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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, 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: INTEGER INFO, LDA, M, N 10: * .. 11: * .. Array Arguments .. 12: DOUBLE PRECISION D( * ), E( * ) 13: COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) 14: * .. 15: * 16: * Purpose 17: * ======= 18: * 19: * ZGEBD2 reduces a complex general m by n matrix A to upper or lower 20: * real bidiagonal form B by a unitary transformation: Q' * A * P = B. 21: * 22: * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 23: * 24: * Arguments 25: * ========= 26: * 27: * M (input) INTEGER 28: * The number of rows in the matrix A. M >= 0. 29: * 30: * N (input) INTEGER 31: * The number of columns in the matrix A. N >= 0. 32: * 33: * A (input/output) COMPLEX*16 array, dimension (LDA,N) 34: * On entry, the m by n general matrix to be reduced. 35: * On exit, 36: * if m >= n, the diagonal and the first superdiagonal are 37: * overwritten with the upper bidiagonal matrix B; the 38: * elements below the diagonal, with the array TAUQ, represent 39: * the unitary matrix Q as a product of elementary 40: * reflectors, and the elements above the first superdiagonal, 41: * with the array TAUP, represent the unitary matrix P as 42: * a product of elementary reflectors; 43: * if m < n, the diagonal and the first subdiagonal are 44: * overwritten with the lower bidiagonal matrix B; the 45: * elements below the first subdiagonal, with the array TAUQ, 46: * represent the unitary matrix Q as a product of 47: * elementary reflectors, and the elements above the diagonal, 48: * with the array TAUP, represent the unitary matrix P as 49: * a product of elementary reflectors. 50: * See Further Details. 51: * 52: * LDA (input) INTEGER 53: * The leading dimension of the array A. LDA >= max(1,M). 54: * 55: * D (output) DOUBLE PRECISION array, dimension (min(M,N)) 56: * The diagonal elements of the bidiagonal matrix B: 57: * D(i) = A(i,i). 58: * 59: * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) 60: * The off-diagonal elements of the bidiagonal matrix B: 61: * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 62: * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 63: * 64: * TAUQ (output) COMPLEX*16 array dimension (min(M,N)) 65: * The scalar factors of the elementary reflectors which 66: * represent the unitary matrix Q. See Further Details. 67: * 68: * TAUP (output) COMPLEX*16 array, dimension (min(M,N)) 69: * The scalar factors of the elementary reflectors which 70: * represent the unitary matrix P. See Further Details. 71: * 72: * WORK (workspace) COMPLEX*16 array, dimension (max(M,N)) 73: * 74: * INFO (output) INTEGER 75: * = 0: successful exit 76: * < 0: if INFO = -i, the i-th argument had an illegal value. 77: * 78: * Further Details 79: * =============== 80: * 81: * The matrices Q and P are represented as products of elementary 82: * reflectors: 83: * 84: * If m >= n, 85: * 86: * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 87: * 88: * Each H(i) and G(i) has the form: 89: * 90: * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 91: * 92: * where tauq and taup are complex scalars, and v and u are complex 93: * vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in 94: * A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in 95: * A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 96: * 97: * If m < n, 98: * 99: * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 100: * 101: * Each H(i) and G(i) has the form: 102: * 103: * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 104: * 105: * where tauq and taup are complex scalars, v and u are complex vectors; 106: * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); 107: * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); 108: * tauq is stored in TAUQ(i) and taup in TAUP(i). 109: * 110: * The contents of A on exit are illustrated by the following examples: 111: * 112: * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 113: * 114: * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 115: * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 116: * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 117: * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 118: * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 119: * ( v1 v2 v3 v4 v5 ) 120: * 121: * where d and e denote diagonal and off-diagonal elements of B, vi 122: * denotes an element of the vector defining H(i), and ui an element of 123: * the vector defining G(i). 124: * 125: * ===================================================================== 126: * 127: * .. Parameters .. 128: COMPLEX*16 ZERO, ONE 129: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), 130: $ ONE = ( 1.0D+0, 0.0D+0 ) ) 131: * .. 132: * .. Local Scalars .. 133: INTEGER I 134: COMPLEX*16 ALPHA 135: * .. 136: * .. External Subroutines .. 137: EXTERNAL XERBLA, ZLACGV, ZLARF, ZLARFG 138: * .. 139: * .. Intrinsic Functions .. 140: INTRINSIC DCONJG, MAX, MIN 141: * .. 142: * .. Executable Statements .. 143: * 144: * Test the input parameters 145: * 146: INFO = 0 147: IF( M.LT.0 ) THEN 148: INFO = -1 149: ELSE IF( N.LT.0 ) THEN 150: INFO = -2 151: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 152: INFO = -4 153: END IF 154: IF( INFO.LT.0 ) THEN 155: CALL XERBLA( 'ZGEBD2', -INFO ) 156: RETURN 157: END IF 158: * 159: IF( M.GE.N ) THEN 160: * 161: * Reduce to upper bidiagonal form 162: * 163: DO 10 I = 1, N 164: * 165: * Generate elementary reflector H(i) to annihilate A(i+1:m,i) 166: * 167: ALPHA = A( I, I ) 168: CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1, 169: $ TAUQ( I ) ) 170: D( I ) = ALPHA 171: A( I, I ) = ONE 172: * 173: * Apply H(i)' to A(i:m,i+1:n) from the left 174: * 175: IF( I.LT.N ) 176: $ CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1, 177: $ DCONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK ) 178: A( I, I ) = D( I ) 179: * 180: IF( I.LT.N ) THEN 181: * 182: * Generate elementary reflector G(i) to annihilate 183: * A(i,i+2:n) 184: * 185: CALL ZLACGV( N-I, A( I, I+1 ), LDA ) 186: ALPHA = A( I, I+1 ) 187: CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA, 188: $ TAUP( I ) ) 189: E( I ) = ALPHA 190: A( I, I+1 ) = ONE 191: * 192: * Apply G(i) to A(i+1:m,i+1:n) from the right 193: * 194: CALL ZLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, 195: $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) 196: CALL ZLACGV( N-I, A( I, I+1 ), LDA ) 197: A( I, I+1 ) = E( I ) 198: ELSE 199: TAUP( I ) = ZERO 200: END IF 201: 10 CONTINUE 202: ELSE 203: * 204: * Reduce to lower bidiagonal form 205: * 206: DO 20 I = 1, M 207: * 208: * Generate elementary reflector G(i) to annihilate A(i,i+1:n) 209: * 210: CALL ZLACGV( N-I+1, A( I, I ), LDA ) 211: ALPHA = A( I, I ) 212: CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA, 213: $ TAUP( I ) ) 214: D( I ) = ALPHA 215: A( I, I ) = ONE 216: * 217: * Apply G(i) to A(i+1:m,i:n) from the right 218: * 219: IF( I.LT.M ) 220: $ CALL ZLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, 221: $ TAUP( I ), A( I+1, I ), LDA, WORK ) 222: CALL ZLACGV( N-I+1, A( I, I ), LDA ) 223: A( I, I ) = D( I ) 224: * 225: IF( I.LT.M ) THEN 226: * 227: * Generate elementary reflector H(i) to annihilate 228: * A(i+2:m,i) 229: * 230: ALPHA = A( I+1, I ) 231: CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1, 232: $ TAUQ( I ) ) 233: E( I ) = ALPHA 234: A( I+1, I ) = ONE 235: * 236: * Apply H(i)' to A(i+1:m,i+1:n) from the left 237: * 238: CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1, 239: $ DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA, 240: $ WORK ) 241: A( I+1, I ) = E( I ) 242: ELSE 243: TAUQ( I ) = ZERO 244: END IF 245: 20 CONTINUE 246: END IF 247: RETURN 248: * 249: * End of ZGEBD2 250: * 251: END