Annotation of rpl/lapack/lapack/dgbbrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
! 2: $ LDQ, PT, LDPT, C, LDC, WORK, INFO )
! 3: *
! 4: * -- LAPACK routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: CHARACTER VECT
! 11: INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
! 12: * ..
! 13: * .. Array Arguments ..
! 14: DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
! 15: $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DGBBRD reduces a real general m-by-n band matrix A to upper
! 22: * bidiagonal form B by an orthogonal transformation: Q' * A * P = B.
! 23: *
! 24: * The routine computes B, and optionally forms Q or P', or computes
! 25: * Q'*C for a given matrix C.
! 26: *
! 27: * Arguments
! 28: * =========
! 29: *
! 30: * VECT (input) CHARACTER*1
! 31: * Specifies whether or not the matrices Q and P' are to be
! 32: * formed.
! 33: * = 'N': do not form Q or P';
! 34: * = 'Q': form Q only;
! 35: * = 'P': form P' only;
! 36: * = 'B': form both.
! 37: *
! 38: * M (input) INTEGER
! 39: * The number of rows of the matrix A. M >= 0.
! 40: *
! 41: * N (input) INTEGER
! 42: * The number of columns of the matrix A. N >= 0.
! 43: *
! 44: * NCC (input) INTEGER
! 45: * The number of columns of the matrix C. NCC >= 0.
! 46: *
! 47: * KL (input) INTEGER
! 48: * The number of subdiagonals of the matrix A. KL >= 0.
! 49: *
! 50: * KU (input) INTEGER
! 51: * The number of superdiagonals of the matrix A. KU >= 0.
! 52: *
! 53: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
! 54: * On entry, the m-by-n band matrix A, stored in rows 1 to
! 55: * KL+KU+1. The j-th column of A is stored in the j-th column of
! 56: * the array AB as follows:
! 57: * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
! 58: * On exit, A is overwritten by values generated during the
! 59: * reduction.
! 60: *
! 61: * LDAB (input) INTEGER
! 62: * The leading dimension of the array A. LDAB >= KL+KU+1.
! 63: *
! 64: * D (output) DOUBLE PRECISION array, dimension (min(M,N))
! 65: * The diagonal elements of the bidiagonal matrix B.
! 66: *
! 67: * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
! 68: * The superdiagonal elements of the bidiagonal matrix B.
! 69: *
! 70: * Q (output) DOUBLE PRECISION array, dimension (LDQ,M)
! 71: * If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
! 72: * If VECT = 'N' or 'P', the array Q is not referenced.
! 73: *
! 74: * LDQ (input) INTEGER
! 75: * The leading dimension of the array Q.
! 76: * LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
! 77: *
! 78: * PT (output) DOUBLE PRECISION array, dimension (LDPT,N)
! 79: * If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
! 80: * If VECT = 'N' or 'Q', the array PT is not referenced.
! 81: *
! 82: * LDPT (input) INTEGER
! 83: * The leading dimension of the array PT.
! 84: * LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
! 85: *
! 86: * C (input/output) DOUBLE PRECISION array, dimension (LDC,NCC)
! 87: * On entry, an m-by-ncc matrix C.
! 88: * On exit, C is overwritten by Q'*C.
! 89: * C is not referenced if NCC = 0.
! 90: *
! 91: * LDC (input) INTEGER
! 92: * The leading dimension of the array C.
! 93: * LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
! 94: *
! 95: * WORK (workspace) DOUBLE PRECISION array, dimension (2*max(M,N))
! 96: *
! 97: * INFO (output) INTEGER
! 98: * = 0: successful exit.
! 99: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 100: *
! 101: * =====================================================================
! 102: *
! 103: * .. Parameters ..
! 104: DOUBLE PRECISION ZERO, ONE
! 105: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 106: * ..
! 107: * .. Local Scalars ..
! 108: LOGICAL WANTB, WANTC, WANTPT, WANTQ
! 109: INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
! 110: $ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
! 111: DOUBLE PRECISION RA, RB, RC, RS
! 112: * ..
! 113: * .. External Subroutines ..
! 114: EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA
! 115: * ..
! 116: * .. Intrinsic Functions ..
! 117: INTRINSIC MAX, MIN
! 118: * ..
! 119: * .. External Functions ..
! 120: LOGICAL LSAME
! 121: EXTERNAL LSAME
! 122: * ..
! 123: * .. Executable Statements ..
! 124: *
! 125: * Test the input parameters
! 126: *
! 127: WANTB = LSAME( VECT, 'B' )
! 128: WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
! 129: WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
! 130: WANTC = NCC.GT.0
! 131: KLU1 = KL + KU + 1
! 132: INFO = 0
! 133: IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
! 134: $ THEN
! 135: INFO = -1
! 136: ELSE IF( M.LT.0 ) THEN
! 137: INFO = -2
! 138: ELSE IF( N.LT.0 ) THEN
! 139: INFO = -3
! 140: ELSE IF( NCC.LT.0 ) THEN
! 141: INFO = -4
! 142: ELSE IF( KL.LT.0 ) THEN
! 143: INFO = -5
! 144: ELSE IF( KU.LT.0 ) THEN
! 145: INFO = -6
! 146: ELSE IF( LDAB.LT.KLU1 ) THEN
! 147: INFO = -8
! 148: ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
! 149: INFO = -12
! 150: ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
! 151: INFO = -14
! 152: ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
! 153: INFO = -16
! 154: END IF
! 155: IF( INFO.NE.0 ) THEN
! 156: CALL XERBLA( 'DGBBRD', -INFO )
! 157: RETURN
! 158: END IF
! 159: *
! 160: * Initialize Q and P' to the unit matrix, if needed
! 161: *
! 162: IF( WANTQ )
! 163: $ CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
! 164: IF( WANTPT )
! 165: $ CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
! 166: *
! 167: * Quick return if possible.
! 168: *
! 169: IF( M.EQ.0 .OR. N.EQ.0 )
! 170: $ RETURN
! 171: *
! 172: MINMN = MIN( M, N )
! 173: *
! 174: IF( KL+KU.GT.1 ) THEN
! 175: *
! 176: * Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
! 177: * first to lower bidiagonal form and then transform to upper
! 178: * bidiagonal
! 179: *
! 180: IF( KU.GT.0 ) THEN
! 181: ML0 = 1
! 182: MU0 = 2
! 183: ELSE
! 184: ML0 = 2
! 185: MU0 = 1
! 186: END IF
! 187: *
! 188: * Wherever possible, plane rotations are generated and applied in
! 189: * vector operations of length NR over the index set J1:J2:KLU1.
! 190: *
! 191: * The sines of the plane rotations are stored in WORK(1:max(m,n))
! 192: * and the cosines in WORK(max(m,n)+1:2*max(m,n)).
! 193: *
! 194: MN = MAX( M, N )
! 195: KLM = MIN( M-1, KL )
! 196: KUN = MIN( N-1, KU )
! 197: KB = KLM + KUN
! 198: KB1 = KB + 1
! 199: INCA = KB1*LDAB
! 200: NR = 0
! 201: J1 = KLM + 2
! 202: J2 = 1 - KUN
! 203: *
! 204: DO 90 I = 1, MINMN
! 205: *
! 206: * Reduce i-th column and i-th row of matrix to bidiagonal form
! 207: *
! 208: ML = KLM + 1
! 209: MU = KUN + 1
! 210: DO 80 KK = 1, KB
! 211: J1 = J1 + KB
! 212: J2 = J2 + KB
! 213: *
! 214: * generate plane rotations to annihilate nonzero elements
! 215: * which have been created below the band
! 216: *
! 217: IF( NR.GT.0 )
! 218: $ CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
! 219: $ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
! 220: *
! 221: * apply plane rotations from the left
! 222: *
! 223: DO 10 L = 1, KB
! 224: IF( J2-KLM+L-1.GT.N ) THEN
! 225: NRT = NR - 1
! 226: ELSE
! 227: NRT = NR
! 228: END IF
! 229: IF( NRT.GT.0 )
! 230: $ CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
! 231: $ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
! 232: $ WORK( MN+J1 ), WORK( J1 ), KB1 )
! 233: 10 CONTINUE
! 234: *
! 235: IF( ML.GT.ML0 ) THEN
! 236: IF( ML.LE.M-I+1 ) THEN
! 237: *
! 238: * generate plane rotation to annihilate a(i+ml-1,i)
! 239: * within the band, and apply rotation from the left
! 240: *
! 241: CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
! 242: $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
! 243: $ RA )
! 244: AB( KU+ML-1, I ) = RA
! 245: IF( I.LT.N )
! 246: $ CALL DROT( MIN( KU+ML-2, N-I ),
! 247: $ AB( KU+ML-2, I+1 ), LDAB-1,
! 248: $ AB( KU+ML-1, I+1 ), LDAB-1,
! 249: $ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
! 250: END IF
! 251: NR = NR + 1
! 252: J1 = J1 - KB1
! 253: END IF
! 254: *
! 255: IF( WANTQ ) THEN
! 256: *
! 257: * accumulate product of plane rotations in Q
! 258: *
! 259: DO 20 J = J1, J2, KB1
! 260: CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
! 261: $ WORK( MN+J ), WORK( J ) )
! 262: 20 CONTINUE
! 263: END IF
! 264: *
! 265: IF( WANTC ) THEN
! 266: *
! 267: * apply plane rotations to C
! 268: *
! 269: DO 30 J = J1, J2, KB1
! 270: CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
! 271: $ WORK( MN+J ), WORK( J ) )
! 272: 30 CONTINUE
! 273: END IF
! 274: *
! 275: IF( J2+KUN.GT.N ) THEN
! 276: *
! 277: * adjust J2 to keep within the bounds of the matrix
! 278: *
! 279: NR = NR - 1
! 280: J2 = J2 - KB1
! 281: END IF
! 282: *
! 283: DO 40 J = J1, J2, KB1
! 284: *
! 285: * create nonzero element a(j-1,j+ku) above the band
! 286: * and store it in WORK(n+1:2*n)
! 287: *
! 288: WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
! 289: AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
! 290: 40 CONTINUE
! 291: *
! 292: * generate plane rotations to annihilate nonzero elements
! 293: * which have been generated above the band
! 294: *
! 295: IF( NR.GT.0 )
! 296: $ CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
! 297: $ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
! 298: $ KB1 )
! 299: *
! 300: * apply plane rotations from the right
! 301: *
! 302: DO 50 L = 1, KB
! 303: IF( J2+L-1.GT.M ) THEN
! 304: NRT = NR - 1
! 305: ELSE
! 306: NRT = NR
! 307: END IF
! 308: IF( NRT.GT.0 )
! 309: $ CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
! 310: $ AB( L, J1+KUN ), INCA,
! 311: $ WORK( MN+J1+KUN ), WORK( J1+KUN ),
! 312: $ KB1 )
! 313: 50 CONTINUE
! 314: *
! 315: IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
! 316: IF( MU.LE.N-I+1 ) THEN
! 317: *
! 318: * generate plane rotation to annihilate a(i,i+mu-1)
! 319: * within the band, and apply rotation from the right
! 320: *
! 321: CALL DLARTG( AB( KU-MU+3, I+MU-2 ),
! 322: $ AB( KU-MU+2, I+MU-1 ),
! 323: $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
! 324: $ RA )
! 325: AB( KU-MU+3, I+MU-2 ) = RA
! 326: CALL DROT( MIN( KL+MU-2, M-I ),
! 327: $ AB( KU-MU+4, I+MU-2 ), 1,
! 328: $ AB( KU-MU+3, I+MU-1 ), 1,
! 329: $ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
! 330: END IF
! 331: NR = NR + 1
! 332: J1 = J1 - KB1
! 333: END IF
! 334: *
! 335: IF( WANTPT ) THEN
! 336: *
! 337: * accumulate product of plane rotations in P'
! 338: *
! 339: DO 60 J = J1, J2, KB1
! 340: CALL DROT( N, PT( J+KUN-1, 1 ), LDPT,
! 341: $ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
! 342: $ WORK( J+KUN ) )
! 343: 60 CONTINUE
! 344: END IF
! 345: *
! 346: IF( J2+KB.GT.M ) THEN
! 347: *
! 348: * adjust J2 to keep within the bounds of the matrix
! 349: *
! 350: NR = NR - 1
! 351: J2 = J2 - KB1
! 352: END IF
! 353: *
! 354: DO 70 J = J1, J2, KB1
! 355: *
! 356: * create nonzero element a(j+kl+ku,j+ku-1) below the
! 357: * band and store it in WORK(1:n)
! 358: *
! 359: WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
! 360: AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
! 361: 70 CONTINUE
! 362: *
! 363: IF( ML.GT.ML0 ) THEN
! 364: ML = ML - 1
! 365: ELSE
! 366: MU = MU - 1
! 367: END IF
! 368: 80 CONTINUE
! 369: 90 CONTINUE
! 370: END IF
! 371: *
! 372: IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
! 373: *
! 374: * A has been reduced to lower bidiagonal form
! 375: *
! 376: * Transform lower bidiagonal form to upper bidiagonal by applying
! 377: * plane rotations from the left, storing diagonal elements in D
! 378: * and off-diagonal elements in E
! 379: *
! 380: DO 100 I = 1, MIN( M-1, N )
! 381: CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
! 382: D( I ) = RA
! 383: IF( I.LT.N ) THEN
! 384: E( I ) = RS*AB( 1, I+1 )
! 385: AB( 1, I+1 ) = RC*AB( 1, I+1 )
! 386: END IF
! 387: IF( WANTQ )
! 388: $ CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
! 389: IF( WANTC )
! 390: $ CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
! 391: $ RS )
! 392: 100 CONTINUE
! 393: IF( M.LE.N )
! 394: $ D( M ) = AB( 1, M )
! 395: ELSE IF( KU.GT.0 ) THEN
! 396: *
! 397: * A has been reduced to upper bidiagonal form
! 398: *
! 399: IF( M.LT.N ) THEN
! 400: *
! 401: * Annihilate a(m,m+1) by applying plane rotations from the
! 402: * right, storing diagonal elements in D and off-diagonal
! 403: * elements in E
! 404: *
! 405: RB = AB( KU, M+1 )
! 406: DO 110 I = M, 1, -1
! 407: CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA )
! 408: D( I ) = RA
! 409: IF( I.GT.1 ) THEN
! 410: RB = -RS*AB( KU, I )
! 411: E( I-1 ) = RC*AB( KU, I )
! 412: END IF
! 413: IF( WANTPT )
! 414: $ CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
! 415: $ RC, RS )
! 416: 110 CONTINUE
! 417: ELSE
! 418: *
! 419: * Copy off-diagonal elements to E and diagonal elements to D
! 420: *
! 421: DO 120 I = 1, MINMN - 1
! 422: E( I ) = AB( KU, I+1 )
! 423: 120 CONTINUE
! 424: DO 130 I = 1, MINMN
! 425: D( I ) = AB( KU+1, I )
! 426: 130 CONTINUE
! 427: END IF
! 428: ELSE
! 429: *
! 430: * A is diagonal. Set elements of E to zero and copy diagonal
! 431: * elements to D.
! 432: *
! 433: DO 140 I = 1, MINMN - 1
! 434: E( I ) = ZERO
! 435: 140 CONTINUE
! 436: DO 150 I = 1, MINMN
! 437: D( I ) = AB( 1, I )
! 438: 150 CONTINUE
! 439: END IF
! 440: RETURN
! 441: *
! 442: * End of DGBBRD
! 443: *
! 444: END
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