * = 'B': both permute and scale.
! 40: *
! 41: * N (input) INTEGER
! 42: * The order of the matrix A. N >= 0.
! 43: *
! 44: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
! 45: * On entry, the input matrix A.
! 46: * On exit, A is overwritten by the balanced matrix.
! 47: * If JOB = 'N', A is not referenced.
! 48: * See Further Details.
! 49: *
! 50: * LDA (input) INTEGER
! 51: * The leading dimension of the array A. LDA >= max(1,N).
! 52: *
! 53: * ILO (output) INTEGER
! 54: * IHI (output) INTEGER
! 55: * ILO and IHI are set to integers such that on exit
! 56: * A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
! 57: * If JOB = 'N' or 'S', ILO = 1 and IHI = N.
! 58: *
! 59: * SCALE (output) DOUBLE PRECISION array, dimension (N)
! 60: * Details of the permutations and scaling factors applied to
! 61: * A. If P(j) is the index of the row and column interchanged
! 62: * with row and column j and D(j) is the scaling factor
! 63: * applied to row and column j, then
! 64: * SCALE(j) = P(j) for j = 1,...,ILO-1
! 65: * = D(j) for j = ILO,...,IHI
! 66: * = P(j) for j = IHI+1,...,N.
! 67: * The order in which the interchanges are made is N to IHI+1,
! 68: * then 1 to ILO-1.
! 69: *
! 70: * INFO (output) INTEGER
! 71: * = 0: successful exit.
! 72: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 73: *
! 74: * Further Details
! 75: * ===============
! 76: *
! 77: * The permutations consist of row and column interchanges which put
! 78: * the matrix in the form
! 79: *
! 80: * ( T1 X Y )
! 81: * P A P = ( 0 B Z )
! 82: * ( 0 0 T2 )
! 83: *
! 84: * where T1 and T2 are upper triangular matrices whose eigenvalues lie
! 85: * along the diagonal. The column indices ILO and IHI mark the starting
! 86: * and ending columns of the submatrix B. Balancing consists of applying
! 87: * a diagonal similarity transformation inv(D) * B * D to make the
! 88: * 1-norms of each row of B and its corresponding column nearly equal.
! 89: * The output matrix is
! 90: *
! 91: * ( T1 X*D Y )
! 92: * ( 0 inv(D)*B*D inv(D)*Z ).
! 93: * ( 0 0 T2 )
! 94: *
! 95: * Information about the permutations P and the diagonal matrix D is
! 96: * returned in the vector SCALE.
! 97: *
! 98: * This subroutine is based on the EISPACK routine CBAL.
! 99: *
! 100: * Modified by Tzu-Yi Chen, Computer Science Division, University of
! 101: * California at Berkeley, USA
! 102: *
! 103: * =====================================================================
! 104: *
! 105: * .. Parameters ..
! 106: DOUBLE PRECISION ZERO, ONE
! 107: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 108: DOUBLE PRECISION SCLFAC
! 109: PARAMETER ( SCLFAC = 2.0D+0 )
! 110: DOUBLE PRECISION FACTOR
! 111: PARAMETER ( FACTOR = 0.95D+0 )
! 112: * ..
! 113: * .. Local Scalars ..
! 114: LOGICAL NOCONV
! 115: INTEGER I, ICA, IEXC, IRA, J, K, L, M
! 116: DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
! 117: $ SFMIN2
! 118: COMPLEX*16 CDUM
! 119: * ..
! 120: * .. External Functions ..
! 121: LOGICAL LSAME
! 122: INTEGER IZAMAX
! 123: DOUBLE PRECISION DLAMCH
! 124: EXTERNAL LSAME, IZAMAX, DLAMCH
! 125: * ..
! 126: * .. External Subroutines ..
! 127: EXTERNAL XERBLA, ZDSCAL, ZSWAP
! 128: * ..
! 129: * .. Intrinsic Functions ..
! 130: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
! 131: * ..
! 132: * .. Statement Functions ..
! 133: DOUBLE PRECISION CABS1
! 134: * ..
! 135: * .. Statement Function definitions ..
! 136: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
! 137: * ..
! 138: * .. Executable Statements ..
! 139: *
! 140: * Test the input parameters
! 141: *
! 142: INFO = 0
! 143: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
! 144: $ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
! 145: INFO = -1
! 146: ELSE IF( N.LT.0 ) THEN
! 147: INFO = -2
! 148: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 149: INFO = -4
! 150: END IF
! 151: IF( INFO.NE.0 ) THEN
! 152: CALL XERBLA( 'ZGEBAL', -INFO )
! 153: RETURN
! 154: END IF
! 155: *
! 156: K = 1
! 157: L = N
! 158: *
! 159: IF( N.EQ.0 )
! 160: $ GO TO 210
! 161: *
! 162: IF( LSAME( JOB, 'N' ) ) THEN
! 163: DO 10 I = 1, N
! 164: SCALE( I ) = ONE
! 165: 10 CONTINUE
! 166: GO TO 210
! 167: END IF
! 168: *
! 169: IF( LSAME( JOB, 'S' ) )
! 170: $ GO TO 120
! 171: *
! 172: * Permutation to isolate eigenvalues if possible
! 173: *
! 174: GO TO 50
! 175: *
! 176: * Row and column exchange.
! 177: *
! 178: 20 CONTINUE
! 179: SCALE( M ) = J
! 180: IF( J.EQ.M )
! 181: $ GO TO 30
! 182: *
! 183: CALL ZSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
! 184: CALL ZSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA )
! 185: *
! 186: 30 CONTINUE
! 187: GO TO ( 40, 80 )IEXC
! 188: *
! 189: * Search for rows isolating an eigenvalue and push them down.
! 190: *
! 191: 40 CONTINUE
! 192: IF( L.EQ.1 )
! 193: $ GO TO 210
! 194: L = L - 1
! 195: *
! 196: 50 CONTINUE
! 197: DO 70 J = L, 1, -1
! 198: *
! 199: DO 60 I = 1, L
! 200: IF( I.EQ.J )
! 201: $ GO TO 60
! 202: IF( DBLE( A( J, I ) ).NE.ZERO .OR. DIMAG( A( J, I ) ).NE.
! 203: $ ZERO )GO TO 70
! 204: 60 CONTINUE
! 205: *
! 206: M = L
! 207: IEXC = 1
! 208: GO TO 20
! 209: 70 CONTINUE
! 210: *
! 211: GO TO 90
! 212: *
! 213: * Search for columns isolating an eigenvalue and push them left.
! 214: *
! 215: 80 CONTINUE
! 216: K = K + 1
! 217: *
! 218: 90 CONTINUE
! 219: DO 110 J = K, L
! 220: *
! 221: DO 100 I = K, L
! 222: IF( I.EQ.J )
! 223: $ GO TO 100
! 224: IF( DBLE( A( I, J ) ).NE.ZERO .OR. DIMAG( A( I, J ) ).NE.
! 225: $ ZERO )GO TO 110
! 226: 100 CONTINUE
! 227: *
! 228: M = K
! 229: IEXC = 2
! 230: GO TO 20
! 231: 110 CONTINUE
! 232: *
! 233: 120 CONTINUE
! 234: DO 130 I = K, L
! 235: SCALE( I ) = ONE
! 236: 130 CONTINUE
! 237: *
! 238: IF( LSAME( JOB, 'P' ) )
! 239: $ GO TO 210
! 240: *
! 241: * Balance the submatrix in rows K to L.
! 242: *
! 243: * Iterative loop for norm reduction
! 244: *
! 245: SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
! 246: SFMAX1 = ONE / SFMIN1
! 247: SFMIN2 = SFMIN1*SCLFAC
! 248: SFMAX2 = ONE / SFMIN2
! 249: 140 CONTINUE
! 250: NOCONV = .FALSE.
! 251: *
! 252: DO 200 I = K, L
! 253: C = ZERO
! 254: R = ZERO
! 255: *
! 256: DO 150 J = K, L
! 257: IF( J.EQ.I )
! 258: $ GO TO 150
! 259: C = C + CABS1( A( J, I ) )
! 260: R = R + CABS1( A( I, J ) )
! 261: 150 CONTINUE
! 262: ICA = IZAMAX( L, A( 1, I ), 1 )
! 263: CA = ABS( A( ICA, I ) )
! 264: IRA = IZAMAX( N-K+1, A( I, K ), LDA )
! 265: RA = ABS( A( I, IRA+K-1 ) )
! 266: *
! 267: * Guard against zero C or R due to underflow.
! 268: *
! 269: IF( C.EQ.ZERO .OR. R.EQ.ZERO )
! 270: $ GO TO 200
! 271: G = R / SCLFAC
! 272: F = ONE
! 273: S = C + R
! 274: 160 CONTINUE
! 275: IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR.
! 276: $ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170
! 277: F = F*SCLFAC
! 278: C = C*SCLFAC
! 279: CA = CA*SCLFAC
! 280: R = R / SCLFAC
! 281: G = G / SCLFAC
! 282: RA = RA / SCLFAC
! 283: GO TO 160
! 284: *
! 285: 170 CONTINUE
! 286: G = C / SCLFAC
! 287: 180 CONTINUE
! 288: IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR.
! 289: $ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190
! 290: F = F / SCLFAC
! 291: C = C / SCLFAC
! 292: G = G / SCLFAC
! 293: CA = CA / SCLFAC
! 294: R = R*SCLFAC
! 295: RA = RA*SCLFAC
! 296: GO TO 180
! 297: *
! 298: * Now balance.
! 299: *
! 300: 190 CONTINUE
! 301: IF( ( C+R ).GE.FACTOR*S )
! 302: $ GO TO 200
! 303: IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
! 304: IF( F*SCALE( I ).LE.SFMIN1 )
! 305: $ GO TO 200
! 306: END IF
! 307: IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
! 308: IF( SCALE( I ).GE.SFMAX1 / F )
! 309: $ GO TO 200
! 310: END IF
! 311: G = ONE / F
! 312: SCALE( I ) = SCALE( I )*F
! 313: NOCONV = .TRUE.
! 314: *
! 315: CALL ZDSCAL( N-K+1, G, A( I, K ), LDA )
! 316: CALL ZDSCAL( L, F, A( 1, I ), 1 )
! 317: *
! 318: 200 CONTINUE
! 319: *
! 320: IF( NOCONV )
! 321: $ GO TO 140
! 322: *
! 323: 210 CONTINUE
! 324: ILO = K
! 325: IHI = L
! 326: *
! 327: RETURN
! 328: *
! 329: * End of ZGEBAL
! 330: *
! 331: END
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