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eigenvectors, some fixing

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Anton Ljungdahl 2024-11-05 22:23:16 +01:00
parent ae39bb793f
commit 1ae83767ff
22 changed files with 608 additions and 718 deletions
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15
TODO.txt Normal file
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//~ TODO
//-
/**
*
[-] Use arena for matrices instead of malloc maybe
[-] Factor l-dependent matrix
[-] Print eigenfunctions for plotting
[-] Solve for more shells
[-] Make some good orbital/atom abstraction
[-] Then try to make hartree fock work maybe.
[-] Need to make a Complex double (Z64) version if I want to go relativistic
[-] Smarter way to multiply bsplines and do gauss integration?
*
*/

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notebook/hf_again.ipynb
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out/bspline1.dat
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out/bspline2.dat
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@ -1,9 +1,9 @@
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out/bsplines_grid.dat
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out/dBsplines_knots.dat
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0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 -1.032291e-28 -3.433940e-14 -7.615385e+00 7.615385e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 -1.032291e-28 -3.433940e-14 -7.615385e+00 7.615385e+00

BIN
out/grid.bin
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Binary file not shown.

184
out/grid.dat
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@ -1,100 +1,100 @@
0.000000e+00
1.000000e-01
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1.300000e-01
2.600000e-01
3.900000e-01
5.200000e-01
6.500000e-01
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1.040000e+00
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2.000000e+00
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2.300000e+00
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1.430000e+00
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34
out/ground_state_coeffs.dat Normal file
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@ -0,0 +1,34 @@
1.133340e-01
2.694214e-01
4.439444e-01
4.595803e-01
4.179690e-01
3.547308e-01
2.883617e-01
2.276047e-01
1.758416e-01
1.336568e-01
1.003007e-01
7.449632e-02
5.486226e-02
4.011609e-02
2.915634e-02
2.108062e-02
1.517274e-02
1.087716e-02
7.770252e-03
5.533336e-03
3.929222e-03
2.782933e-03
1.966331e-03
1.386158e-03
9.748959e-04
6.838729e-04
4.781267e-04
3.326017e-04
2.293545e-04
1.555076e-04
1.017495e-04
6.122882e-05
2.593097e-05
8.380605e-06

68
out/knotpoints.dat
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@ -2,39 +2,39 @@
0.000000e+00
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2
src/hf/bsplines_and_grid.c
View File

@ -268,7 +268,7 @@ set_up_bsplines_at_points_and_write_matrix_F64(Arena *arena)
{
U32 index = get_bspline_index(i, j);
g_bspline_ctx.bsplines[index] = compute_bspline_F64(t[i], j);
g_bspline_ctx.bsplines[index] = compute_dBspline_F64(t[i], j);
g_bspline_ctx.dBsplines[index] = compute_dBspline_F64(t[i], j);
}
}

45
src/hf/hf_base.c
View File

@ -22,6 +22,24 @@ mat_F64(U32 size1, U32 size2)
return out;
}
function Mat_F64
mat_F64_from_data(U32 size1, U32 size2, F64 *data)
{
Mat_F64 out = {0};
out.size1 = size1;
out.size2 = size2;
out.matrix = (F64 **)malloc(size1 * sizeof(F64 *));
out.data = data;
for(U32 i = 0; i < size1; i++)
{
out.matrix[i] = &out.data[i * size2];
}
return out;
}
function void
mat_F64_free(Mat_F64 *mat)
{
@ -175,3 +193,30 @@ print_mat_F64(Mat_F64 *mat)
}
function void
mat_invert_F64(Mat_F64 *mat)
{
S32 n = mat->size1;
S32 lda = n;
S32 info;
S32 *ipiv = (S32*)malloc(n * sizeof(S32));
F64 *work = (F64 *)malloc(n * sizeof(F64));
S32 lwork = n;
dgetrf_(&n, &n, mat->data, &lda, ipiv, &info);
dgetri_(&n, mat->data, &lda, ipiv, work, &lwork, &info);
free(ipiv);
free(work);
}

3
src/hf/hf_base.h
View File

@ -31,12 +31,15 @@ struct GaussLegendre
// Mat_F64 functions
function Mat_F64 mat_F64(U32 size1, U32 size2);
function Mat_F64 mat_F64_from_data(U32 size1, U32 size2, F64 *data);
function void mat_F64_free(Mat_F64 *mat);
function inline void mat_F64_set(Mat_F64 *mat, U32 i, U32 j, F64 val);
function inline F64 mat_F64_get(Mat_F64 *mat, U32 i, U32 j);
function Mat_F64 mat_F64_copy(Mat_F64 *src);
function void print_mat_F64(Mat_F64 *mat);
function void mat_invert_F64(Mat_F64 *mat);
// Gauss-Legendre
function void set_up_gauss_legendre_points(Arena* arena);

123
src/main.c
View File

@ -54,6 +54,15 @@ struct SortPair_F64
U64 original_index;
};
typedef struct Eigensolution_F64 Eigensolution_F64;
struct Eigensolution_F64
{
F64 *eigenvalues_re;
F64 *eigenvalues_im;
Mat_F64 right_eigenvectors;
Mat_F64 left_eigenvectors;
};
global Arena* g_base_arena = 0;
@ -179,13 +188,14 @@ function void EntryPoint(void)
//- Then we generate the BSplines and save them off for reference and debugging.
set_up_bsplines_at_points_and_write_matrix_F64(g_base_arena);
U32 N = g_bspline_ctx.num_knotpoints;
U32 k = g_bspline_ctx.order;
U32 mat_size1 = N - k - 2;
U32 mat_size2 = mat_size1;
Mat_F64 H = mat_F64(mat_size1, mat_size2);
Mat_F64 B = mat_F64(mat_size1, mat_size2);
Mat_F64 H_l = mat_F64(mat_size1, mat_size2);
// This will be the inverse of B, but to start with we construct B.
Mat_F64 B_inv = mat_F64(mat_size1, mat_size2);
// We work in units hbar = 1, bohr radius a0 = 1, electron mass m_e = 1, and charge e = 1,
// and 1/(4piepsilon_0) = 1.
@ -194,11 +204,13 @@ function void EntryPoint(void)
{
ArenaTemp scratch = scratch_get(0, 0);
LOG("Setting up matrices.\n");
String8 setuplog = str8_pushf(scratch.arena,
"Num knotpoints: %i, Bspline order k = %i, Matrix size1 = N-k-2 = %i, size2 = %i\n",
N, k, mat_size1, mat_size2);
LOG(setuplog.str);
/* { */
/* LOG("Setting up matrices.\n"); */
/* String8 setuplog = str8_pushf(scratch.arena, */
/* "Num knotpoints: %i, Bspline order k = %i, Matrix size1 = N-k-2 = %i, size2 = %i\n", */
/* N, k, mat_size1, mat_size2); */
/* LOG(setuplog.str); */
/* } */
F64 *t = g_bspline_ctx.knotpoints;
@ -239,7 +251,7 @@ function void EntryPoint(void)
F64 b = t[knotpoint_idx+1];
F64 prefac = 0.5*(b-a);
// Skip integrating between ghost points
// Only integrate non-zero intervals
if(prefac > 1e-16)
{
for(U32 gq_i = 0; gq_i < g_gauss_legendre.order; gq_i++)
@ -252,7 +264,6 @@ function void EntryPoint(void)
F64 dB_j = compute_dBspline_F64(r, bspl_index_j);
F64 B_i = compute_bspline_F64(r, bspl_index_i);
F64 B_j = compute_bspline_F64(r, bspl_index_j);
//F64 r_inv = r > 0.0 ? 1.0/r : 0.0;
term1 += term_prefac*dB_i*dB_j;
term2 += term_prefac*B_i*B_j / (r * r);
term3 += term_prefac*B_i*B_j/r;
@ -261,68 +272,34 @@ function void EntryPoint(void)
}
}
F64 term_sum = 0.5*term1 + (0.5*(l*(l+1.0))*term2) + (-Z)*term3;
String8 debug = str8_pushf(scratch.arena, "(i=%i,j=%i,t_i=%4.4f,t_i=%4.4f,term1=%.4e,term2=%.4e,term3=%.4e,term_sum=%.4e) \n",
bspl_index_i, bspl_index_j, t[bspl_index_i+k-1],t[bspl_index_j+k-1],term1,term2,term3,term_sum);
LOG(debug.str);
mat_F64_set(&H, i, j, term_sum);
mat_F64_set(&B, i, j, Bmat_term);
F64 H_term_sum = 0.5*term1 + (-Z)*term3;
F64 H_l_term = 0.5*term2;
/* String8 debug = str8_pushf(scratch.arena, "(i=%i,j=%i,t_i=%4.4f,t_i=%4.4f,term1=%.4e,term2=%.4e,term3=%.4e,term_sum=%.4e) \n", */
/* bspl_index_i, bspl_index_j, t[bspl_index_i+k-1],t[bspl_index_j+k-1],term1,term2,term3,term_sum); */
/* LOG(debug.str); */
mat_F64_set(&H, i, j, H_term_sum);
mat_F64_set(&H_l, i, j, H_l_term);
mat_F64_set(&B_inv, i, j, Bmat_term);
//mat_F64_set(&H, i, j, abs_index_diff);
}
//mat_F64_set(&H, i, j, abs_index_diff);
}
LOG("\n");
//LOG("\n");
}
LOG(str8_pushf(scratch.arena, "H.size1=N-k-2=%i, last bspline index=%i \n", H.size1, g_bspline_ctx.num_bsplines-1).str);
//LOG(str8_pushf(scratch.arena, "H.size1=N-k-2=%i, last bspline index=%i \n", H.size1, g_bspline_ctx.num_bsplines-1).str);
scratch_release(scratch);
print_mat_F64(&H);
LOG("\n");
//print_mat_F64(&H);
//LOG("\n");
//print_mat_F64(&B);
}
// Our problem is Hc = EBc, but we want to solve B^-1Hc = Ec,
// so we invert the B matrix and compute the product A = B^-1H before calling zgeev
Mat_F64 B_inv = mat_F64_copy(&B);
Mat_F64 A = mat_F64(H.size1, H.size2);
Arena *mkl_arena = m_make_arena();
//print_mat_F64(&B);
LOG("\n");
//print_mat_F64(&B_inv);
// Invert
{
S32 n = B_inv.size1;
S32 lda = n;
S32 info;
S32 *ipiv = (S32*)malloc(n * sizeof(S32));
F64 *work = (F64 *)malloc(n * sizeof(F64));
S32 lwork = n;
dgetrf_(&n, &n, B_inv.data, &lda, ipiv, &info);
dgetri_(&n, B_inv.data, &lda, ipiv, work, &lwork, &info);
//print_mat_F64(&B_inv);
// Test that inverse worked
/* { */
/* Mat_F64 test = mat_F64(B_inv.size1, B_inv.size2); */
/* cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, */
/* n, n, n, 1.0, B_inv.data, n, B.data, n, 1.0, test.data, n); */
/* LOG("\n"); */
/* print_mat_F64(&test); */
/* mat_F64_free(&test); */
/* } */
free(ipiv);
free(work);
}
// Our problem is Hc = EBc, but we want to solve B^-1Hc = Ec,
// so we invert the B matrix and compute the product A = B^-1H before calling zgeev
Mat_F64 A = mat_F64(H.size1, H.size2);
mat_invert_F64(&B_inv);
// Multiply to get A = B^-1 H
{
@ -333,6 +310,7 @@ function void EntryPoint(void)
print_mat_F64(&A);
}
Eigensolution_F64 eigensolution = {0};
// Solve generalised eigenvalue problem
{
S32 n = A.size1;
@ -349,7 +327,7 @@ function void EntryPoint(void)
F64 *wi = PushArray(mkl_arena, F64, n);
F64 *vl = PushArray(mkl_arena, F64, ldvl*n);
F64 *vr = PushArray(mkl_arena, F64, ldvr*n);
lwork = -1;
F64 *a = A.data;
dgeev("Vectors", "Vectors", &n, a, &lda, wr, wi, vl, &ldvl, vr, &ldvr,
@ -364,27 +342,22 @@ function void EntryPoint(void)
exit( 1 );
}
eigensolution.right_eigenvectors = mat_F64_from_data(ldvr, n, vr);
print_eigenvalues( "Eigenvalues unsorted:", n, wr, wi );
U64 *sorted_indices = PushArray(mkl_arena, U64, n);
sort_and_get_indices_F64(wr, sorted_indices, n);
sort_by_indices_F64(wi, sorted_indices, n);
/* Print eigenvalues */
print_eigenvalues( "Eigenvalues sorted: ", n, wr, wi );
/* Free workspace */
U64 lowest_energy_index = sorted_indices[0];
write_array_F64(str8_lit("D:\\dev\\hf_again\\out\\ground_state_coeffs.dat"), eigensolution.right_eigenvectors.matrix[lowest_energy_index], n, "%13.6e\n");
free( (void*)work );
}
}
//~ TODO
//-
///
/// - Use arena for matrices instead of malloc maybe
/// - Factor l-dependent matrix
/// - Print eigenfunctions for plotting
/// - Solve for more shells
/// - Make some good orbital/atom abstraction
/// - Then try to make hartree fock work maybe.
/// - Need to make a Complex double (Z64) version if I want to go relativistic
}

272
src/test_mkl.c
View File

@ -1,272 +0,0 @@
/*******************************************************************************
* Copyright 2009-2021 Intel Corporation.
*
* This software and the related documents are Intel copyrighted materials, and
* your use of them is governed by the express license under which they were
* provided to you (License). Unless the License provides otherwise, you may not
* use, modify, copy, publish, distribute, disclose or transmit this software or
* the related documents without Intel's prior written permission.
*
* This software and the related documents are provided as is, with no express
* or implied warranties, other than those that are expressly stated in the
* License.
*******************************************************************************/
/*
ZGEEV Example.
==============
Program computes the eigenvalues and left and right eigenvectors of a general
rectangular matrix A:
( -3.84, 2.25) ( -8.94, -4.75) ( 8.95, -6.53) ( -9.87, 4.82)
( -0.66, 0.83) ( -4.40, -3.82) ( -3.50, -4.26) ( -3.15, 7.36)
( -3.99, -4.73) ( -5.88, -6.60) ( -3.36, -0.40) ( -0.75, 5.23)
( 7.74, 4.18) ( 3.66, -7.53) ( 2.58, 3.60) ( 4.59, 5.41)
Description.
============
The routine computes for an n-by-n complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors. The right
eigenvector v(j) of A satisfies
A*v(j)= lambda(j)*v(j)
where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies
u(j)H*A = lambda(j)*u(j)H
where u(j)H denotes the conjugate transpose of u(j). The computed
eigenvectors are normalized to have Euclidean norm equal to 1 and
largest component real.
Example Program Results.
========================
ZGEEV Example Program Results
Eigenvalues
( -9.43,-12.98) ( -3.44, 12.69) ( 0.11, -3.40) ( 5.76, 7.13)
Left eigenvectors
( 0.24, -0.18) ( 0.61, 0.00) ( -0.18, -0.33) ( 0.28, 0.09)
( 0.79, 0.00) ( -0.05, -0.27) ( 0.82, 0.00) ( -0.55, 0.16)
( 0.22, -0.27) ( -0.21, 0.53) ( -0.37, 0.15) ( 0.45, 0.09)
( -0.02, 0.41) ( 0.40, -0.24) ( 0.06, 0.12) ( 0.62, 0.00)
Right eigenvectors
( 0.43, 0.33) ( 0.83, 0.00) ( 0.60, 0.00) ( -0.31, 0.03)
( 0.51, -0.03) ( 0.08, -0.25) ( -0.40, -0.20) ( 0.04, 0.34)
( 0.62, 0.00) ( -0.25, 0.28) ( -0.09, -0.48) ( 0.36, 0.06)
( -0.23, 0.11) ( -0.10, -0.32) ( -0.43, 0.13) ( 0.81, 0.00)
*/
#include <stdlib.h>
#include <stdio.h>
#include <mkl.h>
/* Auxiliary routines prototypes */
function void
print_matrix_cmplx16( char* desc, int m, int n, MKL_Complex16* a, int lda );
#ifndef TEST_ZGEEV_N
#define TEST_ZGEEV_N 4
#endif
/* Main program */
static void
test_mkl_zgeev(void) {
/* Locals */
int N = TEST_ZGEEV_N;
MKL_INT n = N, lda = N, ldvl = N, ldvr = N, info, lwork;
MKL_Complex16 wkopt;
MKL_Complex16* work;
/* Local arrays */
/* rwork dimension should be at least 2*n */
double rwork[2*TEST_ZGEEV_N];
MKL_Complex16 w[TEST_ZGEEV_N], vl[TEST_ZGEEV_N*TEST_ZGEEV_N], vr[TEST_ZGEEV_N*TEST_ZGEEV_N];
MKL_Complex16 a[TEST_ZGEEV_N*TEST_ZGEEV_N] = {
{-3.84, 2.25}, {-0.66, 0.83}, {-3.99, -4.73}, { 7.74, 4.18},
{-8.94, -4.75}, {-4.40, -3.82}, {-5.88, -6.60}, { 3.66, -7.53},
{ 8.95, -6.53}, {-3.50, -4.26}, {-3.36, -0.40}, { 2.58, 3.60},
{-9.87, 4.82}, {-3.15, 7.36}, {-0.75, 5.23}, { 4.59, 5.41}
};
/* Executable statements */
LOG( " ZGEEV Example Program Results\n" );
/* Query and allocate the optimal workspace */
lwork = -1;
zgeev( "Vectors", "Vectors", &n, a, &lda, w, vl, &ldvl, vr, &ldvr,
&wkopt, &lwork, rwork, &info );
lwork = (MKL_INT)wkopt.real;
work = (MKL_Complex16*)malloc( lwork*sizeof(MKL_Complex16) );
/* Solve eigenproblem */
zgeev( "Vectors", "Vectors", &n, a, &lda, w, vl, &ldvl, vr, &ldvr,
work, &lwork, rwork, &info );
/* Check for convergence */
if( info > 0 ) {
LOG( "The algorithm failed to compute eigenvalues.\n" );
exit( 1 );
}
/* Print eigenvalues */
print_matrix_cmplx16( "Eigenvalues", 1, n, w, 1 );
/* Print left eigenvectors */
print_matrix_cmplx16( "Left eigenvectors", n, n, vl, ldvl );
/* Print right eigenvectors */
print_matrix_cmplx16( "Right eigenvectors", n, n, vr, ldvr );
/* Free workspace */
free( (void*)work );
//exit( 0 );
} /* End of ZGEEV Example */
/* Auxiliary routine: printing a matrix */
function void
print_matrix_cmplx16( char* desc, int m, int n, MKL_Complex16* a, int lda ) {
ArenaTemp scratch = scratch_get(0,0);
int i, j;
String8 newline = str8_lit("\n");
String8 header = str8_pushf(scratch.arena, "\n %s\n", desc );
LOG(header.str);
//printf("\n %s \n", desc);
for( i = 0; i < m; i++ ) {
for( j = 0; j < n; j++ ) {
String8 outstr = str8_pushf(scratch.arena, " (%6.2f,%6.2f)", a[i+j*lda].real, a[i+j*lda].imag );
LOG(outstr.str);
//printf(" (%6.2f,%6.2f)", a[i+j*lda].real, a[i+j*lda].imag );
}
LOG(newline.str);
//printf("\n");
}
}
/*******************************************************************************
* Copyright 2009-2021 Intel Corporation.
*
* This software and the related documents are Intel copyrighted materials, and
* your use of them is governed by the express license under which they were
* provided to you (License). Unless the License provides otherwise, you may not
* use, modify, copy, publish, distribute, disclose or transmit this software or
* the related documents without Intel's prior written permission.
*
* This software and the related documents are provided as is, with no express
* or implied warranties, other than those that are expressly stated in the
* License.
*******************************************************************************/
/*
DSYEVD Example.
==============
Program computes all eigenvalues and eigenvectors of a real symmetric
matrix A using divide and conquer algorithm, where A is:
6.39 0.13 -8.23 5.71 -3.18
0.13 8.37 -4.46 -6.10 7.21
-8.23 -4.46 -9.58 -9.25 -7.42
5.71 -6.10 -9.25 3.72 8.54
-3.18 7.21 -7.42 8.54 2.51
Description.
============
The routine computes all eigenvalues and, optionally, eigenvectors of an
n-by-n real symmetric matrix A. The eigenvector v(j) of A satisfies
A*v(j) = lambda(j)*v(j)
where lambda(j) is its eigenvalue. The computed eigenvectors are
orthonormal.
If the eigenvectors are requested, then this routine uses a divide and
conquer algorithm to compute eigenvalues and eigenvectors.
Example Program Results.
========================
DSYEVD Example Program Results
Eigenvalues
-17.44 -11.96 6.72 14.25 19.84
Eigenvectors (stored columnwise)
-0.26 0.31 -0.74 0.33 0.42
-0.17 -0.39 -0.38 -0.80 0.16
-0.89 0.04 0.09 0.03 -0.45
-0.29 -0.59 0.34 0.31 0.60
-0.19 0.63 0.44 -0.38 0.48
*/
#include <stdlib.h>
#include <stdio.h>
#include <mkl.h>
/* Auxiliary routines prototypes */
extern void print_matrix( char* desc, int m, int n, double* a, int lda );
/* Parameters */
#define TEST_DSYEVD_N 5
#define LDA TEST_DSYEVD_N
/* Main program */
function void
test_mkl_dsyevd(void) {
/* Locals */
MKL_INT n = TEST_DSYEVD_N, lda = LDA, info, lwork, liwork;
MKL_INT iwkopt;
MKL_INT* iwork;
double wkopt;
double* work;
/* Local arrays */
double w[TEST_DSYEVD_N];
double a[LDA*TEST_DSYEVD_N] = {
6.39, 0.00, 0.00, 0.00, 0.00,
0.13, 8.37, 0.00, 0.00, 0.00,
-8.23, -4.46, -9.58, 0.00, 0.00,
5.71, -6.10, -9.25, 3.72, 0.00,
-3.18, 7.21, -7.42, 8.54, 2.51
};
/* Executable statements */
LOG( " DSYEVD Example Program Results\n" );
/* Query and allocate the optimal workspace */
lwork = -1;
liwork = -1;
dsyevd( "Vectors", "Upper", &n, a, &lda, w, &wkopt, &lwork, &iwkopt,
&liwork, &info );
lwork = (MKL_INT)wkopt;
work = (double*)malloc( lwork*sizeof(double) );
liwork = iwkopt;
iwork = (MKL_INT*)malloc( liwork*sizeof(MKL_INT) );
/* Solve eigenproblem */
dsyevd( "Vectors", "Upper", &n, a, &lda, w, work, &lwork, iwork,
&liwork, &info );
/* Check for convergence */
if( info > 0 ) {
LOG( "The algorithm failed to compute eigenvalues.\n" );
exit( 1 );
}
/* Print eigenvalues */
print_matrix( "Eigenvalues", 1, n, w, 1 );
/* Print eigenvectors */
print_matrix( "Eigenvectors (stored columnwise)", n, n, a, lda );
/* Free workspace */
free( (void*)iwork );
free( (void*)work );
} /* End of DSYEVD Example */
/* Auxiliary routine: printing a matrix */
void print_matrix( char* desc, int m, int n, double* a, int lda ) {
ArenaTemp scratch = scratch_get(0, 0);
int i, j;
//printf( "\n %s\n", desc );
String8 header = str8_pushf(scratch.arena, "\n %s\n", desc);
LOG(header.str);
for( i = 0; i < m; i++ ) {
for( j = 0; j < n; j++ )
{
String8 out = str8_pushf(scratch.arena, " %6.2f", a[i+j*lda] );
LOG(out.str);
//printf( " %6.2f", a[i+j*lda] );
}
LOG(str8_lit("\n").str);
}
}