00001 SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) 00002 * 00003 * -- LAPACK auxiliary routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 INTEGER K, LDA, LDT, LDY, N, NB 00010 * .. 00011 * .. Array Arguments .. 00012 REAL A( LDA, * ), T( LDT, NB ), TAU( NB ), 00013 $ Y( LDY, NB ) 00014 * .. 00015 * 00016 * Purpose 00017 * ======= 00018 * 00019 * SLAHRD reduces the first NB columns of a real general n-by-(n-k+1) 00020 * matrix A so that elements below the k-th subdiagonal are zero. The 00021 * reduction is performed by an orthogonal similarity transformation 00022 * Q' * A * Q. The routine returns the matrices V and T which determine 00023 * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. 00024 * 00025 * This is an OBSOLETE auxiliary routine. 00026 * This routine will be 'deprecated' in a future release. 00027 * Please use the new routine SLAHR2 instead. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * N (input) INTEGER 00033 * The order of the matrix A. 00034 * 00035 * K (input) INTEGER 00036 * The offset for the reduction. Elements below the k-th 00037 * subdiagonal in the first NB columns are reduced to zero. 00038 * 00039 * NB (input) INTEGER 00040 * The number of columns to be reduced. 00041 * 00042 * A (input/output) REAL array, dimension (LDA,N-K+1) 00043 * On entry, the n-by-(n-k+1) general matrix A. 00044 * On exit, the elements on and above the k-th subdiagonal in 00045 * the first NB columns are overwritten with the corresponding 00046 * elements of the reduced matrix; the elements below the k-th 00047 * subdiagonal, with the array TAU, represent the matrix Q as a 00048 * product of elementary reflectors. The other columns of A are 00049 * unchanged. See Further Details. 00050 * 00051 * LDA (input) INTEGER 00052 * The leading dimension of the array A. LDA >= max(1,N). 00053 * 00054 * TAU (output) REAL array, dimension (NB) 00055 * The scalar factors of the elementary reflectors. See Further 00056 * Details. 00057 * 00058 * T (output) REAL array, dimension (LDT,NB) 00059 * The upper triangular matrix T. 00060 * 00061 * LDT (input) INTEGER 00062 * The leading dimension of the array T. LDT >= NB. 00063 * 00064 * Y (output) REAL array, dimension (LDY,NB) 00065 * The n-by-nb matrix Y. 00066 * 00067 * LDY (input) INTEGER 00068 * The leading dimension of the array Y. LDY >= N. 00069 * 00070 * Further Details 00071 * =============== 00072 * 00073 * The matrix Q is represented as a product of nb elementary reflectors 00074 * 00075 * Q = H(1) H(2) . . . H(nb). 00076 * 00077 * Each H(i) has the form 00078 * 00079 * H(i) = I - tau * v * v' 00080 * 00081 * where tau is a real scalar, and v is a real vector with 00082 * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in 00083 * A(i+k+1:n,i), and tau in TAU(i). 00084 * 00085 * The elements of the vectors v together form the (n-k+1)-by-nb matrix 00086 * V which is needed, with T and Y, to apply the transformation to the 00087 * unreduced part of the matrix, using an update of the form: 00088 * A := (I - V*T*V') * (A - Y*V'). 00089 * 00090 * The contents of A on exit are illustrated by the following example 00091 * with n = 7, k = 3 and nb = 2: 00092 * 00093 * ( a h a a a ) 00094 * ( a h a a a ) 00095 * ( a h a a a ) 00096 * ( h h a a a ) 00097 * ( v1 h a a a ) 00098 * ( v1 v2 a a a ) 00099 * ( v1 v2 a a a ) 00100 * 00101 * where a denotes an element of the original matrix A, h denotes a 00102 * modified element of the upper Hessenberg matrix H, and vi denotes an 00103 * element of the vector defining H(i). 00104 * 00105 * ===================================================================== 00106 * 00107 * .. Parameters .. 00108 REAL ZERO, ONE 00109 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00110 * .. 00111 * .. Local Scalars .. 00112 INTEGER I 00113 REAL EI 00114 * .. 00115 * .. External Subroutines .. 00116 EXTERNAL SAXPY, SCOPY, SGEMV, SLARFG, SSCAL, STRMV 00117 * .. 00118 * .. Intrinsic Functions .. 00119 INTRINSIC MIN 00120 * .. 00121 * .. Executable Statements .. 00122 * 00123 * Quick return if possible 00124 * 00125 IF( N.LE.1 ) 00126 $ RETURN 00127 * 00128 DO 10 I = 1, NB 00129 IF( I.GT.1 ) THEN 00130 * 00131 * Update A(1:n,i) 00132 * 00133 * Compute i-th column of A - Y * V' 00134 * 00135 CALL SGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, 00136 $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) 00137 * 00138 * Apply I - V * T' * V' to this column (call it b) from the 00139 * left, using the last column of T as workspace 00140 * 00141 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) 00142 * ( V2 ) ( b2 ) 00143 * 00144 * where V1 is unit lower triangular 00145 * 00146 * w := V1' * b1 00147 * 00148 CALL SCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) 00149 CALL STRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ), 00150 $ LDA, T( 1, NB ), 1 ) 00151 * 00152 * w := w + V2'*b2 00153 * 00154 CALL SGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), 00155 $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) 00156 * 00157 * w := T'*w 00158 * 00159 CALL STRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT, 00160 $ T( 1, NB ), 1 ) 00161 * 00162 * b2 := b2 - V2*w 00163 * 00164 CALL SGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ), 00165 $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) 00166 * 00167 * b1 := b1 - V1*w 00168 * 00169 CALL STRMV( 'Lower', 'No transpose', 'Unit', I-1, 00170 $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) 00171 CALL SAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) 00172 * 00173 A( K+I-1, I-1 ) = EI 00174 END IF 00175 * 00176 * Generate the elementary reflector H(i) to annihilate 00177 * A(k+i+1:n,i) 00178 * 00179 CALL SLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1, 00180 $ TAU( I ) ) 00181 EI = A( K+I, I ) 00182 A( K+I, I ) = ONE 00183 * 00184 * Compute Y(1:n,i) 00185 * 00186 CALL SGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, 00187 $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) 00188 CALL SGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA, 00189 $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) 00190 CALL SGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, 00191 $ ONE, Y( 1, I ), 1 ) 00192 CALL SSCAL( N, TAU( I ), Y( 1, I ), 1 ) 00193 * 00194 * Compute T(1:i,i) 00195 * 00196 CALL SSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) 00197 CALL STRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT, 00198 $ T( 1, I ), 1 ) 00199 T( I, I ) = TAU( I ) 00200 * 00201 10 CONTINUE 00202 A( K+NB, NB ) = EI 00203 * 00204 RETURN 00205 * 00206 * End of SLAHRD 00207 * 00208 END
1.7.2