Matrices array operations

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

Finally, most doubly or triply subscripted array operations can execute as a single vector instruction on the ASC. To demonstrate the hardware capabilities of the ASC,the vector dot product matrix multiplication instruction, which utilizes one of the most powerful pieces of hardware on the ASC, is compared to similar code on an IBM 360/91 and the CDC 7600 and Cyber 174. Table IV lists the Fortran pattern, which is recognized by the ASC compiler and collapsed into a single vector dot product instruction, the basic instructions required and the hardware speeds obtained when executing the same matrix operations on all four machines. Since many vector instructions in a CP pipe produce one result every clock cycle (80 nanoseconds), ordinary vector multiplications and additions (together) execute at the rate of 24 million floating point operations per second (MFLOPS). For the vector dot product instruction however, each output value produced represents a multiplication and an addition. Thus, vector dot product on the ASC attains a speed of 48 million floating point operations per second. 

The blue OLEDs were based on DPVBi [13,17,35,60,61,65,69]. The OLEDs were prepared as a small encapsulated matrix array of 2 x 2 mm square pixels resulting from perpendicular stripes of etched ITO and evaporated A1 for back-detection, as described above [15,17]. A typical encapsulated array, with 4 x 10 pixels lit simultaneously, is shown in Fig. 3.14. The OLEDs were operated in a dc mode with a forward bias of 9-20 V, or in a pulsed mode with a forward bias of up to 35 V. The photodetector (PD) was a photomultiplier tube (PMT). 

NOTE Because A. B is carried out on elemental level, it requires that both A and B have the same size (see Sec. 2.2). Matrix addition and subtraction are defined based on an element-by-element array operation. 

T. F. Brody, F. C. Luo, D. H. Vavies, and E. W. Greeneich, Operational Characteristics of a 6 X 6 Inch, TFT Matrix Array, Liquid Crystal Display, Digest 1974 Soc. for Information Display International Symp., San Diego, Calif., p. 166. 

Dhe detector array is a bulk 10X64 pixel Si As hybrid device manufac- d by Hu es Aircraft Co. The detector is assembled from a wafer of tnide-doped silicon material bump-bonded with a matrix of indium con- s to a Hughes CRC-310 direct readout (DRO) integrated circuit mul-exer dup.The array operates as 10 linear rows of detectors in parallel, ii 10 readout chains. The pixel size is 100X100 /aa with a separation of /xm center to center,and the operating temperature of the array is 10-12 The optical system of the camera consisting of two convex ZnSe lenses, ndes a pixel field of view onto the array of 1.23 /pix. 

By definition, a numerical matrix is a rectangular array of numbers (termed elements ) enclosed by square brackets [ ]. Matrices can be used to organize information such as size versus cost in a grocery department, or they may be used to simplify the problems associated with systems or groups of linear equations. Later in this chapter we will introduce the operations involved for linear equations (see Table 2-1 for common symbols used). 

In an Excel spreadsheet, the [TRANSPOSE function can be applied to an array of data. For this, we need to become familiar with the two most important rules to perform matrix operations in Excel ... 

In Excel, mathematical operations of one or more cells can be dragged to other cells. Since a cell represents one element of an array or matrix, the effect will be an element-wise matrix calculation. Thus, addition and subtraction of matrices are straightforward. An example ... 

What about the converse does any linear transformation determine a matrix This question raises two issues. First, if the domain is infinite-dimensional, the question is more complicated. Mathematicians usually reserve the word matrix for a finite-dimensional matrix (i.e., an array with a finite number of rows and colunms). Physicists often use matrix to denote a linear transformation between infinite-dimensional spaces, where mathematicians would usually prefer to say linear transformation. Second, even in finitedimensional spaces, one must specify bases in domain and target space to determine the entries in a matrix. We discuss this issue in more detail in Section 2.5 for the special case of linear operators. 

We will also need to shift occasionally from one basis to another. This can be done by a matrix multiply, so in the representation as an array it looks precisely as if an operator had been applied. 

The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian hF—h+G(R)z becomes h+A + G R), and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople et al.73b gives a good introduction to the method, including a discussion of the computational errors likely to be involved. 

The eigenvalue of Qz is N — 2 for a state of gN. We can now consider that the identical components of g1 and g together form a quasi-spin tensor of rank 1/ 2, whose array of ranks we can now indicate by writing G(l - - -]. The e, operators can be broken down into parts that have well-defined quasi-spin ranks however, it turns out that e2 is a quasi-spin scalar, which can be used to explain some similar matrix elements of e2 in g 2 and g 4 [10]. 

Figure 2 shows the code for the spin-echo pulse sequence. In lines 1-7, NMR object variables are declared. In lines 9-16, these objects are assigned values or specific NMR operations. A lactate spin system is read in from a file using a standard GAMMA text file format. Lines 20-24 are the code for the actual 90°Y-delay-180°Y-delay-acquire pulse sequence. In line 26, the density matrix is parsed into a transition table for the specified observation operator. And the "write results" function in line 27 converts the GAMMA transition table into three arrays of ppm, area and phase values one value for each line found in the transition table. This is a fairly trivial example, and the use of ideal pulses is often not sufficient to account for real-world artefacts in a simulation, but it shows how the object-oriented style of coding results in short amounts of code that is easy to read and comprehend. When compiled with the Visual Studio C++ compiler on a... 

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