Multilin arrays

Much of numerical computation in chemistry revolves about numerical multilinear algebra. By this term I denote the manipulation of arrays whose dimension may exceed two. Mindful of Richard and Ledgard s admonition (16) that language design should never be overly ambitious, I propose only data structures, operations, and syntax for programming multilinear algebra. Multilin exceeds Bohlender s Pascal extension in two ways its provision for more than two dimensions, and its explicit declaration of data representations. 

In addition to the usual array representation which stores A(i,j,k) as A (((i-l)n+j-l)o+k), where A is dimensioned (m,n,o) and A1, (mno), Multilin provides representations taking advantage of sparsity and symmetry. Patterned sparsity such as that exhibited by a band diagonal matrix can be treated through another matrix. Thus a band matrix, A, dimensioned (n,r), with k lower... 

The power of Multilin revolves about its implicit loop variables and its summation convention. These variables denote what operations are to be performed on the arrays, not how they are to be implemented for particular representations. For example, assume an array V to be declared identically to GAMMA above. The trace of the product of these arrays is denoted by "V(I,J,K,L) GAMMA(K,L,I,J) there is no mention of inequalities on index ranges, or factors of 2 Re or 4 Re due to those inequalities. 

Often a broadband laser (for example, a pulsed dye laser without etalons), or a multiline laser (for example, a CO2 or CO laser without grating) may simultaneously cover several absorption lines of different molecules. In such cases the reflected beam is sent to a polychromator with a diode array or an optical multichannel analyzer (OMA, Vol. 1, Sect. 4.5). If a fraction of the laser power Po(< ) is imaged onto the upper part of the OMA detector and the transmitted power onto the lower part (insert in Fig. 10.17), electronic difference and ratio recording allows the simultaneous determination of the concentrations V/ of all absorbing species. A retroreflector arrangement is feasible for measurements at low altitudes above ground, where buildings or chimneys can support the construction. Examples are measurements of fluorine concentrations in an aluminum plant [1452], or the detection of different constituents in the chimney emission of power plants, such as NOjc and SOj components [1453]. Often, ammonia is added to the exhaust of power stations in order to reduce the amount of NO emission. In such cases, the optimum... 

Often a broadband laser (for example, a pulsed dye laser without etalons), or a multiline laser (for example, a CO2 or CO laser without grating) may simultaneously cover several absorption lines of different molecules. In such cases the reflected beam is sent to a polychromator with a diode array or an optical multichannel analyzer (OMA, Sect. 4.5). If a fraction of the laser power Po(co) is imaged onto the upper part of the OMA detector and the... 

A definite absorption path length L can be realized with a retroreflector, placed at a distance L/2 from the source. Another possibility is to place the laser in an airplane (or satellite) and the receiver on the ground. Figure 14.4 illustrates a possible experimental setup. The laser output is expanded by a telescope in order to decrease the divergence of the beam. The retro-reflector reflects the beam exactly back into a beam splitter and into a polychromator. With a diode array at the spectrometer output (optical multichannel analyzer see Sect.4.5.9) the whole spectral range of the multiline laser can be recorded simultaneously. This diode array also allows measure-... 

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